{"text":"\\section{Introduction}\n\nThe $d$-dimensional semi-discrete wave equation is given as the following infinite system of second order ordinary differential equations: \n\\begin{equation}\\label{u0}\n\\frac{d^2}{dt^2} u(t)[k] \n-a^2 \\sum_{j=1}^d \\(u(t)[k+e_j] - 2u(t)[k] + u(t)[k-e_j]\\) =0,\\;\\;\n  (t,k)\\in \\R \\times  \\Z^d, \n\\end{equation}\nwhere $u(t)=\\{u(t)[k]\\}_{k\\in \\Z^d}$ and \n$a$ is a positive constant, $\\{e_j,\\ldots,e_d\\}$ is the standard basis of $\\R^d$. \n\\eqref{u0} is a discretization with respect to the space variables for the following $d$-dimensional wave equation: \n\\begin{equation}\\label{w}\n  \\pa_t^2 u(t,x) - a^2 \\sum_{j=1}^d \\pa_{x_j}^2 u(t,x) = 0,\\;\\;\n  (t,x)\\in \\R \\times \\R^d, \n\\end{equation}\nwhere $a$ describes the propagation speed of the wave. \nOne of the most important properties for the wave equation \\eqref{w} is the following equality which is called the energy conservation: \n\\begin{equation}\\label{ec}\n\\tilde{E}(t):=\\int_{\\R^d} |\\pa_t u(t,x)|^2\\,dx\n     +a^2 \\sum_{j=1}^d \\int_{\\R^d} |\\pa_{x_j} u(t,x)|^2\\,dx\n  \\equiv \\tilde{E}(t_0),\\;\\;\n  t,t_0\\in \\R. \n\\end{equation}\nHowever, if the propagation speed $a$ depends on time variable, then \\eqref{ec} does not hold in general. \nOn the contrary, the existence of a solution in the Sobolev space may not be valid if $a(t)$ has singularities like non-Lipschitz continuity or degeneration (see \\cite{CDS,CJS}); \nthus time dependent propagation speed is possible to give a crucial influence to the property of the wave equation. \n\nLet us consider the following Cauchy problem for the wave equation with time dependent propagation speed: \n\\begin{equation}\\label{wC}\n\\begin{cases}\n\\ds{\\pa_t^2 u(t,x) - a(t)^2 \\sum_{j=1}^d \\pa_{x_j}^2 u(t,x) = 0}, \n  & (t,x)\\in \\R_+ \\times \\R^d, \\\\\nu(0,x)=u_0(x),\\;\\; (\\pa_t u)(0,x)=u_1(x), & x\\in \\R^d, \n\\end{cases}\n\\end{equation}\nwhere $\\R_+:=[0,\\infty)$ and \n$a(t)$ satisfies \n\\begin{equation}\\label{a0a1}\n  a_0 \\le a(t) \\le a_1\n\\end{equation}\nfor some positive constants $a_0$ and $a_1$. \nSince the energy conservation does not hold for \\eqref{wC}, \nwe introduce the generalized energy conservation, which is abbreviated as GEC, \nas the following uniform equivalence of the energy with respect to $t$: \n\\begin{equation}\n\\label{tGEC}\n  \\tilde{E}(0) \\lesssim \\tilde{E}(t) \\lesssim \\tilde{E}(0),\n  \\;\\; t \\in \\R_+,\n\\end{equation} \nwhere $f \\lesssim g$ with positive functions $f$ and $g$ denotes that \nthere exists a positive constant $C$ such that $f \\le Cg$. \nWe will use the notations $g \\gtrsim f$ and $f \\simeq g$ if \n$f\\lesssim g$ and $g\\lesssim f \\lesssim g$ hold, respectively. \nMoreover, we will use $C$ to denote a generic positive constant. \n\nLet $a\\in C^m(\\R_+)$ with $m\\ge 1$ and $(u_0,u_1)\\in H^2\\times H^1$. \nThen a unique time global classical solution of \\eqref{wC} exists, and the following energy estimate holds: \n\\begin{equation}\\label{tE-a'L1}\n  \\tilde{E}(t) \\le \\exp\\(\\frac{2}{a_0}\\int^t_0 |a'(s)|\\,ds\\) \\tilde{E}(0),\n  \\;\\; t\\in \\R_+. \n\\end{equation}\nIt follows that GEC holds if $a'\\in L^1(\\R_+)$. \nOn the other hand, the success or failure of GEC depends on the properties of $a(t)$ and the initial data if $a'\\not\\in L^1((0,\\infty))$. \nIndeed, the following result is known: \n\n\\begin{theorem}[\\cite{RS}]\\label{Thm-RS}\n\\begin{itemize}\n\\item[(i)] \nIf $a\\in C^2(\\R_+)$, $|a'(t)|\\lesssim (1+t)^{-1}$ and $|a''(t)|\\lesssim (1+t)^{-2}$, then GEC is established for the Cauchy problem \\eqref{wC}. \n\\item[(ii)] \nFor any positive and monotone increasing function $\\nu(t)$ satisfying \n$\\lim_{t\\to\\infty}\\nu(t)=\\infty$, the conditions \n$|a'(t)|\\lesssim \\nu(t)(1+t)^{-1}$ and $|a''(t)|\\lesssim \\nu(t)(1+t)^{-2}$ \ndoes not necessarily conclude GEC. \n\\end{itemize}\n\\end{theorem}\n\n\\begin{remark}\n\\eqref{tE-a'L1} is derived by the estimate\n\\begin{equation}\\label{tE'}\n  \\tilde{E}'(t)=2a'(t)a(t) \\sum_{j=1}^d \\int_{\\R^d} |\\pa_{x_j} u(t,x)|^2\\,dx\n  \\le \\frac{2|a'(t)|}{a(t)} \\tilde{E}(t)\n\\end{equation}\nand Gronwall's inequality. \nWe observe from the first equality of \\eqref{tE'} that $\\tilde{E}(t)$ increases and decreases if $a'(t)>0$ and $a'(t)<0$, respectively. \nThat is, time dependent propagation speed causes increase or decrease for the energy. \nFurthermore, we notice that the second inequality is not taken account the sign of $a'(t)$. \nActually, \\eqref{tE'} is obtained with assuming both $a'(t)>0$ and $a'(t)<0$ increase the energy, but Theorem 1.1 (i) is derived with considering some cancellation of the energy which is caused by changing sign of $a'(t)$. \n\\end{remark}\n\nAccording to Theorem \\ref{Thm-RS}, the oscillation speed of $a(t)$, which is described by the order of $|a'(t)|$, is crucial for GEC, and the order of threshold is $(1+t)^{-1}$. \nHowever, GEC is not determined only the order of $|a'(t)|$. \nIndeed, under the additional assumptions to $a(t)$ below, GEC is possible even if $|a'(t)|\\lesssim (1+t)^{-1}$ does not hold. \n\n\\begin{description}\n\\item[(H1)]\nThere exists a positive constant $a_\\infty$ such that either of the following estimates hold: \n\\begin{equation}\\label{stbc-}\n  \\int^\\infty_t|a(s)-a_\\infty|\\,ds \\lesssim (1+t)^{\\al}\n  \\;\\text{ for }\\;\n  \\al < 0\n\\end{equation}\nor\n\\begin{equation}\\label{stbc}\n  \\int^t_0|a(s)-a_\\infty|\\,ds \\lesssim (1+t)^{\\al}\n  \\;\\text{ for }\\;\n  0\\le \\al \\le 1.\n\\end{equation}\n\\item[(H2)]\n$a \\in C^m(\\R_+)$ with $m\\ge 1$ and the following estimates hold for $\\be<1$: \n\\begin{equation}\\label{akc}\n  \\left|a^{(k)}(t)\\right| \\lesssim (1+t)^{-k\\be},\\;\\; k=1,\\ldots,m. \n\\end{equation}\n\\end{description}\n\nThen the following theorem is established: \n\\begin{theorem}[\\cite{H07}]\\label{Thm-H07}\nIf $m\\ge 2$, $a(t)$ satisfies {\\rm (H1)} and {\\rm (H2)} for \n\\begin{equation}\\label{albem}\n  \\be \\ge \\al + \\frac{1-\\al}{m},\n\\end{equation}\nthen GEC is established for the Cauchy problem \\eqref{wC}. \nOn the other hand, if $\\be<\\al$, then GEC does not hold in general. \n\\end{theorem}\n\n\\begin{remark}\nThe right hand side of \\eqref{albem} is smaller as $m$ larger or $\\al$ smaller. That is, the restriction on $\\be$ can be weaker if the differentiability of $a(t)$ is higher or the restriction of (H1) is stronger. \nActually, the Theorem \\ref{Thm-H07} does not conclude the optimality of the condition \\eqref{albem}, but the limit case $m=\\infty$ is nearly optimal for GEC. \n\\end{remark}\n\n\\begin{remark}\nSince the estimate \\eqref{stbc} with $\\al=1$ is trivial, \nTheorem \\ref{Thm-RS} can be considered a special case of Theorem \\ref{Thm-H07} without (H1). \n\\end{remark}\n\n\nLet us consider the following initial boundary value problem: \n\\begin{equation}\\label{wIBV}\n\\begin{cases}\n\\ds{\\pa_t^2 u(t,x) - a^2 \\sum_{j=1}^d \\pa_{x_j}^2 u(t,x) = 0}, \n  & (t,x)\\in \\R_+ \\times \\Om, \\\\\nu(t,x)=0, & (t,x)\\in \\R_+\\times \\pa\\Om,\\\\\nu(0,x)=u_0(x),\\;\\; \\(\\pa_t u\\)(0,x)=u_1(x), & x\\in \\pa\\Om, \n\\end{cases}\n\\end{equation}\nwhere $\\Om$ is a bounded domain of $\\R^d$ with smooth boundary $\\pa\\Om$. \nThen the following energy conservation corresponding to \\eqref{ec} for \\eqref{wC} is established:  \n\\begin{equation*\n\\tilde{E}(t):=\\int_{\\Om} |\\pa_t u(t,x)|^2\\,dx\n     +a^2 \\sum_{j=1}^d \\int_{\\Om} |\\pa_{x_j} u(t,x)|^2\\,dx\n\\equiv \\tilde{E}(t_0). \n\\end{equation*}\nMoreover, if $a$ depends on time variable, then the following theorem, which is corresponding to Theorem \\ref{Thm-H07} in bounded domain, is established: \n\\begin{theorem}[\\cite{H10}]\\label{Thm-H10}\nIf $m\\ge 2$ and $a=a(t)$ satisfies {\\rm (H2)} for \n\\begin{equation}\\label{albem-bdd}\n  \\be \\ge \\frac{1}{m},\n\\end{equation}\nthen GEC is established for the initial boundary problem \\eqref{wIBV}. \n\\end{theorem}\n\nThe condition \\eqref{albem-bdd} in Theorem \\ref{Thm-H10} corresponds to \\eqref{albem} in Theorem \\ref{Thm-H07} with $\\al=0$. \nThis implies that GEC is established without the assumption (H1) even though the oscillation speed is faster on the problem \\eqref{wIBV}. \nBriefly, (H1) and (H2) are required for the estimate of the solution in the time-frequency space for the low and the high frequency part, respectively. \nHowever, (H1) is not necessary for the problem \\eqref{wIBV} because the influence to the low frequency part can be neglected since the largest Dirichlet eigenvalue of the Laplace operator $\\sum_{j=1}^d \\pa_{x_j}^2$ is strictly negative. \nOne of our main interest of the present paper is how the properties (H1) and (H2) for the time dependent propagation speed $a(t)$ relate to the energy estimate for semi-discrete wave equation \\eqref{u0}. \n\nThere are many studies on the discretized wave equations with constant propagation speed, but not many results are known for time dependent propagation model, especially in the case that the propagation speed $a(t)$ is singular to collapse GEC. In \\cite{CR}, an approximation of the discretized wave equation with respect to time variable is studied in the case that $a(t)$ is degenerate and oscillating which was studied in \\cite{CJS}. The result is not directly related to the studies of the present paper; indeed, no approximation to the continuous model will be discussed, but discretization is a useful approach to study the influence of the time dependent propagation speed to the energy of the solution in time-frequency spaces. \n\n\n\n\\section{Discretization and discrete-time Fourier transformation}\n\nLet us consider a discretized model of the Cauchy problem for the wave equation \\eqref{wC} with respect to the space variables $x$. \n\nFor $f = \\{f[k]\\}_{k\\in \\Z^d}$ and $j=1,\\ldots,d$, we denote \nthe forward and the backward difference operators $D_j^+$ and $D_j^-$ by \n\\begin{equation*}\n  D_j^+ f := \\{f[k+e_j]-f[k]\\}_{k\\in\\Z^d}\n  \\;\\text{ and }\\;\n  D_j^- f := \\{f[k]-f[k-e_j]\\}_{k\\in\\Z^d},\n  \\;\\; k\\in \\Z^d,\n\\end{equation*}\nrespectively. \nThen the discrete Laplace operator in $\\Z^d$ is given by \n$\\sum_{j=1}^d D_j^+ D_j^-$, that is, \n\\begin{equation*}\n  \\sum_{j=1}^d D_j^+ D_j^- f[k]\n =\\sum_{j=1}^d \\(f[k+e_j]-2f[k]+f[k-e_j]\\). \n\\end{equation*}\n\nFor the solution $u(t,x)$ of \\eqref{wC}, we consider the $d$-dimensional infinite matrix valued function $u(t)=\\{u(t)[k]\\}_{k\\in \\Z^d}$ as a sampled of $u(t,x)$ on the lattice $\\Z^d$. \nThen a discretized model of \\eqref{wC} is given as the following initial value problem for an infinite system of ordinary differential equations, \nwhich is called the semi-discrete wave equation with time dependent propagation speed: \n\\begin{equation}\\label{u}\n\\begin{cases}\n\\ds{\n  u''(t)[k] \n -a(t)^2 \\sum_{j=1}^d D_j^+ D_j^- u(t)[k] =0, }\n  & (t,k)\\in \\R_+ \\times \\Z^d, \n\\\\[3mm]\n\\ds{\n  u(0)[k]=u_0[k],\\;\\; u'(0)[k]=u_1[k], }\n  &k\\in \\Z^d.\n\\end{cases}\n\\end{equation}\nThen we define the total energy for the solution of \\eqref{u} by \n\\begin{equation*\n  E(t):=\n  \\sum_{k\\in \\Z^d}\\left|u'(t)[k]\\right|^2\n +a(t)^2 \\sum_{j=1}^d \\sum_{k\\in \\Z^d}\\left|D_j^+ u(t)[k]\\right|^2. \n\\end{equation*}\nEvidently, the energy conservation $E(t)\\equiv E(0)$ is valid if the propagation speed $a$ is a constant, but it does not hold in general for variable propagation speed. \nTherefore, the following energy estimate corresponding to \\eqref{tGEC} can be considered: \n\\begin{equation}\\label{GEC}\n  E(t) \\simeq E(0).\n\\end{equation}\nHere \\eqref{GEC} will be also denoted by GEC. \n\nIt is usual to study the energy estimate of \\eqref{wC} and \\eqref{wIBV} in the time-frequency spaces by introducing Fourier transformation and Fourier coefficients with respect to the space variables of the solution rather than the solutions $u(t,x)$ themselves. \nTherefore, we introduce the discrete-time Fourier transformation to study the energy estimate of the solution for \\eqref{u}. \n\n\\begin{definition}\nFor $f=\\{f[k]\\}_{k\\in \\Z^d} \\in l^2(\\Z^d)$ we define the discrete-time Fourier transformation $\\cF_{\\Z^d}[f](\\th)$ by\n\\begin{equation*}\n  \\cF_{\\Z^d}[f](\\th):=\\sum_{k\\in \\Z^d} e^{-\\i k\\cd\\th} f[k],\n  \\;\\; \\th=(\\th_1,\\ldots,\\th_d) \\in \\R^d,\n\\end{equation*}\nwhere $k \\cdot \\th = \\sum_{j=1}^d k_j \\th_j$ and $k=(k_1,\\ldots,k_d)$. \nWe denote that $\\cF_{\\Z^d}[f]=\\hat{f}$ and \n$\\sum_{k\\in \\Z^d}=\\sum_{k}$ without any confusion. \n\\end{definition}\n\nSince the discrete-time Fourier transformation $\\hat{f}(\\th)$ is the $d$-dimensional Fourier series with the Fourier coefficient $\\{f[-k]\\}_{k\\in \\Z^d}$, \n$\\hat{f}$ is a $2\\pi$-periodic function in $\\R^d$. \nThat is, the following equality holds:  \n\\begin{equation*}\n  \\hat{f}(\\th+2k\\pi)=\\hat{f}(\\th)\n\\end{equation*}\nfor any $k\\in \\Z^d$ and $\\th\\in \\T^d$, where $\\T:=[-\\pi,\\pi]$. \nTherefore, we will restrict the domain of $\\cF_{\\Z^d}[\\cd]$ on $\\T^d$. \n\nHere we introduce some lemmas for the discrete-time Fourier transformation, \nwhere the proofs will be introduced in Appendix. \n\\begin{lemma}\\label{TDFT}\nIf $f \\in l^1(\\Z^d)$ then the following equalities are established \nfor $j=1,\\ldots,d$: \n\\begin{equation}\\label{TDFT2}\n  \\cF_{\\Z^d}[D_j^+ f](\\th) = \\(e^{\\i\\th_j}-1\\) \\hat{f}(\\th)\n\\end{equation}\nand\n\\begin{equation}\\label{TDFT3}\n \\cF_{\\Z^d}[D_j^+D_j^- f](\\th) \n  = -4\\(\\sin\\frac{\\th_j}{2}\\)^2\\hat{f}(\\th).\n\\end{equation}\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lemm-Parseval}\nIf $f \\in l^2(\\Z^d)$ then\nthe following Parseval's type equality is established: \n\\begin{equation*\n  \\sum_{k\\in \\Z^d}|f[k]|^2\n =\\frac{1}{(2\\pi)^d}\\int_{\\T^d}|\\hat{f}(\\th)|^2\\,d\\th. \n\\end{equation*}\n\\end{lemma}\n\nFor $\\th\\in \\T^d$ we define $\\xi=\\xi(\\th)$ by \n\\begin{equation}\\label{xith}\n  \\xi(\\th)=(\\xi_1(\\th_1),\\ldots,\\xi_d(\\th_d)),\\;\\;\n  \\xi_j(\\th_j):=2\\sin\\frac{\\th_j}{2}\n  \\;\\;(j=1,\\ldots,d).\n\\end{equation}\nBy the discrete-time Fourier transformation, \\eqref{u} is reduced to the following problem: \n\\begin{equation}\\label{hu}\n\\begin{cases}\n\\partial_t^2 \\hat{u}(t,\\th) \n+a(t)^2 |\\xi(\\th)|^2 \\hat{u}(t,\\th) =0, &\n  (t,\\th)\\in \\R_+ \\times \\T^d, \n\\\\\n  \\hat{u}(0,\\th)=\\hat{u}_0(\\th),\\;\\;\n  \\(\\partial_t \\hat{u}\\)(0,\\th)=\\hat{u}_1(\\th), \n  & \\th\\in \\T^d,\n\\end{cases}\n\\end{equation}\nwhere $\\hat{u}(t,\\th)=\\sum_{k} e^{-\\i k\\cd\\th}u(t)[k]$. \n\nFor the solution $\\hat{u}(t,\\th)$ of \\eqref{hu}, we define the energy density function $\\cE(t,\\th)$ by \n\\begin{equation*\n  \\cE(t,\\th):=\n  \\left|\\pa_t \\hat{u}(t,\\th)\\right|^2\n +a(t)^2 |\\xi(\\th)|^2 \\left|\\hat{u}(t,\\th)\\right|^2.\n\\end{equation*}\nBy Lemma \\ref{lemm-Parseval}, the total energy $E(t)$ is represented by $\\cE(t,\\th)$ as follows: \n\\begin{lemma}\\label{lemm-E-cE}\nIf $u'(t), D_j^+ u(t)\\in l^2(\\Z^d)$ $(j=1,\\ldots,d)$, \nthen the following equality is established: \n\\begin{equation}\\label{EcE}\n  E(t) = \\frac{1}{(2\\pi)^d}\\int_{\\T^d} \\cE(t,\\th)\\,d\\th.\n\\end{equation}\n\\end{lemma}\n\nIn the Cauchy problem of the wave equation \\eqref{wC}, we shall call it a continuous model, not only the total energy $\\tE(t)$ but also the energy density $\\tcE(t,\\xi)$: \n\\begin{equation*\n  \\tcE(t,\\xi):=\n  \\left|\\pa_t \\cF_{\\R^d}[u](t,\\xi)\\right|^2\n +a^2 |\\xi|^2 \\left|\\cF_{\\R^d}[u](t,\\xi)\\right|^2\n\\end{equation*}\nis conserved if the propagation speed $a$ is a constant, \nwhere $\\cF_{\\R^d}[u](t,\\xi)$ denotes the Fourier transformation of $u(t,x)$ with respect to the space variable $x$. \nHowever, the energy density is not conserved for time dependent propagation speed in general. \nIndeed, time dependent propagation speed has the effect of changing the total energy with respect to $t$, and it considered to be a phenomenon that caused by transition of energy across frequencies. \nBasically, the behavior of $\\tcE(t,\\xi)$ is determined by the properties of $a(t)$, and the assumptions for $a(t)$ in Theorem \\ref{Thm-H07} ensure the estimate $\\tcE(t,\\xi)\\simeq \\tcE(0,\\xi)$. \nOn the other hand, the negative results for \\eqref{tGEC} are implied from the unboundedness of $\\tcE(t,\\xi)$. \nIn particular, the non-existence result in the Sobolev space is derived from the increase in the energy in high frequency part which is provided from the non-Lipschitz continuity with very fast oscillation of $a(t)$. \n\nThe energy density $\\cE(t,\\th)$ for the solution of the semi-discrete model \\eqref{hu} is also conserved if the propagation speed $a$ is a constant. \nOn the other hand, if the propagation speed is variable, then the estimate \n\\begin{equation}\\label{GECcE}\n  \\cE(t,\\th) \\simeq \\cE(0,\\th)\n\\end{equation}\ndoes not hold in general as in the case of continuous model, and thus GEC may not be established. \nIt will be natural that the estimate \\eqref{GECcE} is established if $a(t)$ satisfies the same assumptions of Theorem \\ref{Thm-RS} and Theorem \\ref{Thm-H07}. \nHowever, we may expect to prove \\eqref{GECcE} under some weaker assumptions to $a(t)$, because the range of $|\\xi|$ for $\\tcE(t,\\xi)$ is $[0,\\infty)$, but the range of $|\\xi(\\th)|$ for $\\cE(t,\\xi)$ is $[0,2\\sqrt{d}]$. In the other words, unlike the continuous model, the solution cannot have high-frequency energy above a certain level because the solution of semi-discrete model has a finite resolution. \nHere we note that the situation of high-frequency energy for the semi-discrete model is the corresponding to the low-frequency energy for the initial boundary value problem \\eqref{wIBV}. \n\n\\section{Main results}\nLet us generalize the properties \\eqref{stbc} of (H1) and (H2) by positive and monotone increasing functions $\\Th(t)$ and $\\Xi(t)$ on $\\R_+$ as follows: \n\n\\begin{description}\n\\item[(H1*)]\nThere exists a positive constant $a_\\infty$ such that the following estimate holds: \n\\begin{equation}\\label{stb}\n  \\int^t_0|a(s)-a_\\infty|\\,ds \\le \\Th(t).\n\\end{equation}\n\\item[(H2*)]\n$a \\in C^m(\\R_+)$ with $m\\ge 1$ and the following estimates hold for some positive constants $C_k$:\n\\begin{equation}\\label{ak}\n  \\left|a^{(k)}(t)\\right| \\le C_k \\Xi(t)^{-k},\\;\\; k=1,\\ldots,m. \n\\end{equation}\n\\end{description}\n\nFor $\\Th(t)$ and $\\Xi(t)$ we introduce the following conditions corresponding to the conditions $\\al\\le \\be$ and \\eqref{albem}: \n\\begin{description}\n\\item[(H3*)] \nThere exists a positive constant $C_0$ such that \n\\begin{equation}\\label{ThXi}\n\\Th(t) \\le C_0 \\Xi(t).\n\\end{equation}\n\\item[(H4*)]\nFor $m\\ge 2$, $\\Xi(t)^{-m}\\in L^1(\\R_+)$ and the following estimate holds: \n\\begin{equation}\\label{int_Cm}\n\\sup_{t\\ge 0}\\left\\{\\Th(t)^{m-1}\\int_t^\\infty \\Xi(s)^{-m}\\,ds\n\\right\\}<\\infty.\n\\end{equation}\n\\end{description}\n\nThen our first theorem is given as follows: \n\n\\begin{theorem}\\label{Thm1}\nFor the initial value problem of semi-discrete wave equation \\eqref{u}, GEC is established if any of the following \n{\\rm(i)} to {\\rm (iii)} is satisfied: \n\\begin{itemize}\n\\item[{\\rm (i)}]\n{\\rm (H1*)} with $\\sup_{t\\ge 0}\\{\\Th(t)\\}<\\infty$. \n\\item[{\\rm (ii)}]\n{\\rm (H2*)} with $m=1$ and $\\Xi(t)^{-1} \\in L^1(\\R_+)$. \n\\item[{\\rm (iii)}]\n{\\rm (H1*)}, {\\rm (H2*)}, {\\rm (H3*)} and {\\rm (H4*)}. \n\\end{itemize}\n\\end{theorem}\n\nIf we restrict ourselves $\\Th(t)=(1+t)^\\al$ and $\\Xi(t)=(1+t)^\\be$ with $m\\be>1$, then \\eqref{stb}, \\eqref{ak} and \\eqref{int_Cm} \nare the same as \\eqref{stbc}, \\eqref{akc} and \\eqref{albem}, respectively. \nMoreover, \\eqref{ThXi} is valid if \\eqref{albem} holds. \nComparing the assumptions of $a(t)$ in Theorem \\ref{Thm1} with \nTheorem \\ref{Thm-H07} we observe the followings: \n\n\\begin{itemize}\n\\item\nIf $\\al>0$, then GEC is established under the same assumptions for $a(t)$. \n\\item\nThough GEC does not hold for \\eqref{wC} in general if $a(t)$ is not Lipschitz continuous, Theorem \\ref{Thm1} concludes GEC without any assumption of the continuity for $a(t)$ if $\\al=0$. \n\\end{itemize}\n\n\n\\begin{example}\\label{Ex1}\nLet $\\chi \\in C^m(\\R_+)$ be a positive and periodic function. \nFor non-negative constants $p$, $q$ and $r$ we define $a \\in C^m(\\R_+)$ by \n\\begin{equation*}\n  a(t)=1 + (1+t)^{-p} \\chi\\((1+t)^q \\(\\log(e+t)\\)^r\\).\n\\end{equation*}\nThen we see the followings: \n\\begin{equation*}\n  \\int^t_0|a(s)-1|\\,ds \\le \\Th(t) \\simeq\n  \\begin{cases}\n  1 & (p>1), \\\\\n  \\log(e+t) & (p=1), \\\\\n  (1+t)^{-p+1} & (p<1).\n  \\end{cases}\n\\end{equation*}\nMoreover, for any $k=1,\\ldots,m$ we have \n\\begin{align*}\n  \\left|a^{(k)}(t)\\right| \n  \\lesssim \\: & \n  (1+t)^{-p}\\((1+t)^{q-1} \\(\\log(e+t)\\)^{r} \\)^k\n\\\\\n  \\le \\: & \n  \\((1+t)^{\\frac{p}{m}-q+1} \\(\\log(e+t)\\)^{-r} \\)^{-k}\n\\end{align*}\nfor $q>0$ and \n\\begin{align*}\n  \\left|a^{(k)}(t)\\right| \n  \\lesssim  \\: &\n  (1+t)^{-p}\\((1+t)^{-1} \\(\\log(e+t)\\)^{r-1} \\)^k\n\\\\\n  \\le \\: &\n  \\((1+t)^{\\frac{p}{m}+1} \\(\\log(e+t)\\)^{-r+1} \\)^{-k}\n\\end{align*}\nfor $q=0$. \nHence we set \n\\begin{equation}\n  \\Xi(t)=\\begin{cases}\n    (1+t)^{\\frac{p}{m}-q+1} \\(\\log(e+t)\\)^{-r} & (q>0),\\\\\n    (1+t)^{\\frac{p}{m}+1} \\(\\log(e+t)\\)^{-r+1} & (q=0). \n  \\end{cases}\n\\end{equation}\nBy Theorem \\ref{Thm1}, we have GEC if any of the following holds: \n\\begin{itemize}\n\\item $p>1$ (by (i)). \n\\item $m=1$ and $q<p$ (by (ii)). \n\\item $m \\ge 2$, $0<p=q$ and $r=0$ (by (iii)).\n\\item $m \\ge 2$, $p=q=0$ and $0<r\\le 1$ (by (iii)).\n\\end{itemize}\n\\end{example}\n\n\\begin{example\nLet $\\chi_\\ve \\in C^m(\\R_+)$ with $m \\ge 2$ be a $2\\pi$-periodic function with a positive small parameter $\\ve$ satisfying \n$\\chi_\\ve(\\tau)=0$ near $\\tau=0$ and \n\\begin{equation}\n  \\sup_{\\ve>0}\\max_{\\tau\\in[0,2\\pi]}\n  \\left\\{\\frac{1}{\\ve}\n  \\left|\\frac{d^k}{d\\tau^k}\\chi_\\ve(\\tau)\\right|\\right\\}\n  <\\infty\\quad(k=0,1,\\ldots,m).\n\\end{equation}\nFor a positive large constant $\\eta$ and positive constants $\\al$, $\\be$ and $\\ka$ satisfying \n\\begin{equation}\\label{pqka}\n  \\al \\le \\ka \\le 1\n  \\;\\text{ and }\\;\n  \\al+\\frac{1-\\al}{m} \\le \\be \\le \\ka+\\frac{\\ka-\\al}{m},\n\\end{equation}\nwe define the sequences $\\{t_j\\}_{j=1}^\\infty$, $\\{\\ve_j\\}_{j=1}^\\infty$, $\\{\\rho_j\\}_{j=1}^\\infty$ and $\\{\\nu_j\\}_{j=1}^\\infty$ by \n\\begin{equation*}\n  t_j:=\\eta^j,\\;\\;\n  \\ve_j:=t_j^{\\al-\\ka},\\;\\;\n  \\rho_j:=\\eta^{-1}t_j^\\ka\n  \\;\\text{ and }\\;\n  \\nu_j:=\\left[t_j^{-\\be+\\ka+\\frac{\\ka-\\al}{m}}+1\\right]. \n\\end{equation*}\nThen we define $a(t)$ by \n\\begin{equation*}\n  a(t):=\\begin{cases}\n  \\sqrt{1+\\chi_{\\ve_j}\\(\\nu_j\\rho_j^{-1}(t-t_j)\\)} \n    & \\text{ for } \\; t\\in [t_j-\\rho_j,t_j+\\rho_j]\\;\\; (j=1,2,\\ldots),\\\\\n  1 & \\text{ for } \\; t\\in \\R_+ \\setminus \\bigcup_{j=1}^\\infty [t_j-\\rho_j,t_j+\\rho_j].\n\\end{cases}\n\\end{equation*}\nwhere $[\\;]$ denotes the Gauss symbol. \nHere we note that \n\\begin{align*}\n  t_j + \\rho_j + \\rho_{j+1}\n= \\eta^j \\(1 + \\eta^{-j(1-\\ka) -1} + \\eta^{-(j+1)(1-\\ka)}\\)\n\\le 3 \\eta^j \\le \\eta^{j+1} = t_{j+1}\n\\end{align*}\nfor $\\eta \\ge 3$, it follows that \n$t_j + \\rho_j \\le t_{j+1} - \\rho_{j+1}$. \nLet $t\\in[t_{j-1}+\\rho_{j-1},t_j+\\rho_j]$. \nThen we have \n\\begin{equation}\n  \\int^{t}_0 |a(s)-1|\\,ds\n  \\lesssim \\sum_{k=1}^j \\ve_k \\rho_k\n  =\\eta^{-1} \\sum_{k=1}^j \\eta^{k \\al} \\simeq t_j^\\al\n  \\simeq (1+t)^\\al\n\\end{equation}\nand\n\\begin{align*}\n  \\left|a^{(k)}(t)\\right|\n  \\lesssim \\ve_j\\(\\nu_j \\rho_j^{-1}\\)^k \n  \\simeq t_j^{-k\\be-(\\ka-\\al)\\(1-\\frac{k}{m}\\)}\n  \\le t_j^{-k\\be}\n  \\simeq (1+t)^{-k\\be}, \n\\end{align*}\nit follows that \\eqref{stb}, \\eqref{ak} and \\eqref{ThXi} are established with \n\\begin{equation}\n  \\Th(t) \\simeq (1+t)^\\al\\;\\text{ and }\\;\n  \\Xi(t) = (1+t)^\\be.\n\\end{equation}\nMoreover, by \\eqref{pqka} and noting $\\al+(1-\\al)\/m>1\/m$, we have \n$\\Xi(t)^{-m}\\in L^1(\\R_+)$ and \n\\begin{align*}\n  \\Th(t)^{m-1}\\int^\\infty_t \\Xi(s)^{-m}\\,ds\n  \\lesssim (1+t)^{\\al(m-1)-m\\be+1} \\lesssim 1,\n\\end{align*}\nthus \\eqref{int_Cm} is established. \nTherefore, $a(t)$ satisfies (iii) of Theorem \\ref{Thm1}. \n\\end{example}\n\nIf $a \\not\\in C^1(\\R_+)$, then \\eqref{wC} is not $L^2$ well-posed in general, hence the energy estimate $\\tE(t) \\lesssim \\tE(0)$ cannot be expected. \nHowever, $\\tE(t)$ is not necessarily unbounded if \\eqref{wC} is not $L^2$ well-posed. \nFor example, if $a(t)$ is a H\\\"older continuous function, then \\eqref{wC} is not $L^2$ well-posed in general but the Gevrey well-posed. \nThat is, if the initial data are functions of the Gevrey class of suitable order, then the solution is a function of the Gevrey class, too; \nhence $\\tE(t)$ is bounded (see \\cite{CDS}). \nIf $a(t)$ does not satisfy \\eqref{albem}, then GEC does not hold in general. \nHowever, if the initial data is a function of the Gevrey class of suitable order and \\eqref{stbc-} holds, then there exists a positive constant $C(u_0,u_1)$, which depends on not only the initial energy $\\tE(0)$ but also a norm of the initial data in the Gevrey class, such that the total energy $\\tE(t)$ is bounded as follows (see \\cite{EFH}): \n\\begin{equation}\\label{tBE}\n  \\tE(t) \\le C(u_0,u_1). \n\\end{equation}\n\nFaster oscillation of $a(t)$, which is described as smaller $\\be$ not to satisfy \\eqref{albem}, may increase the high frequency part of the energy for the solution of \\eqref{wC} and cause GEC to collapse. \nHowever, since the high frequency part of the initial energy is small with the initial data in the Gevrey class, $\\tE(t)$ can stay bounded against the effect from the faster oscillation of $a(t)$. \nOur second theorem concludes a corresponding estimate to \\eqref{tBE} for the solution of the semi-discrete wave equation \\eqref{u} under some weaker assumptions than that for $a(t)$ in Theorem \\ref{Thm1}. \n\n\nFor a positive and strictly increasing function $\\La(t)$ on $\\R_+$ satisfying $\\lim_{t\\to\\infty}\\La(t)=\\infty$ such that \n\\begin{equation}\\label{La-infty}\n  \\frac{\\Th(t)}{\\La(t)\n  \\text{ is monotone increasing and }\n  \\lim_{t\\to\\infty}\\frac{\\Th(t)}{\\La(t)}=\\infty,\n\\end{equation}\nwe introduce the following conditions that are alternative to (H3*) and (H4*): \n\\begin{description}\n\\item[(H5*)]\nThere exists a positive constant $C_0$ such that \n\\begin{equation*\n  \\La(t)\\le C_0 \\Xi(t). \n\\end{equation*}\n\\item[(H6*)]\nFor $m\\ge 2$, $\\Xi(t)^{-m}\\in L^1(\\R_+)$ and the following estimate holds:  \n\\begin{equation*\n  \\sup_{t\\ge 0}\\left\\{\\La(t)^m \\Th(t)^{-1}\\int^\\infty_t \\Xi(s)^{-m}\\,ds\n  \\right\\}<\\infty.\n\\end{equation*}\n\\end{description}\n\nHere we note that if $\\La(t) \\simeq \\Th(t)$, then (H5*) and (H6*) are coincide with (H3*) and (H4*), respectively. \nOn the other hand, it is possible that (H6*) holds, but (H4*) does not hold if \\eqref{La-infty} is valid. \nOur second theorem is given as follows: \n\n\\begin{theorem}\\label{Thm2}\nLet $m\\ge 2$. \nIf $\\Th(t)$, $\\Xi(t)$ and $\\La(t)$ satisfy {\\rm (H1*)}, {\\rm (H2*)}, \\eqref{La-infty}, {\\rm (H5*)} and {\\rm (H6*)}, then there exists a positive constant $N_0$ such that the following estimate is established: \n\\begin{equation*\n  \\cE(t,\\th) \\lesssim \n  \\exp\\(2|\\xi(\\th)| \\Th\\(\\La^{-1}\\(\\frac{N_0}{|\\xi(\\th)|}\\)\\)\\)\n  \\cE(0,\\th). \n\\end{equation*}\nConsequently, denoting \n\\begin{equation*}\n  U(N,u_0,u_1):=\n  \\int_{\\T^d}\n  \\exp\\(2|\\xi(\\th)| \\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\\)\n  \\cE(0,\\th)\n  \\,d\\th\n\\end{equation*}\nfor $N>0$, we have the followings: \n\\begin{itemize}\n\\item[{\\rm (i)}] \nIf $U(N,u_0,u_1)<\\infty$ holds for any $N\\ge 1$, then there exists a positiv constant $N_0$ such that the energy estimate \n\\begin{equation}\\label{Ebdd}\n  E(t) \\lesssim U(N_0,u_0,u_1)\n\\end{equation} \nis established. \n\\item[{\\rm (ii)}] \nThere exists a positive constant $N_0$, and the energy estimate\n\\eqref{Ebdd} is established for any $(u_0,u_1)$ satisfying \n$U(N_0,u_0,u_1)<\\infty$. \n\\end{itemize}\n\n\\end{theorem}\n\n\\begin{remark}\nDenoting $t=\\La^{-1}(r^{-1})$ for $r>0$, we have \n\\begin{equation}\\label{La-infty2}\n  \\mu(r):=r \\Th\\(\\La^{-1}\\(\\frac{1}{r}\\)\\)\n  =\\frac{\\Th(t)}{\\La(t)}\n  \\nearrow \\infty \\quad (r \\to +0)\n\\end{equation}\nby \\eqref{La-infty} since $\\La^{-1}(r^{-1})$ is positive and strictly decreasing with respect to $r$. \nIt follows that \n\\begin{equation*}\n  \\lim_{\\th \\to 0}\n  |\\xi(\\th)|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\n =\\lim_{|\\xi|\\to 0}N\\mu\\(\\frac{N}{|\\xi|}\\) = \\infty. \n\\end{equation*}\nOn the other hand, if $\\Th(t) \\lesssim \\La(t)$, then we have\n\\[\n  \\sup_{\\th\\in\\T^d\\setminus\\{0\\}}\\left\\{\n  |\\xi(\\th)|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\n  \\right\\}<\\infty, \n\\]\nit follows that $U(N,u_0,u_1) \\simeq E(0)$ by Lemma \\ref{lemm-E-cE}. \nTherefore, \\eqref{La-infty} is a reasonable assumption for the case that Theorem \\ref{Thm1} cannot be applied. \n\\end{remark}\n\nSince \\eqref{La-infty} provides \n\\[\n  \\lim_{|\\th| \\to 0} |\\xi(\\th)|\\Th\\(\\La^{-1}\\(\\frac{1}{|\\xi(\\th)|}\\)\\)=\\infty, \n\\]\nthe condition\n\\begin{equation}\\label{est_Thm2}\nU(N,u_0,u_1)<\\infty\n\\end{equation}\nin Theorem \\ref{Thm2} requires approximately that the $|\\xi(\\th)|\\hat{u}_0(\\th)$ and $\\hat{u}_1(\\th)$ degenerate at $\\th=0$ in an appropriate order which is determined by $\\Th(t)$ and $\\Xi(t)$. \nThe following examples of $\\Th(t)$, $\\Xi(t)$ and $u_1$ provide \\eqref{est_Thm2} for $d=1$ and $u_0=0$. \n\n\\begin{example}\\label{Ex1Thm2}\nLet $a(t)$ be defined in Example \\ref{Ex1} with $m\\ge 2$, $p=q=0$, \n$r \\ge 1$ and $|\\chi| \\le 1$. \nThen we have \n\\begin{equation*}\n  \\Th(t) = 1+t, \\;\\;\n  \\Xi(t) \\simeq (1+t)\\(\\log(e+t)\\)^{-r+1}, \n\\end{equation*}\nand (H4*) does not hold. \nLet us define $\\La(t)$ by \n\\begin{align*}\n  \\La(t):= \\: & (1+t)\\(\\log(e+t)\\)^{-r+1}\n = \\(\\frac{1+t}{(1+t)^{-m+1}\\(\\log(e+t)\\)^{m(r-1)}}\\)^{\\frac{1}{m}}\n\\\\\n \\simeq \\: &  \\(\\frac{\\Th(t)}{\\int^\\infty_t \\Xi(s)^{-m}\\,ds}\\)^{\\frac{1}{m}}. \n\\end{align*}\nThen \\eqref{La-infty}, (H5*) are (H6*) are valid. \nNoting that $\\La^{-1}(\\tau) \\simeq \\tau (\\log \\tau)^{r-1}$ for $\\tau\\ge e$, \nthere exists a positive constant $M$ such that \n\\begin{align*}\n  \\exp\\(2 |\\xi(\\th)| \\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\\)\n  \\lesssim\n  \\exp\\(2M N \\(\\log\\frac{N}{|\\xi(\\th)|}\\)^{r-1}\\)\n\\end{align*}\nfor any $N\\ge 2e$ on $\\T\\setminus\\{0\\}$. \nFor $M_0>0$, we define $u_1=\\{u_1[k]\\}_{k\\in \\Z}$ by \n\\begin{equation*\n  u_1[k]:=\\frac{1}{2\\pi}\\int_\\T\n  \\(\\sin^2 \\frac{\\th}{2}\\)^{\\frac{M_0}{2}} e^{\\i k \\th}\\,d\\th, \n\\end{equation*}\nit follows that\n\\begin{align*}\n  \\cE(0,\\th)\n =\\(\\sin^2 \\frac{\\th}{2}\\)^{M_0}\n =\\(\\frac{N}{2}\\)^{2M_0} \n  \\exp\\(-2M_0\\log \\frac{N}{|\\xi(\\th)|} \\). \n\\end{align*}\nThen we have \n\\begin{align*}\n  U(N,u_0,u_1)\n  \\lesssim \\int_\\T \n  \\exp\\(2M N\\(\\log\\frac{N}{|\\xi(\\th)|}\\)^{r-1}\n       -2M_0\\log \\frac{N}{|\\xi(\\th)|} \\)\n  \\,d\\th.\n\\end{align*}\nIf $1<r<2$ then $U(N,u_0,u_1)<\\infty$ for any $N\\ge 2e$, hence Theorem \\ref{Thm2} (i) can be applied. \nIf $r=1$ then $U(N_0,u_0,u_1)<\\infty$ for $M_0 \\ge MN_0$, hence Theorem \\ref{Thm2} (ii) can be applied if $M_0$ is large enough. \n\\end{example}\n\n\\begin{example}\\label{Ex2Thm2}\nLet $a(t)$ be defined in Example \\ref{Ex1} with $m \\ge 1$, $0\\le p<q<1$ and $r=0$. \nThen we have \n\\begin{equation*}\n  \\Th(t)\\simeq (1+t)^{1-p},\\;\\;\n  \\Xi(t) \\simeq (1+t)^{\\frac{p}{m}-q+1}, \n\\end{equation*}\nand (H4*) dose not hold. \nLet us define $\\La(t)$ by \n\\begin{equation*}\n  \\La(t):=(1+t)^{1-q}\n =\\(\\frac{(1+t)^{1-p}}{(1+t)^{-p-m(1-q)+1}}\\)^{\\frac{1}{m}}\n \\simeq \\(\\frac{\\Th(t)}{\\int^\\infty_t \\Xi(s)^{-m}\\,ds}\\)^{\\frac{1}{m}}. \n\\end{equation*}\nThen \\eqref{La-infty}, (H5*) are (H6*) are valid. \nNoting that $\\Th(\\La^{-1}(\\tau)) \\simeq \\tau^{(1-p)\/(1-q)}$ for $\\tau\\ge 1$, there exists a positive constant $M_0$ such that\n\\begin{align*}\n  |\\xi(\\th)| \\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi(\\th)|}\\)\\)\n  \\le\n  M_0 N^{\\frac{1-p}{1-q}}|\\xi(\\th)|^{-\\frac{q-p}{1-q}}\n\\end{align*}\nfor any $N\\ge 2$ on $\\T\\setminus\\{0\\}$. \nFor $\\rho>0$ and $\\ka \\ge (q-p)\/(1-q)$ we define $u_1=\\{u_1[k]\\}_{k\\in \\Z}$ by \n\\begin{equation*}\n  u_1[k]:=\\frac{1}{2\\pi}\\int_{\\T}\n  \\exp\\(-\\rho\\(\\sin^2\\frac{\\th}{2}\\)^{-\\frac{\\ka}{2}}\\)\n  e^{\\i k \\th}\\,d\\th,\n\\end{equation*}\nit follows that\n\\begin{align*}\n  \\cE(0,\\th)\n =\\exp\\(-2 \\rho \\(\\sin^2 \\frac{\\th}{2}\\)^{-\\frac{\\ka}{2}}\\)\n =\\exp\\(-2^{\\ka+1}\\rho|\\xi(\\th)|^{-\\ka}\\). \n\\end{align*}\nThen we have \n\\begin{align*}\n  U(N,u_0,u_1)\n  \\le\n  \\int_\\T \\exp\\(\n  -2|\\xi(\\th)|^{-\\ka}\\(\n  2^{\\ka}\\rho-M_0 N^{\\frac{1-p}{1-q}}|\\xi(\\th)|^{\\ka-\\frac{q-p}{1-q}}\n  \\)\\)\\,d\\th.\n\\end{align*}\nIf $\\ka > (q-p)\/(1-q)$, then $U(N,u_0,u_1)<\\infty$ for any large $N$, hence Theorem \\ref{Thm2} (i) can be applied. \nIf $\\ka = (q-p)\/(1-q)$ then $U(N_0,u_0,u_1)<\\infty$ for \n$\\rho \\ge 2^{-\\ka}M_0 N_0^{(1-p)\/(1-q)}$, hence Theorem \\ref{Thm2} (ii) can be applied if $\\rho$ is large enough. \n\\end{example}\n\n\n\nBy Theorem \\ref{Thm2}, Example \\ref{Ex2-Thm3} and Lemma \\ref{Lemma1-Thm3} we have the following corollary: \n\\begin{corollary}\nFor $\\nu>1$ and $\\rho>0$ we define the Gevrey classes $\\ga_\\rho^\\nu$ and $\\ga_\\infty^\\nu$ by \n\\begin{equation*}\n  \\ga^\\nu_\\rho:=\\left\\{f\\in C^\\infty\\(\\T^d\\)\\;;\\;\n\\sup_{\\th\\in\\T^d}\\left\\{\\left|\\pa_\\th^\\al f(\\th)\\right|\n  \\frac{\\rho^{|\\al|}}{\\al!^\\nu}\\right\\}<\\infty\\right\\},\n\\;\\;\n  \\ga_\\infty^\\nu:=\\bigcup_{\\rho>0} \\ga_\\rho^\\nu,\n\\end{equation*}\nrespectively. \nLet {\\rm (H1*)} and {\\rm (H2*)} hold for \n$\\Th(t)\\lesssim (1+t)^{1-p}$ and $\\Xi(t)=(1+t)^{\\frac{p}{m}-q+1}$ \nwith $m\\ge 1$ and $0\\le p < q<1$. \n\\begin{itemize}\n\\item[{\\rm (i)}]\nIf $\\xi_j \\hat{u}_0,\\, \\hat{u}_1 \\in \\ga_\\infty^\\nu$ \n$(j=1,\\ldots,d)$ with \n$\\nu<(1-p)\/(q-p)$ and \n\\begin{equation}\\label{eq1-cor}\n  \\(\\pa_\\th^\\al \\xi_j \\hat{u}_0\\)(0) = 0\\;\\;(j=1,\\ldots,d),\\;\\;\n  \\(\\pa_\\th^\\al \\hat{u}_1\\)(0) = 0,\\;\\;\n  \\al\\in \\N_0^d,\n\\end{equation}\nthen there exists a positive constant $N_0$ such that $U(N_0,u_0,u_1)<\\infty$ and \n\\eqref{Ebdd} is established. \n\\item[{\\rm (ii)}] \nThere exist positive constants $\\rho$ and $N_0$ such that \nfor any $\\xi_j \\hat{u}_0,\\, \\hat{u}_1 \\in \\ga_\\rho^\\nu$ \n$(j=1,\\ldots,d)$ with $\\nu=(1-p)\/(q-p)$ satisfying \\eqref{eq1-cor}, \n$U(N_0,u_0,u_1)<\\infty$ and \\eqref{Ebdd} is established.  \n\\end{itemize}\n\\end{corollary}\n\n\n\nLet us consider the conditions for \\eqref{est_Thm2} to the initial data $u_0$ and $u_1$ themselves instead of $\\hat{u}_0$ and $\\hat{u}_1$. \nIn order to describe the conditions, we introduce the logarithmic convexity and the associated functions for sequences.\nThe sequence of positive real numbers $\\{M_j\\}=\\{M_j\\}_{j=0}^\\infty$ is called logarithmically convex if the following estimate holds: \n\\begin{equation*}\n  \\frac{M_{j}}{M_{j-1}} \\le \\frac{M_{j+1}}{M_{j}},\\;\\;\n  j\\in \\N.\n\\end{equation*}\nFor a logarithmically convex sequence $\\{M_j\\}$, we define the associated function $T[\\{M_j\\}](\\tau)$ on $\\R_+$ by \n\\begin{equation*}\n  T[\\{M_j\\}](\\tau):=\\sup_{j\\in \\N}\\left\\{\\frac{\\tau^j}{M_j}\\right\\}. \n\\end{equation*}\nThen our third theorem is given as follows: \n\n\\begin{theorem}\\label{Thm3}\nLet $\\Th(t)$, $\\Xi(t)$ and $\\La(t)$ satisfy the same assumptions in Theorem \\ref{Thm3}, and $\\{M_j\\}$ be a logarithmically convex sequence \nsatisfying \n\\begin{equation}\\label{eq1-Thm3}\n  L\\(N,\\{M_j\\}\\):=\n  \\inf_{\\tau \\ge 1}\\left\\{\n  \\frac{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)}\n       {\\exp\\(\\tau^{-1}\\Th\\(\\La^{-1}(N \\tau\\)\\)}\n  \\right\\}>0\n\\end{equation}\nfor a positive real number $N$. \n\\begin{itemize}\n\\item[{\\rm (i)}] \nIf $L(N,\\{M_j\\})>0$ for any $N>0$, and $(u_0,u_1)$ satisfies \n\\begin{equation}\\label{eq2-Thm3}\n  \\sum_{k\\in \\Z^d} k^{\\al} D_j^+ u_0[k]=0\n  \\;\\;(j=1,\\ldots,d), \\;\\;\n  \\sum_{k\\in \\Z^d} k^\\al u_1[k]=0, \\;\\;\n  \\al\\in \\N_0^d:=(\\N\\cup\\{0\\})^d\n\\end{equation}\nand \n\\begin{equation}\\label{eq3-Thm3}\n  \\sup_{k\\in \\N^d}\\left\\{\n  \\(\\sum_{j=1}^d|D_j^+ u_0[k]|+ |u_1[k]|\\)|k|^{d+1} T[\\{M_j\\}]\\(\\rho^{-1}|k|\\)\n  \\right\\} <\\infty \n\\end{equation}\nfor a positive constant $\\rho$, \nthen there exists a positive constant $N_0$ such that \n$U(N_0,u_0,u_1)<\\infty$ and the energy estimate \\eqref{Ebdd} is established. \n\\item[{\\rm (ii)}]\nIf the following estimate holds: \n\\begin{equation}\\label{eq4-Thm3}\n  \\lim_{N\\to\\infty} \\inf_{\\tau\\ge \\frac{1}{2\\sqrt{d}}}\n  \\left\\{\\frac{\\Th\\(\\La^{-1}(N\\tau)\\)}{N\\Th\\(\\La^{-1}(\\tau)\\)}\\right\\}\n  =\\infty,\n\\end{equation}\nthen there exist positive constants $\\rho$ and $N_0$ such that \nfor any $(u_0,u_1)$ satisfying \\eqref{eq2-Thm3} and \\eqref{eq3-Thm3}, \n$U(N_0,u_0,u_1)<\\infty$ and \\eqref{Ebdd} is established.  \n\\end{itemize}\n\\end{theorem}\n\nLet us introduce some examples of the choice of $\\{M_j\\}$ in Theorem \\ref{Thm3}. \n\\begin{example\nLet $a(t)$ be define in Example \\ref{Ex1} and Example \\ref{Ex1Thm2}. \nIf $\\{M_j\\} = \\{j! \\exp(b j^{\\si})\\}$ for $b>0$ and $\\si>1$, then there exists a positive constant $b^\\ast$ such that \n\\begin{align*}\n  T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)\n  \\gtrsim \\exp\\(b^\\ast \\(\\log\\tau\\)^{\\frac{\\si}{\\si-1}}\\)\n\\end{align*}\nfor any $\\tau\\ge 1$ (see \\cite{M}). \nBy the consideration of Example \\ref{Ex1Thm2}, there exists a positive constant $M$ such that\n\\begin{align*}\n  \\frac{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)}\n       {\\exp\\(\\tau^{-1}\\Th\\(\\La^{-1}(N \\tau\\)\\)}\n  \\gtrsim \n  \\exp\\(b^\\ast \\(\\log\\tau\\)^{\\frac{\\si}{\\si-1}}\n       -M\\log(N\\tau)^{r-1}\\). \n\\end{align*}\nTherefore, for any $N>0$, \n\\eqref{eq1-Thm3} is valid for $\\si>1$ if $r\\le 2$, and for $(r-1)\/(r-2)>\\si>1$ if $r>2$. \nHere we note that \\eqref{eq4-Thm3} does not hold. \n\\end{example}\n\n\n\\begin{example}\\label{Ex2-Thm3}\nLet $a(t)$ be define in Example \\ref{Ex1} and Example \\ref{Ex2Thm2}. \nIf $\\{M_j\\} = \\{j!^{\\nu}\\}$ for $\\nu>1$, then there exists a positive constant $q$ such that \n\\begin{align*}\n  T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)\n  \\gtrsim \\tau^{-\\frac12}\\exp\\((\\nu-1) \\tau^{\\frac{1}{\\nu-1}}\\)\n\\end{align*}\nfor any $\\tau\\ge 1$ (see \\cite{M}). \nBy the consideration of Example \\ref{Ex1Thm2}, there exists a positive constant $M$ such that\n\\begin{align*}\n  \\frac{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right](\\tau)}\n       {\\exp\\(\\tau^{-1}\\Th\\(\\La^{-1}(N \\tau)\\)\\)}\n  \\gtrsim \n  \\tau^{-\\frac12}\\exp\\((\\nu-1) \\tau^{\\frac{1}{\\nu-1}}\n       -M N^{\\frac{1-p}{1-q}}\\tau^{\\frac{q-p}{1-q}}\\). \n\\end{align*}\nTherefore, \\eqref{eq1-Thm3} is valid for any $N>0$ with $\\nu<(1-p)\/(q-p)$, \nthat is, $1\/(\\nu-1)>(q-p)\/(1-q)$, and thus Theorem \\ref{Thm3} (i) can be applied. \nIf $\\nu=(1-p)\/(q-p)$ then \\eqref{eq1-Thm3} is valid for \n$N \\le ((\\nu-1)\/M)^{(1-q)\/(1-p)}$. \nNoting that $\\Th(\\La^{-1}(N\\tau))\/(N\\Th(\\La^{-1}(\\tau)))\\simeq N^{(q-p)\/(1-q)}\\to\\infty$ as $N\\to\\infty$, Theorem \\ref{Thm3} (ii) can be applied. \n\\end{example}\n\n\n\\section{Proof of Theorem \\ref{Thm1}}\n\\subsection{Proof of Theorem \\ref{Thm1} (i)}\nLet $\\xi=\\xi(\\th)$ be defined by \\eqref{xith}. \nDenoting \n\\begin{equation*}\n  v=v(t,\\xi)=\\hat{u}(t,\\th),\n\\end{equation*}\nthe equation of \\eqref{hu} is represented as follows: \n\\begin{equation}\\label{v}\n\\partial_t^2 v(t,\\xi) + a(t)^2 |\\xi|^2 v(t,\\xi) =0, \\;\\;\n  (t,\\xi)\\in \\R_+ \\times [-2,2]^d. \n\\end{equation}\nHere we denote $\\cE(t,\\th)$ by $\\cE(t,\\xi)$: \n\\begin{align*}\n  \\cE(t,\\xi) = |\\pa_t v(t,\\xi)|^2 + a(t)^2 |\\xi|^2 |v(t,\\xi)|^2 \n\\end{align*}\nwithout any confusion. \nWe define $\\cE_\\infty(t,\\xi)$ by \n\\begin{equation}\n  \\cE_\\infty(t,\\xi):=|\\pa_t v(t,\\xi)|^2 + a_\\infty^2 |\\xi|^2 |v(t,\\xi)|^2. \n\\end{equation}\nThen we have \n\\begin{align*}\n  \\pa_t \\cE_\\infty(t,\\xi)\n=\\: &2\\(a_\\infty^2-a(t)^2\\)|\\xi|^2 \\Re\\(\\pa_t v\\, \\ol{v}\\) \n\\\\\n\\le\\: &\n  \\frac{\\left|a_\\infty^2-a(t)^2\\right||\\xi|}{a_\\infty} \n  \\cE_\\infty(t,\\xi)\n\\le\n  \\frac{2a_1}{a_0} \\left|a(t)-a_\\infty\\right||\\xi|\n  \\cE_\\infty(t,\\xi).\n\\end{align*}\nBy Gronwall's inequality and \\eqref{stb}, we have \n\\begin{align*}\n  \\cE_\\infty(t,\\xi)\n  \\le\\:& \\exp\\(\\frac{2a_1}{a_0}\\Th(t)|\\xi|\\) \\cE_\\infty(0,\\xi)\n\\\\\n  \\le\\:& \\exp\\(\\frac{4\\sqrt{d} \\, a_1}{a_0}\\sup_{t\\ge 0}\\{\\Th(t)\\}\\) \n  \\cE_\\infty(0,\\xi) \n  \\simeq \\cE_\\infty(0,\\xi) \n\\end{align*}\nfor any $(t,\\xi)\\in \\R_+ \\times [-2,2]^d$. \nAnalogously, we have \n\\begin{equation*}\n  \\cE_\\infty(t,\\xi) \\ge \n  \\exp\\(-\\frac{4\\sqrt{d} \\, a_1}{a_0}\\sup_{t\\ge 0}\\{\\Th(t)\\}\\) \n  \\cE_\\infty(0,\\xi)\n  \\simeq \\cE_\\infty(0,\\xi). \n\\end{equation*}\nTherefore, by Lemma \\ref{lemm-E-cE} and noting the estimate: \n\\begin{equation}\\label{cEinfcE}\n  \\cE_\\infty(t,\\xi) \\simeq \\cE(t,\\xi)\n\\end{equation}\ndue to \\eqref{a0a1}, we have \n\\begin{equation*}\n  E(t) \\simeq \\int_{\\T^d} \\cE_\\infty(t,\\xi(\\th))\\,d\\th\n  \\simeq  \\int_{\\T^d} \\cE_\\infty(0,\\xi(\\th))\\,d\\th\n  \\simeq E(0).\n\\end{equation*}\n\\qed\n\n\\begin{remark}\nIt is crucial for the proof of Theorem \\ref{Thm1} (i) that $\\sup_{\\th\\in\\T}\\{|\\xi(\\th)|\\}<\\infty$ which comes from the characteristics of the discrete model. \n\\end{remark}\n\n\\subsection{Proof of Theorem \\ref{Thm1} (ii)}\n\nBy \\eqref{ak} we have \n\\begin{align*}\n  \\pa_t \\cE(t,\\xi)\n = 2a'(t)a(t)|\\xi|^2 |v(t,\\xi)|^2\n \\le \\frac{2C_1}{a_0}\\Xi(t)^{-1} \\cE(t,\\xi). \n\\end{align*}\nTherefore, by Gronwall's inequality we have \n\\begin{equation*}\n  \\cE(t,\\xi)\n  \\le \\exp\\(\\frac{2C_1}{a_0}\\left\\|\\Xi(\\cd)^{-1}\\right\\|_{L^1(\\R_+)}\\) \n  \\cE(0,\\xi)\n  \\simeq \\cE(0,\\xi). \n\\end{equation*}\nAnalogously, we have \n\\begin{equation*}\n  \\cE(t,\\xi) \\ge \n  \\exp\\(-\\frac{2C_1}{a_0}\\left\\|\\Xi(\\cd)^{-1}\\right\\|_{L^2(\\R_+)}\\) \n  \\cE(0,\\xi)\n  \\simeq \\cE(0,\\xi).  \n\\end{equation*}\nThus the proof is concluded by using Lemma \\ref{lemm-E-cE}. \n\\qed\n\n\\subsection{Proof of Theorem \\ref{Thm1} (iii)}\n\\subsubsection{Zones}\nWe can suppose that $\\lim_{t\\to\\infty}\\Th(t)=\\infty$; otherwise we have GEC by Theorem \\ref{Thm1} (i). \nLet $N$ be a large constant depends on $a_0$, $a_1$ and $C_k$ ($k=0,\\cdots,m$), to be chosen later. \nWe define $T_0$ and $t_\\xi$ for $\\xi \\in [-2,2]^d$ by \n\\[\n  T_0:=\\max\\left\\{t\\ge 0\\;;\\; \\Th(t) = \\frac{N}{2\\sqrt{d}}\\right\\}\n\\]\nand\n\\[\n  t_\\xi:=\\max\\left\\{t\\ge T_0\\;;\\; \\Th(t)|\\xi| = N\\right\\},\n\\]\nrespectively. \nThen we divide the region $\\R_+\\times [-2,2]^d$ by $(t_\\xi,\\xi)$ into the pseudo-differential zone $Z_\\Psi$ and the hyperbolic zone $Z_H$ as follows: \n\\begin{equation*\n  Z_\\Psi:=\\left\\{(t,\\xi)\\in \\R_+\\times [-2,2]^d\\;;\\; t \\le t_\\xi \\right\\}\n\\end{equation*}\nand\n\\begin{equation*\n  Z_H:=\\left\\{(t,\\xi)\\in \\R_+\\times [-2,2]^d \\;;\\; t \\ge t_\\xi \\right\\},\n\\end{equation*}\nrespectively. \nWe shall estimate $\\cE(t,\\xi)$ in each zones by different methods. \n\n\\subsubsection{Estimate in $Z_\\Psi$}\nLet $(t,\\xi) \\in Z_\\Psi$, that is, $\\Th(t)|\\xi| \\le N$. \nBy the same way as the method for the proof of Theorem \\ref{Thm1} (i), we have\n\\begin{align}\\label{estcE0-ZP}\n  \\cE_\\infty(t,\\xi)\n  \\le \\exp\\(\\frac{2a_1}{a_0}\\Th(t)|\\xi|\\) \\cE_\\infty(0,\\xi)\n  \\le \\exp\\(\\frac{2a_1 N}{a_0}\\) \\cE_\\infty(0,\\xi) \n\\end{align}\nand\n\\begin{align*}\n  \\cE_\\infty(t,\\xi)\n  \\ge \\exp\\(-\\frac{2a_1 N}{a_0}\\) \\cE_\\infty(0,\\xi). \n\\end{align*}\nTherefore, by \\eqref{cEinfcE} we have \n\\begin{equation}\\label{estcE-ZP}\n  \\cE(t,\\xi)\\simeq \\cE(0,\\xi), \\;\\; t \\le t_\\xi.\n\\end{equation}\n\n\\subsubsection{Estimate in $Z_H$}\n\nLet us consider the following first order system:\n\\begin{equation}\\label{V}\n\\pa_t V=A V,\\;\\;\nV=\\begin{pmatrix}v_1 \\\\ v_2 \\end{pmatrix},\\;\\;\nA=\\begin{pmatrix} \\phi(t,\\xi) & \\ol{r(t,\\xi)} \\\\ \n  r(t,\\xi) & \\ol{\\phi(t,\\xi)} \\end{pmatrix}. \n\\end{equation}\nFor a complex valued function $\\phi$, we denote the real and the imaginary parts of $\\phi$ by $\\phi_\\Re$ and $\\phi_\\Im$, respectively. \nThen we have the following lemma: \n\\begin{lemma}\\label{lemm-estV}\nFor any $t,\\tilde{t} \\in \\R$ satisfying $\\tilde{t} < t$ \nthe solution $V=V(t,\\xi)$ of \\eqref{V} satisfies the following estimates: \n\\begin{equation}\\label{estV}\n  \\|V(t,\\xi)\\|_{\\C^2}^2\n  \\begin{cases}\n  \\ds{\\le  \\exp\\(2\\int^t_{\\tilde{t}} \\phi_\\Re(s,\\xi)\\,ds\n        + 2\\int^t_{\\tilde{t}} |r(s,\\xi)|\\,ds\\)\n  \\|V(t_0,\\xi)\\|_{\\C^2}^2},\n\\\\[3ex]\n  \\ds{\\ge \\exp\\(2\\int^t_{\\tilde{t}} \\phi_\\Re(s,\\xi)\\,ds\n        - 2\\int^t_{\\tilde{t}} |r(s,\\xi)|\\,ds\\) \n  \\|V(t_0,\\xi)\\|_{\\C^2}^2}.\n  \\end{cases}\n\\end{equation}\n\n\\end{lemma}\n\\begin{proof}\nBy Cauchy-Schwarz inequality, we have \n\\begin{align*}\n  \\pa_t \\|V\\|_{\\C^2}^2\n&=2\\Re\\(\\pa_t V,V\\)_{\\C^2}\n =2\\Re\\(\\begin{pmatrix} \\phi & 0 \\\\ 0 & \\ol{\\phi} \\end{pmatrix}V, V\\)_{\\C^2}\n +2\\Re\\(\\begin{pmatrix} 0 & \\ol{r} \\\\ r & 0 \\end{pmatrix}V, V\\)_{\\C^2}\n\\\\\n&=2\\phi_\\Re\\|V\\|_{\\C^2}^2 + 4\\Re\\(rv_1\\ol{v_2}\\)\n\\begin{cases}\n\\le 2\\(\\phi_\\Re + |r|\\)\\|V\\|_{\\C^2}^2,\n\\\\\n\\ge 2\\(\\phi_\\Re - |r|\\)\\|V\\|_{\\C^2}^2.\n\\end{cases}\n\\end{align*}\nThus we have \\eqref{estV} by Gronwall's inequality. \n\\end{proof}\n\nSince the equation of \\eqref{v} is reduced to the following first order system:\n\\begin{equation}\\label{V1}\n  \\pa_t V_1=A_1 V_1,\\;\\;\n  V_1:=\\begin{pmatrix}\n    \\pa_t v + \\i a(t)|\\xi|v \\\\ \\pa_t v -\\i a(t)|\\xi|v\n  \\end{pmatrix},\n  \\;\\;\n  A_1:=\\begin{pmatrix} \\phi_1 & \\ol{r_1} \\\\ r_1 & \\ol{\\phi_1} \\end{pmatrix},\n\\end{equation}\nwhere \n\\begin{align*}\n  r_1:=-\\frac{a'(t)}{2a(t)}=\\ol{r_1}\n  \\;\\text{ and }\\;\n  \\phi_1=\\frac{a'(t)}{2a(t)}+\\i a(t)|\\xi|,\n\\end{align*}\nby Lemma \\ref{lemm-estV} and noting the equality \n\\begin{equation}\\label{cEV1}\n  \\|V_1(t,\\xi)\\|_{\\C^2}^2=2\\cE(t,\\xi),\n\\end{equation}\nwe have \n\\begin{equation}\\label{estV1}\n  \\cE(t,\\xi)\n\\begin{cases}\n\\ds{\\le\n\\frac{a(t)}{a(t_0)}\n  \\exp\\(\\int^t_{\\tilde{t}} \\frac{|a'(s)|}{a(s)}\\,ds\\) \n  \\cE(t_0,\\xi)}, \n\\\\[3ex] \n\\ds{\\ge\n\\frac{a(t)}{a(t_0)}\n  \\exp\\(-\\int^t_{\\tilde{t}} \\frac{|a'(s)|}{a(s)}\\,ds\\) \n  \\cE(t_0,\\xi)}. \n\\end{cases}\n\\end{equation}\nIndeed, \\eqref{GEC} is concluded from the estimate \\eqref{estV1} \nif $\\Xi(t)^{-1}\\in L^1(\\R_+)$, but not for $\\Xi(t)\\not\\in L^1(\\R_+)$. \nTherefore, we introduce the following idea, which is called refined diagonalization procedure taking account the properties \\eqref{stb} and \\eqref{ak} with $m\\ge 2$. \nA basic idea of this method was introduced in \\cite{Yg}, and improved in \\cite{H07} to take the benefit of the property \\eqref{stb}. \nThe refined diagonalization procedure can be understood to construct a $2 \\times 2$ regular matrix $M=M(t,\\xi)$ in $Z_H$ satisfies the following properties: \n\\begin{itemize}\n\\item\n$M$ is defined in $Z_H$. \n\\item\n$V:=M V_1$ satisfies $\\|V(t,\\xi)\\|_{\\C^2}^2 \\simeq \\cE(t,\\xi)$. \n\\item\n$V$ is a solution of \\eqref{V}. \n\\item\n$\\int^t_{\\tilde{t}}\\phi_\\Re(s,\\xi)\\,ds$ is bounded in $Z_H$. \n\\item\n$\\int^t_{\\tilde{t}}|r(s,\\xi)|\\,ds$ is bounded in $Z_H$. \n\\end{itemize}\nIndeed, if there exists such a matrix $M$, then we immediately have \\eqref{GEC} from \\eqref{estcE-ZP} and \\eqref{estV} with $\\tilde{t}=t_\\xi$. \n\n\nFor $p\\in \\N_0$ and $q, r \\in \\Z$ we introduce the symbol-like class \n$S^{(p)}\\{q,r\\}$ as the set of functions $f(t,\\xi)$ satisfy the following estimates in $Z_H$: \n\\begin{equation*\n  \\left|\\pa_t^k f(t,\\xi)\\right|\\lesssim \n  |\\xi|^q \\Xi(t)^{-r-k}\n  \\;\\; (k=0,\\ldots,p).\n\\end{equation*}\nThen we have the following usual algebraic properties and a hierarchy of the classes in $Z_H$:\n\\begin{lemma}\\label{symbol}\nThe following properties are established:\n\n\\begin{itemize}\n\\item[\\rm (i)]\nIf $f\\in S^{(p)}\\{q,r\\}$ with $p\\ge1$, then $f\\in S^{(p-1)}\\{q,r\\}$ and $\\pa_t f\\in S^{(p-1)}\\{q,r+1\\}$.\n\n\\item[\\rm (ii)]\nIf $f_1\\in S^{(p)}\\{q_1,r_1\\}$ and $f_2\\in S^{(p)}\\{q_2,r_2\\}$, then\n$f_1f_2\\in S^{(p)}\\{q_1+q_2,r_1+r_2\\}$.\n\\item[\\rm (iii)]\nIf $f\\in S^{(p)}\\{q,r\\}$, then $f \\in S^{(p)}\\{q+1,r-1\\}$.\n\\item[\\rm (iv)]\nIf $f\\in S^{(p)}\\{-q,q\\}$ with $q\\ge 1$, then for any $\\ve>0$ there exists a positive constant $N_0$ such that $|f(t,\\xi)|\\le \\ve$ for any $N \\ge N_0$. \n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n(i) and (ii) are trivial from the definition of $S^{(p)}\\{q,r\\}$. \nLet $f(t,\\xi)\\in S^{(p)}\\{q,r\\}$.\nBy (\\ref{ThXi}) we have\n\\begin{align*}\n  \\left|\\pa_t^k f(t,\\xi)\\right|\n  \\lesssim\\: & |\\xi|^q \\Xi(t)^{-r-k}\n  \\le N^{-1} |\\xi|^{q+1} \\frac{\\Th(t)}{\\Xi(t)} \\Xi(t)^{-(r-1)-k}\n\\\\\n  \\le\\:& N^{-1} C_0 |\\xi|^{q+1} \\Xi(t)^{-(r-1)-k}\n\\end{align*}\nfor $k=0,\\ldots,p$. \nIt follows that $f(t,\\xi)\\in S^{(p)}\\{q+1,r-1\\}$; thus (iii) is proved. \nIf $f(t,\\xi)\\in S^{(p)}\\{-q,q\\}$, then by \\eqref{ThXi} we have \n\\begin{align*}\n  |f(t,\\xi)|\n  \\le C|\\xi|^{-q}\\Xi(t)^{-q}\n  \\le C N^{-q}\\(\\frac{\\Th(t)}{\\Xi(t)}\\)^{q}\n  \\le C C_0^q N^{-q} \\le \\ve\n\\end{align*}\nby putting $N_0=C_0(C\\ve^{-1})^{1\/q}$. \n\\end{proof}\n\nBy Lemma \\ref{symbol} we have the following lemma: \n\\begin{lemma}\\label{symbol2}\nThere exists a positive constant $N_0$ such that the following properties are established for any $N \\ge N_0$:\n\\begin{itemize}\n\\item[\\rm (i)] \nIf $f(t,\\xi)\\in S^{(p)}\\{0,0\\}$ and $f(t,\\xi)\\gtrsim 1$, \nthen $1\/f(t,\\xi)\\in S^{(p)}\\{0,0\\}$. \n\\item[\\rm (ii)] \nIf $f(t,\\xi)\\in S^{(p)}\\{1,0\\}$ and $f(t,\\xi)\\gtrsim |\\xi|$, \nthen $1\/f(t,\\xi)\\in S^{(p)}\\{-1,0\\}$. \n\\item[\\rm (iii)] \nIf $f(t,\\xi)\\in S^{(p)}\\{-q,q\\}$ with $q \\ge 1$, \nthen $\\sqrt{1-|f(t,\\xi)|^2}\\in S^{(p)}\\{0,0\\}$. \n\\end{itemize} \n\\end{lemma}\n\\begin{proof}\nSince (i) is proved by the same way as the proof of (ii), we prove (ii). \nBy applying Fa\\`a di Bruno's formula:  \n\\begin{equation*\n  \\frac{d^k}{dt^k}F(G(x))\n =k!\\sum_{h=1}^k \n  F^{(h)}(G(t))\n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}}\n  \\,\n  \\prod_{l=1}^k \\frac{1}{h_l!\\, l!^{h_l}}\\(G^{(l)}(t)\\)^{h_l} \n\\end{equation*}\nwith $F(G)=1\/G$ and $G(t)=f(t,\\xi)$, and noting \n$|F^{(h)}(G)|\\lesssim 1\/|G|^{h+1}$, we have \n\\begin{align*}\n  \\left|\\pa_t^k \\frac{1}{f(t,\\xi)}\\right|\n\\lesssim\\:&\n  \\sum_{h=1}^k \\frac{1}{f(t,\\xi)^{h+1}}\n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}}\n  \\,\n  \\prod_{l=1}^k \\left|\\pa_t^l f(t,\\xi)\\right|^{h_l}\n\\\\\n\\lesssim\\:&\n  \\sum_{h=1}^k |\\xi|^{-h-1}\n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}} \\,\n  \\prod_{l=1}^k \\(|\\xi| \\Xi(t)^{-l}\\)^{h_l}\n\\\\\n\\simeq\\:&\n  |\\xi|^{-1} \\Xi(t)^{-k}\n\\end{align*}\nfor $k=0,\\ldots,p$; thus (ii) is proved.  \nBy Lemma \\ref{symbol} (iv) we can suppose that \n$\\sqrt{1-|f(t,\\xi)|^2} \\ge 1\/2$. \nBy applying Fa\\`a di Bruno's formula with \n$F(G)=\\sqrt{1-G}$ and $G(t)=|f(t,\\xi)|^2 \\in S^{(p)}\\{-2q,2q\\}$, \nand noting the estimates \n\\begin{align*}\n  |F^{(h)}(G)| \\lesssim \\frac{F(G)}{(1-G)^h} \\le 4^h\n\\end{align*}\nfor $h=0,1,\\ldots$, we have \n\\begin{align*}\n  \\left|\\pa_t^k \\sqrt{1-|f(t,\\xi)|^2}\\right|\n\\lesssim\\:&\n  \\sum_{h=1}^k 4^h \n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}}\n  \\,\n  \\prod_{l=1}^k \\left|\\pa_t^l |f(t,\\xi)|^2\\right|^{h_l}\n\\\\\n\\lesssim\\:&\n  \\sum_{h=1}^k \n  \\sum_{\\substack{h_1 + 2h_2 + \\cdots + n h_n=k \\\\ \n                  h_1+h_2+\\cdots + h_k = h}}\n  \\,\n  \\prod_{l=1}^k \\(|\\xi|^{-2q}\\Xi(t)^{-2q-l}\\)^{h_l}\n\\\\\n\\lesssim\\:&\n  \\sum_{h=1}^k |\\xi|^{-2qh}\\Xi(t)^{-2qh-k}\n\\le \\Xi(t)^{-k} \\sum_{h=1}^k \\(\\frac{C_0}{N}\\)^{2qh}\n\\\\\n\\simeq\\:& \\Xi(t)^{-k} \n\\end{align*}\nfor $k=1,\\ldots,p$; thus (iii) is prove. \n\\end{proof}\n\n\nAn eigenvalue of $A_1$ is represented by \n\\begin{equation*}\n  \\la_1=\\phi_{1\\Re} + \\i \\phi_{1\\Im}\\sqrt{1 -\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}. \n\\end{equation*}\nBy \\eqref{ak}, \\eqref{ThXi} and choosing $N>C_0 C_1 a_0^{-2}\/2$, we have \n\\begin{align*}\n  \\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2\n\\le \\(\\frac{C_1}{2a_0^2|\\xi|\\Xi(t)}\\)^2\n\\le \\(\\frac{C_1}{2a_0^2N}\\frac{\\Th(t)}{\\Xi(t)}\\)^2\n\\le \\(\\frac{C_0 C_1}{2a_0^2N}\\)^2<1.\n\\end{align*}\nIt follows that the other eigenvalue of $A_1$ is given by $\\ol{\\la_1}$. \nTherefore, a diagonalizer $M_1$ for $A_1$ is formally given by \n\\begin{equation*}\n  M_1=\\begin{pmatrix} 1 & \\ol{\\de_1} \\\\ \\de_1 & 1 \\end{pmatrix},\n  \\quad\n  \\de_{1}=\\frac{\\la_{1}-\\phi_1}{\\ol{r_1}}. \n\\end{equation*}\nHere we note that the following lemma is established which ensures the invertibility of $M_1$: \n\\begin{lemma}\\label{symbol_de1}\nThere exists a positive constant $N_0$ such that \n$r_1\\in S^{(m-1)}\\{0,1\\}$, $1\/\\phi_{1\\Im}\\in S^{(m)}\\{-1,0\\}$\nand $\\de_1\\in S^{(m-1)}\\{-1,1\\}$ for any $N \\ge N_0$. \n\\end{lemma}\n\\begin{proof}\nThe first two properties are trivial. \nLet $N_0$ be large enough so that Lemma \\ref{symbol} and Lemma \\ref{symbol2} can be applied. \nBy Lemma \\ref{symbol} (ii) and (iv), we have \n$(|r_1|\/\\phi_{1\\Im})^2 \\in S^{(m-1)}\\{-2,2\\}$ and \n$(|r_1|\/\\phi_{1\\Im})^2 \\le 1\/2$. \nBy Lemma \\ref{symbol2} (iii) we have \n\\begin{align*}\n  \\sqrt{1-\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}+1\\in S^{m-1}\\{0,0\\},\n\\end{align*}\nit follows from Lemma \\ref{symbol2} (i) that \n\\begin{align*}\n  \\frac{1}{\\sqrt{1-\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}+1}\n  \\in S^{(m-1)}\\{0,0\\}. \n\\end{align*}\nTherefore, by Lemma \\ref{symbol} (ii) and the first two properties of Lemma \\ref{symbol_de1}, we have \n\\begin{align*}\n  \\de_1\n=\\frac{\\i \\phi_{1\\Im}}{\\ol{r_1}}\n  \\(\\sqrt{1-\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}-1\\)\n=-\\frac{\\i r_1}{\\phi_{1\\Im}}\n  \\frac{1}{\\sqrt{1-\\(\\frac{|r_1|}{\\phi_{1\\Im}}\\)^2}+1}\n\\in S^{(m-1)}\\{-1,1\\}. \n\\end{align*}\n\\end{proof}\n\nWe define $A_2=A_2(t,\\xi)$ by\n\\begin{align*}\n  A_2:=\\begin{pmatrix} \\phi_2 & \\ol{r_2} \\\\ r_2 & \\ol{\\phi_2} \\end{pmatrix},\n  \\;\\;\n  r_2:=-\\frac{(\\de_1)_t}{1-|\\de_1|^2},\\;\\;\n  \\phi_2:=\\la_1+\\frac{\\ol{\\de_1}(\\de_1)_t}{1-|\\de_1|^2}. \n\\end{align*}\nThen we see that $r_2 \\in S^{(m-2)}\\{-1,2\\}$ by the lemmas for the symbol classes above. \nNoting the equalities \n\\begin{align*}\n  M_1^{-1} \\(A_1 - \\pa_t I\\)M_1\n=\\begin{pmatrix} \\la_{1} & 0 \\\\ 0 & \\ol{\\la_{1}} \\end{pmatrix}\n -M_1^{-1} \\(\\pa_t M_1\\) -\\pa_t I\n=A_2 - \\pa_t I,\n\\end{align*}\n\\eqref{V1} is reduced to the following system: \n\\begin{equation*\n  \\pa_t V_2 = A_2 V_2,\\quad\n  V_2:=M_1^{-1}V_1.\n\\end{equation*}\nHere we remark the followings: \n\\begin{itemize}\n\\item \n$M_1$ is a diagonalizer for $A_1$ but not for $A_1-\\pa_t I$, \nthat is, $A_1 - M_1^{-1}(\\pa_t M_1)$ is not diagonal. \n\\item \nSince $r_1 \\in S^{(m-2)}\\{0,1\\}$ and $r_2 \\in S^{(m-2)}\\{-1,2\\}$, \n$M_1$ is a diagonalizer for $A_1-\\pa_t I$ modulo $S^{(m-2)}\\{0,1\\}$. \n\\item\nThe structure in which both the diagonal and the off-diagonal entries are complex conjugate are conserved by the diagonalization procedure due to $M_1$. \n\\end{itemize}\nTherefore, one can carry out the same diagonalization procedure for $A_2-\\pa_t I$ if $m\\ge 3$. \nGenerally, we have the following lemma, which is the essential of the refined diagonalization procedure. \n\\begin{lemma}\\label{ref_diag}\nLet $k$ be a positive integer satisfying $k<m$, $A_k$ be given by\n\\begin{equation*}\n  A_k=\\begin{pmatrix} \\phi_k & \\ol{r_k} \\\\ r_k & \\ol{\\phi_k} \\end{pmatrix},\n\\end{equation*}\nwhere $r_k$ and $\\phi_{k\\Im}$ satisfy \n\\begin{equation}\\label{rphik}\n  r_k\\in S^{(m-k)}\\{-k+1,k\\},\\;\\;\n  \\phi_{k\\Im} \\in S^{(m-k)}\\{1,0\\}\n  \\;\\text{ and }\\;\n  |\\phi_{k\\Im}| \\gtrsim |\\xi|. \n\\end{equation}\nThen there exists a positive constant $N_k$ such that the following properties are established for any $N \\ge N_k$: \n\\begin{itemize}\n\\item[\\rm (i)]\n$A_k$ has complex conjugate eigenvalues $\\la_k$ and $\\ol{\\la_k}$. \n\\item[\\rm (ii)]\nThe following matrix $M_k$ is invertible: \n\\begin{equation*}\n  M_k:=\\begin{pmatrix} 1 & \\ol{\\de_k} \\\\ \\de_k & 1 \\end{pmatrix},\n  \\quad\n  \\de_k:=\\frac{\\la_k-\\phi_k}{\\ol{r_k}}, \n\\end{equation*}\nmoreover, we have \n\\begin{equation}\\label{dek}\n  \\de_k \\in S^{(m-k)}\\{-k,k\\}. \n\\end{equation}\n\\item[\\rm (iii)]\n$A_{k+1}:=M_k^{-1}A_kM_k - M_k^{-1}(\\pa_t M_k)$ is represented as follows: \n\\begin{equation*}\n  A_{k+1}=\\begin{pmatrix}\n  \\phi_{k+1} & \\ol{r_{k+1}} \\\\ r_{k+1} & \\ol{\\phi_{k+1}} \\end{pmatrix},\n\\end{equation*}\nwhere $r_{k+1}$ and $\\phi_{k+1}=\\phi_{(k+1)\\Re}+\\i\\phi_{(k+1)\\Im}$ are given by \n\\begin{equation}\\label{rk+1}\n  r_{k+1}=-\\frac{(\\de_k)_t}{1-|\\de_k|^2},\n\\end{equation}\n\\begin{equation*\n  \\phi_{(k+1)\\Re} =\\phi_{k\\Re}-\\frac{\\pa_t \\log\\(1-|\\de_{k}|^2\\)}{2} \n\\end{equation*}\nand\n\\begin{equation*\n  \\phi_{(k+1)\\Im}\n =\\phi_{k\\Im}\\sqrt{1-\\(\\frac{|r_k|}{\\phi_{k\\Im}}\\)^2}\n   -\\Im\\(\\ol{\\de_k}r_{k+1}\\).\n\\end{equation*}\n\\item[\\rm (iv)]\n$r_{k+1}$ and $\\phi_{(k+1)\\Im}$ satisfy the followings: \n\\begin{equation*\n  r_{k+1}\\in S^{(m-k-1)}\\{-k,k+1\\},\n  \\quad\n  \\phi_{(k+1)\\Im} \\in S^{(m-k-1)}\\{1,0\\},\n  \\;\\text{ and }\\;\n  \\phi_{(k+1)\\Im} \\gtrsim |\\xi|.\n\\end{equation*}\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nLet $N \\ge N_k$ with a large constant $N_k$ so that Lemma \\ref{symbol} and Lemma \\ref{symbol2} can be applied. \nAn eigenvalue of $A_k$ is represented by \n\\begin{equation*}\n  \\la_k=\\phi_{k\\Re}+\\i\\phi_{k\\Im}\\sqrt{1-\\(\\frac{|r_k|}{\\phi_{k\\Im}}\\)^2}. \n\\end{equation*}\nBy \\eqref{rphik}, Lemma \\ref{symbol} (ii) and Lemma \\ref{symbol2} (ii), \nwe have \n\\begin{equation}\\label{rkphik}\n  \\(\\frac{|r_{k}|}{\\phi_{k\\Im}}\\)^2 \\in S^{(m-k)}\\{-2k,2k\\}. \n\\end{equation}\nIt follows from Lemma \\ref{symbol} (iv) that the other eigenvalue of \n$A_k$ is given by $\\ol{\\la_k}$. \nBy \\eqref{rphik}, \\eqref{rkphik} and using the same way as the proof of Lemma \\ref{symbol_de1}, we have \\eqref{dek}. \nMoreover, $M_k$ is invertible since $|\\de_k|^2 \\le 1\/2$ by Lemma \\ref{symbol} (iv). \n(iii) is prove by direct computations. \nBy applying Lemma \\ref{symbol2} (i) with $f(t,\\xi)=1-|\\de_k|^2\\ge 1\/2$, \nwe have $(1-|\\de_k|^2)^{-1}\\in S^{(m-k)}\\{0,0\\}$. \nTherefore, by Lemma \\ref{symbol} (i) and \\eqref{rk+1}, we have \n$r_{k+1}\\in S^{(m-k-1)}\\{-k,k+1\\}$, \nhence we have $\\phi_{(k+1)\\Im} \\in S^{m-k-1}\\{1,0\\}$ \nby Lemma \\ref{symbol} (ii), (iii), Lemma \\ref{symbol2} (iii), \n\\eqref{dek} and \\eqref{rkphik}. \nNoting \\eqref{rkphik}, $\\Im(\\ol{\\de_k}r_{k+1})\/\\phi_{k\\Im} \\in S^{(m-k-1)}\\{-2k-1,2k+1\\}$ and Lemma \\ref{symbol} (iv), we can suppose that \n\\begin{align*}\n  \\(\\frac{|r_{k}|}{\\phi_{k\\Im}}\\)^2 \\le \\frac14\n  \\;\\text{ and }\\;\n  \\frac{\\Im\\(\\ol{\\de_k}r_{k+1}\\)}{\\phi_{k\\Im}}\\le \\frac14. \n\\end{align*}\nIt follows that \n\\begin{align*}\n  \\phi_{(k+1)\\Im}\n =\\phi_{k\\Im}\\(\n  \\sqrt{1-\\(\\frac{|r_k|}{\\phi_{k\\Im}}\\)^2}\n   -\\frac{\\Im\\(\\ol{\\de_k}r_{k+1}\\)}{\\phi_{k\\Im}}\\)\n \\ge \\frac{\\sqrt{3}-1}{2}\\phi_{k\\Im}\n  \\gtrsim |\\xi|. \n\\end{align*}\n\\end{proof}\n\nBy Lemma \\ref{ref_diag} we have the following proposition: \n\\begin{proposition}\nThere exists a positive constant $N_m$ such that the following estimate is established for any $N \\ge N_m$ in $Z_H$: \n\\begin{equation}\n  \\cE(t,\\xi) \\simeq \\cE(t_\\xi,\\xi). \n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nLet $M_k$ ($k=1,\\ldots,m-1$) be defined by Lemma \\ref{ref_diag}. \nThen we can suppose that $|\\de_k(t,\\xi)|^2 \\le 1\/2$ for $k=1,\\ldots,m-1$. \nDenoting $\\cM_{m-1} = M_1\\cdots M_{m-1}$ and $V_m=\\cM_{m-1}^{-1}V_1$, we have \n\\begin{equation*}\n  \\(\\pa_t - A_m\\) V_m\n =\\cM_{m-1}^{-1} \\(\\pa_t - A_1\\)\\cM_{m-1} V_m\n =\\cM_{m-1}^{-1} \\(\\pa_t - A_1\\) V_1 = 0\n\\end{equation*}\nby \\eqref{V1}. \nTherefore, by \\eqref{estV} we have \n\\begin{equation*}\n  \\|V_m(t,\\xi)\\|_{\\C^2}^2\n\\begin{cases}\n  \\ds{\\le \\exp\\(2\\int^t_{t_\\xi} \\phi_{m\\Re}(s,\\xi)\\,ds\n        + 2\\int^t_{t_\\xi} |r_m(s,\\xi)|\\,ds\\) \n  \\|V_m(t_\\xi,\\xi)\\|_{\\C^2}^2}, \n\\\\[3ex]\n  \\ds{\\ge \\exp\\(2\\int^t_{t_\\xi} \\phi_{m\\Re}(s,\\xi)\\,ds\n        - 2\\int^t_{t_\\xi} |r_m(s,\\xi)|\\,ds\\) \n  \\|V_m(t_\\xi,\\xi)\\|_{\\C^2}^2}. \n\\end{cases}\n\\end{equation*}\nBy Lemma \\ref{ref_diag} (iii) and (iv) we have \n\\begin{align*}\n  \\exp\\(2\\int^t_{t_\\xi} \\phi_{m\\Re}(s,\\xi)\\,ds\\)\n&=\\exp\\(2\\int^t_{t_\\xi} \\phi_{1\\Re}(s,\\xi)\\,ds \n  + \\sum_{k=1}^{m-1} \\log\\frac{1-|\\de_k(t,\\xi)|^2}{1-|\\de_k(t_\\xi,\\xi)|^2}\\)\n\\\\\n&= \\frac{a(t)}{a(t_\\xi)}\n   \\prod_{k=1}^{m-1}\\frac{1-|\\de_k(t,\\xi)|^2}{1-|\\de_k(t_\\xi,\\xi)|^2}\n\\\\\n&\\begin{cases}\n\\le\n  \\(2^{m-1}a_0^{-1} a_1\\),\n  \\\\\n\\ge  \\(2^{m-1}a_0^{-1} a_1\\)^{-1}\n\\end{cases}\n\\end{align*}\nand\n\\begin{align*}\n  \\exp\\(2\\int^t_{t_\\xi} |r_m(s,\\xi)|\\,ds\\)\n\\le\\:&\\exp\\(C|\\xi|^{-m+1} \\int^t_{t_\\xi} \\Xi(s)^{-m}\\,ds \\)\n\\\\\n\\le\\:&\\exp\\(CN^{-m+1}\\Th(t_\\xi)^{m-1} \\int^\\infty_{t_\\xi} \\Xi(s)^{-m}\\,ds \\)\n\\simeq 1\n\\end{align*}\nby \\eqref{int_Cm}. \nIt follows that\n\\begin{equation}\\label{estVm}\n  \\|V_m(t,\\xi)\\|_{\\C^2}^2 \\simeq \\|V_m(t_\\xi,\\xi)\\|_{\\C^2}^2\n\\end{equation}\nin $Z_H$. \nTherefore, noting the estimates: \n\\begin{align*}\n  \\|V_k\\|_{\\C^2}^2\n =\\|M_k V_{k+1}\\|_{\\C^2}^2\n  \\le \\(1+|\\de_k|\\)^2 \\|V_{k+1}\\|_{\\C^2}^2 \n  \\le \\frac{3+2\\sqrt{2}}{2}\\|V_{k+1}\\|_{\\C^2}^2\n\\end{align*}\nand\n\\begin{align*}\n  \\|V_k\\|_{\\C^2}^2\n\\ge \\(1-|\\de_k|\\)^2 \\|V_{k+1}\\|_{\\C^2}^2 \n\\ge \\frac{3-2\\sqrt{2}}{2}\\|V_{k+1}\\|_{\\C^2}^2\n\\end{align*}\nfor $k=1,\\ldots,m-1$, it follows that \n\\begin{equation*\n  \\(\\frac{3 - 2\\sqrt{2}}{2}\\)^{m-1}\\|V_m\\|_{\\C^2}^2\n  \\le \\|V_1\\|_{\\C^2}^2\n  \\le \\(\\frac{3 + 2\\sqrt{2}}{2}\\)^{m-1}\\|V_m\\|_{\\C^2}^2,\n\\end{equation*}\nand \\eqref{cEV1}, we have \n\\begin{equation}\\label{estcE-ZH}\n  \\cE(t,\\xi) \\simeq \\|V_m(t,\\xi)\\|_{\\C^2}^2 \n  \\simeq \\|V_m(t_\\xi,\\xi)\\|_{\\C^2}^2\n  \\simeq \\cE(t_\\xi,\\xi).\n\\end{equation}\n\\end{proof}\n\nCombining the estimates \\eqref{estcE-ZP}, \\eqref{estcE-ZH} and \nusing Lemma \\ref{lemm-Parseval}, the proof of Theorem \\ref{Thm1} (iii) is concluded. \n\n\n\\section{Proof of Theorem \\ref{Thm2}}\n\nFor a large constant $\\tN$, we define $\\tilde{T}_0$ and $\\tt_\\xi$ for $\\xi \\in [-2,2]^d$ by \n\\[\n  \\tilde{T}_0:=\\max\\left\\{t\\ge 0\\;;\\; \\La(t) = \\frac{\\tN}{2\\sqrt{d}}\\right\\}\n  \\;\\text{ and }\\;\n  \\tt_\\xi:=\\max\\left\\{t\\ge \\tilde{T}\\;;\\; \\La(t)|\\xi| = \\tN\\right\\},\n\\]\nrespectively. \nThen we define the zones $\\tZ_H$ and $\\tZ_\\Psi$ by \n\\begin{equation*\n  \\tZ_\\Psi:=\\left\\{(t,\\xi)\\in  \\R_+\\times [-2,2]^d \\;;\\; t \\ge \\tt_\\xi \\right\\}\n\\end{equation*}\nand\n\\begin{equation*\n  \\tZ_H:=\\left\\{(t,\\xi)\\in  \\R_+\\times [-2,2]^d \\;;\\; t\\le \\tt_\\xi \\right\\},\n\\end{equation*}\nrespectively. \n\nLet $t\\le \\ttx$. \nBy the same way to derive the estimate \\eqref{estcE0-ZP} and using \\eqref{La-infty2}, we have \n\\begin{align*}\n  \\cE(t,\\xi)\n  \\lesssim\\:& \\exp\\(\\frac{2a_1}{a_0}\\Th\\(\\ttx\\)|\\xi|\\)\\cE(0,\\xi)\n\\\\\n  =\\: &\\exp\\(\\frac{2a_1 \\tN}{a_0}\\frac{|\\xi|}{\\tN}\n  \\Th\\(\\La^{-1}\\(\\frac{\\tN}{|\\xi|}\\)\\)\\)\n  \\cE(0,\\xi) \n\\\\\n  \\le\\: &\\exp\\(\\frac{2a_1 \\tN}{a_0} \\frac{|\\xi|}{N}\n  \\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\)\\)\n  \\cE(0,\\xi)\n\\\\\n  \\le\\: &\\exp\\(|\\xi|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\)\\)\n  \\cE(0,\\xi) \n\\end{align*}\nfor any $N \\ge 2a_1 \\tN \/ a_0$. \n\nLet $t\\ge \\ttx$. \nNoting that $\\ttx \\ge \\tx$ with $\\tN=N$, Lemma \\ref{ref_diag} is valid in $\\tZ_H$ for any large $\\tN$. \nTherefore, by (H6*) there exist positive constants $C$ and $N$ such that\n\\begin{align*}\n  \\int^t_{\\ttx} |r_m(s,\\xi)|\\,ds\n  \\le \\: &\n  C|\\xi|^{-m+1} \\int^\\infty_{\\ttx} \\Xi(s)^{-m}\\,ds\n\\\\\n = \\: & CN^{-m}  \n  \\La\\(\\ttx\\)^{m} \\Th(\\ttx)^{-1} \\int^\\infty_{\\ttx} \\Xi(s)^{-m}\\,ds\n  \\: |\\xi|\\Th(\\ttx) \n\\\\\n \\le \\: & C\n  N^{-m} \n  \\sup_{t\\ge 0}\\left\\{\n    \\La(t)^{m} \\Th(t)^{-1} \\int^\\infty_{t} \\Xi(s)^{-m}\\,ds\n  \\right\\}\n  |\\xi|\\Th\\(\\La^{-1}\\(\\frac{\\tN}{|\\xi|}\\)\\) \n\\\\\n \\le \\: & \n  |\\xi|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\). \n\\end{align*}\nBy \\eqref{cEV1} and the same way to derive the estimate \n\\eqref{estVm}, we have the following estimate:\n\\begin{equation*\n  \\cE(t,\\xi)\n  \\lesssim \\exp\\(|\\xi|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\)\\) \\cE(\\ttx,\\xi)\n\\end{equation*}\nfor any $t \\ge \\ttx$. \nCombining the estimates of $\\cE(t,\\xi)$ in $\\tZ_\\Psi$ and $\\tZ_H$, we have \n\\begin{align*}\n  \\cE(t,\\xi) \\lesssim \\exp\\(2|\\xi|\\Th\\(\\La^{-1}\\(\\frac{N}{|\\xi|}\\)\\)\\). \n\\end{align*}\n\\qed\n\n\\section{Proof of Theorem \\ref{Thm3}}\n\nThe following proposition is essential for the proof of Theorem \\ref{Thm3}. \n\n\\begin{proposition}\\label{Prop-Thm3}\nLet $\\{M_j\\}$ be a logarithmically convex sequence. \nIf $v=\\{v[k]\\}_{k\\in \\Z^d}$ satisfies \n\\begin{equation}\\label{eq1-Prop-Thm3}\n  \\sum_{k\\in \\Z^d} k^\\al v[k] = 0 \n\\end{equation}\nfor any $\\al\\in \\N_0^d$ and there exists a positive constant $\\rho$ such that \n\\begin{equation}\\label{eq2-Prop-Thm3}\n  \\sup_{k\\in \\N^d}\\left\\{\n   |v[k]| |k|^{d+1} T[\\{M_j\\}]\\(\\rho^{-1}|k|\\) \n  \\right\\} <\\infty, \n\\end{equation}\nthen the following estimate is established: \n\\begin{equation*}\n  \\sup_{\\th\\in \\T^d\\setminus\\{0\\}}\n  \\left\\{\n  \\left|\\hat{v}(\\th)\\right|T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\n  \\(\\frac{1}{d\\rho|\\th|}\\)\n  \\right\\}<\\infty.\n\\end{equation*}\n\\end{proposition}\n\nIn order to prove Proposition \\ref{Prop-Thm3}, we introduce the following lemma: \n\\begin{lemma}\\label{Lemma1-Thm3}\nIf $f\\in C^\\infty(\\T^d)$ satisfies \n$(\\pa_\\th^\\al f)(0)=0$ and \n\\begin{equation}\\label{eq-Lemma1-Thm3}\n  \\left|\\pa_\\th^\\al f(\\th)\\right| \\le C M_{|\\al|} \\rho^{|\\al|}\n\\end{equation}\nfor any $\\al \\in \\N_0^d$ with positive constants $C$, $\\rho$ and a logarithmically convex sequence $\\{M_j\\}$, then the following estimate is established: \n\\begin{equation*}\n  \\sup_{\\th\\in \\T^d\\setminus\\{0\\}}\\left\\{\n  T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\\(\\frac{1}{d \\rho|\\th|}\\)\n  |f(\\th)|\n  \\right\\}<\\infty.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\nFor any $\\al\\in \\N_0^\\al$, there exists $\\ze \\in \\T^d$ such that \n$f(\\th)=(\\pa_\\th^\\al f)(\\ze) \\th^\\al\/\\al!$ by Taylor's theorem. \nBy \\eqref{eq-Lemma1-Thm3} and the inequalities $|\\th^\\al|\\le|\\th|^{|\\al|}$ and \n$|\\al|! \\le d^{|\\al|}\\al!$ we have\n\\begin{align*}\n  |f(\\th)|\n  \\le C \\frac{M_{|\\al|}\\rho^{|\\al|}}{\\al!} \\left|\\th^\\al\\right|\n  \\le C \\frac{M_{|\\al|}}{|\\al|!} \\(d\\rho|\\th|\\)^{|\\al|}.\n\\end{align*}\nTherefore, we have \n\\begin{align*}\n  |f(\\th)| \\le C\\inf_{j\\in \\N_0}\\left\\{\n  \\frac{M_j}{j!}\\(d\\rho |\\th|\\)^j\\right\\}\n =\\frac{C}{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\n  \\(\\frac{1}{d \\rho|\\th|}\\)}.\n\\end{align*}\n\\end{proof}\n\n\\noindent\n{\\it Proof of Proposition \\ref{Prop-Thm3}.}\\; \nLet $\\al\\in \\N_0^d$. \nBy \\eqref{eq1-Prop-Thm3}, we have \n\\begin{align*}\n  \\pa_\\th^\\al\\hat{v}(\\th)|_{\\th=0}\n=\\sum_{k} (-ik)^\\al e^{-ik\\cd\\th} v[k]\\Bigr|_{\\th=0}\n=(-i)^{|\\al|}\\sum_{k} k^\\al v[k] = 0.\n\\end{align*}\nLet $\\al\\neq 0$. By \\eqref{eq2-Prop-Thm3}, there exists a positive constant $C$ such that\n\\begin{align*}\n  \\left|\\pa_\\th^\\al \\hat{v}(\\th)\\right|\n  \\le \\: & \\sum_{k} \\left|(-ik)^\\al e^{-ik\\cd\\th} v[k]\\right|\n  \\le C \\sum_{k\\in \\Z^d\\setminus\\{0\\}}\n    |k|^{-d-1+|\\al|}\\(T[\\{M_j\\}]\\(\\rho^{-1}|k|\\)\\)^{-1}\n\\\\\n  \\le \\: & C\\sum_{k\\in \\Z^d\\setminus\\{0\\}}\n  |k|^{-d-1+|\\al|}\\(\\frac{\\(\\rho^{-1}|k|\\)^{|\\al|}}{M_{|\\al|}}\\)^{-1}\n  \\simeq M_{|\\al|} \\rho^{|\\al|}.\n\\end{align*}\nTherefore, by Lemma \\ref{Lemma1-Thm3} we have\n\\begin{align*}\n  \\left|\\hat{v}(\\th)\\right|\n  \\lesssim \n  \\frac{1}{T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\n  \\(\\frac{1}{d \\rho|\\th|}\\)}. \n\\end{align*}\n\\qed\n\n\\noindent\n{\\it Proof of Theorem \\ref{Thm3} {\\rm (i)}.}\\; \nBy \\eqref{La-infty2}, \\eqref{eq1-Thm3}, $|\\th| \\le \\pi|\\xi(\\th)|\/2$, \nand applying Proposition \\ref{Prop-Thm3} with \n$v=D_j^+ u_0$ $(j=1,\\ldots,d)$ and $v=u_1$, we have \n\\begin{align*}\n  \\cE(0,\\th)\n\\simeq \\: &\n  \\left|\\hat{u}_1(\\th)\\right|^2\n +\\sum_{j=1}^d\\left|\\widehat{D_j^+ u_0}(\\th)\\right|^2\n\\lesssim\n  T\\left[\\left\\{\\frac{M_j}{j!}\\right\\}\\right]\n  \\(\\frac{1}{d \\rho|\\th|}\\)^{-2}\n\\\\\n\\le \\: &\n  L(N,\\{M_j\\})^{-2}\n  \\exp\\(-2d\\rho|\\th|\\Th\\(\\La^{-1}\\(\\frac{N}{d\\rho|\\th|}\\)\\)\\)\n\\\\\n = \\: &L(N,\\{M_j\\})^{-2}\\exp\\(-2N\\mu\\(\\frac{d\\rho|\\th|}{N}\\)\\)\n\\\\\n\\le \\: &\n  L(N,\\{M_j\\})^{-2}\\exp\\(-2N\\mu\\(\\frac{\\pi d\\rho|\\xi(\\th)|}{2N}\\)\\). \n\\end{align*}\nBy Theorem \\ref{Thm2} there exists a positive constant $\\tN_0 \\ge 1$ such that \n\\begin{equation}\\label{tcEtcE0_tN0}\n  \\cE(t,\\th) \\lesssim \n  \\exp\\(2\\tN_0\\mu\\(\\frac{|\\xi(\\th)|}{\\tN_0}\\)\\) \\cE(0,\\th). \n\\end{equation}\nIf \\eqref{eq1-Thm3} holds for any $N\\ge 1$, \nthen putting $N_0=\\max\\{\\tN_0,\\pi d\\rho \\tN_0\/2\\}$ and noting \n\\begin{align*}\n\\tN_0 \\mu\\(\\frac{|\\xi|}{\\tN_0}\\)-N_0\\mu\\(\\frac{\\pi d\\rho|\\xi|}{2N_0}\\)\n\\begin{cases}\n\\ds{=\\tN_0\\(1 -\\frac{\\pi d \\rho}{2}\\)\\mu\\(\\frac{|\\xi|}{\\tN_0}\\)} \n  \\le 0 & \\text{ for }\\; \\dfrac{\\pi d\\rho}{2} \\ge 1,\n\\\\[3ex]\n\\ds{  \\le\n  \\tN_0 \\mu\\(\\frac{|\\xi|}{\\tN_0}\\)-N_0\\mu\\(\\frac{|\\xi|}{N_0}\\)\n  =0}  &  \\text{ for }\\; \\dfrac{\\pi d\\rho}{2} < 1,\n\\end{cases}\n\\end{align*}\nwe have $\\cE(t,\\th) \\lesssim L(N_0,\\{M_j\\})^{-2}$. \nIt follows that \n\\[\n  U(N_0,u_0,u_1)\\lesssim (2\\pi)^d L(N_0,\\{M_j\\})^{-2} <\\infty, \n\\]\nand thus \\eqref{Ebdd} is established. \n\\qed\n\n\\noindent\n{\\it Proof of Theorem \\ref{Thm3} {\\rm (ii)}.}\\; \nBy the same way for the proof of (i), there exist positive constants $N_0\\ge 1$ and $N_1$ such that $L(N_1,\\{M_j\\})>0$ and \n\\begin{align*}\n  U(N_0,u_0,u_1)\n  \\lesssim \\: &\n  L(N_1,\\{M_j\\})^{-2}\n\\\\\n  & \\: \\times  \\int_{\\T^d}\n  \\exp\\(\n   2N_0\\mu\\(\\frac{|\\xi(\\th)|}{N_0}\\)\n  -2N_1\\mu\\(\\frac{\\pi d\\rho|\\xi(\\th)|}{2N_1}\\)\\)\\,d\\th. \n\\end{align*}\nBy \\eqref{eq4-Thm3} there exists a positive constant $N$ such that \n\\begin{equation*}\n  \\frac{\\mu\\(\\frac{1}{N}\\frac{|\\xi(\\th)|}{N_0}\\)}\n       {\\mu\\(\\frac{|\\xi(\\th)|}{N_0}\\)}\n =\\frac{\\Th\\(\\La^{-1}\\(N \\frac{N_0}{|\\xi(\\th)|}\\)\\)}\n       {N\\Th\\(\\La^{-1}\\(\\frac{N_0}{|\\xi(\\th)|}\\)\\)}\n \\ge \\frac{N_0}{N_1}. \n\\end{equation*}\nTherefore, putting $\\rho=2N_1\/(\\pi d N_0 N)$ we have \n\\begin{equation*}\n  N_0 \\mu\\(\\frac{|\\xi(\\th)|}{N_0}\\)\n  \\le\n  N_1 \\mu\\(\\frac{\\pi d \\rho|\\xi(\\th)|}{2N_1}\\),\n\\end{equation*}\nit follows that $U(N_0,u_0,u_1)<\\infty$, and thus \\eqref{Ebdd} is established. \n\\qed\n\n\n\\section{Appendix}\n\\subsection{Proof of Lemma \\ref{TDFT}}\nSince $f \\in l^1(\\Z^d)$, we have\n\\begin{align*}\n  \\sum_k e^{-\\i k\\cd \\th} f[k \\pm e_j]\n=e^{\\pm \\i e_j \\cd \\th} \\sum_k e^{-\\i(k \\pm e_j)\\cd \\th} f[k \\pm e_j]\n   = e^{\\pm \\i\\th_j} \\sum_k e^{-\\i k\\cd \\th}f[k]. \n\\end{align*}\nHence we have \n\\begin{align*}\n  \\cF_{\\Z^d}[D_j^+ f](\\th)\n =\\sum_{k} e^{-\\i k\\cd\\th}\\(f[k+e_j]-f[k]\\)\n =\\(e^{\\i\\th_j} - 1\\) \\cF_{\\Z^d}[f](\\th),\n\\end{align*}\nthat is, \\eqref{TDFT2} is valid. \nSimilarly, we have \n\\begin{align*}\n  \\cF_{\\Z^d}[D_j^- f](\\th)\n =\\(1-e^{-\\i\\th_j}\\) \\cF_{\\Z^d}[f](\\th).\n\\end{align*}\nTherefore, we have \\eqref{TDFT3} as follows:\n\\begin{align*}\n \\cF_{\\Z^d}[D_j^+ D_j^- f](\\th)\n= \\: &\\(e^{\\i\\th_j} - 1\\) \\cF_{\\Z^d}[D_j^- f](\\th) \n=\\(e^{\\i\\th_j} - 1\\)\\(1-e^{-\\i\\th_j}\\) \\cF_{\\Z^d}[f](\\th) \n\\\\\n= \\: &\\(e^{\\i\\frac{\\th_j}{2}}-e^{-\\i\\frac{\\th_j}{2}}\\)^2 \\hat{f}(\\th)\n =-4\\(\\sin\\frac{\\th_j}{2}\\)^2 \\hat{f}(\\th). \n\\end{align*}\n\\qed\n\n\\subsection{Proof of Lemma \\ref{lemm-Parseval}}\nFrom the definition of $\\hat{f}$ we have \n\\begin{align*}\n  \\int_{\\T^d}|\\hat{f}(\\th)|^2\\,d\\th\n= \\: &\\int_{\\T^d}\n  \\(\\sum_{k} e^{-\\i k\\cd\\th} f[k]\\)\n  \\ol{\\(\\sum_{l} e^{-\\i l\\cd\\th} f[l]\\)}\\,d\\th\n\\\\\n= \\: & \\sum_{k,l} f[k]\\ol{f[l]} \\int_{\\T}  e^{-\\i(k-l)\\cd\\th} \\,d\\th \n\\\\\n= \\: &(2\\pi)^d \\sum_{k,l} \\de_{kl} f[k] \\ol{f[l]}\n=(2\\pi)^d \\sum_{k} |f[k]|^2, \n\\end{align*}\nwhere $\\de_{kl}$ denotes the Kronecker delta. \n\\qed\n\n\\subsection{Proof of Lemma \\ref{lemm-E-cE}}\nBy Lemma \\ref{TDFT} and Lemma \\ref{lemm-Parseval}, we have \n\\[\n  (2\\pi)^d\\sum_{k\\in\\Z^d}|u'(t)[k]|^2\n =\\int_{\\T^d}\\left|\\cF_{\\Z^d}[u'(t)](\\th)\\right|^2\\,d\\th\n =\\int_{\\T^d}\\left|\\pa_t \\hat{u}(t,\\th)\\right|^2\\,d\\th,\n\\]\nand\n\\begin{align*}\n  (2\\pi)^d\\sum_{j=1}^d \\sum_{k\\in\\Z^d}|D_j^+ u(t)[k]|^2\n= \\: &\\sum_{j=1}^d \n  \\int_{\\T^d} \\left| \\cF_{\\Z^d}[D_j^+ u(t)](\\th)\\right|^2\\,d\\th\n\\\\\n= \\: &\\sum_{j=1}^d \n  \\int_{\\T^d} \\left| \\(e^{\\i\\th_j}-1\\)\\hat{u}(t,\\th)\\right|^2\\,d\\th\n\\\\\n= \\: &\\int_{\\T^d} \\sum_{j=1}^d \\xi_j(\\th)^2|\\hat{u}(t,\\th)|^2\\,d\\th.\n\\end{align*}\nThus we have \\eqref{EcE}. \n\\qed\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nUnderstanding the structure of wave functions in Fock space is a hot topic in many-body physics, intimately related to the concepts of ergodicity and thermalization in complex interacting systems.\nThe natural way to think of the interaction is as a cause of transitions between single-particle states, which are no longer eigenstates of a quantum system in the presence of interaction.\nSinge-particle excitations decay into three-particle states (two electrons and one hole), which then further decay into five-particle states, etc. The simplest way to characterize the process of quantum-mechanical spreading of an excitation in Fock space is to study its energy relaxation rate (inverse lifetime).\n\n\nThe study of the inelastic relaxation in a confined geometry has been pioneered by Sivan, Imry and Aronov (SIA) \\cite{SIA}, who calculated the quasiparticle lifetime in a diffusive quantum dot with chaotic electron dynamics.\nWorking within the conventional Fermi golden rule (FGR) picture, they calculated the relaxation rate of an excitation with the energy $\\varepsilon$, induced by the screened Coulomb interaction with the small momentum transfer:\n\\begin{equation}\n  \\gamma_0(\\varepsilon)\n  \\sim\n  \\lambda^2 \\Delta (\\varepsilon\/E_{\\text{Th}})^2 ,\n\\label{SIA}\n\\end{equation}\nwhere $\\Delta$ is the mean single-particle level spacing in the dot, and $E_{\\text{Th}}= D\/L^2$ is the Thouless energy determined by the inverse time of electron diffusion across the system ($D$ is the diffusion coefficient, and $L$ is the typical size of the quantum dot).\nFor generality, in Eq.~(\\ref{SIA}) we introduced the dimensionless interaction strength $\\lambda$, taking its maximal value, $\\lambda=1$, for the screened Coulomb potential.\nThe result (\\ref{SIA}) was derived in the hot-electron regime (negligible temperature, $T\\ll\\varepsilon$) under the assumption of the zero-dimensional geometry, $\\varepsilon \\ll E_{\\text{Th}}$.\nIt provides the diffusive contribution to the relaxation rate originating from the processes with momentum transfer of the order of the inverse system size, $1\/L$. The total relaxation rate is then the sum of $\\gamma_0(\\varepsilon)$ given by Eq.~(\\ref{SIA}) and the Fermi-liquid contribution due to the processes with large-momentum transfer, $\\gamma_*(\\varepsilon)\\sim \\varepsilon^2\/E_F$ ($E_F$ is the Fermi energy) \\cite{FL}. The latter can be neglected for sufficiently large quantum dots, $L\\gg k_Fl^2$ ($k_F$ is the Fermi momentum, and $l$ is the mean free path) \\cite{SIA,Blanter}, that will be assumed thereafter.\n\nRemarkably, Eq.~(\\ref{SIA}) derived in the zero-dimensional limit, $\\varepsilon \\ll E_{\\text{Th}}$, shows that in this regime the single-particle spectrum is well resolved, $\\gamma_0(\\varepsilon)\\ll\\Delta$. Though this conclusion is in a good agreement with the experimental results on tunneling spectroscopy of a disordered quantum dot~\\cite{SMM}, it raises the question of consistency of the derivation.\nIndeed, the FGR can be safely applied only in the case of continuous spectrum, while the result (\\ref{SIA}) implies that it is not. This subtle point was recognized already by SIA, who argued that their approach might be correct as summation over many final states is performed.\nThis idea was elaborated in a seminal paper by Altshuler, Gefen, Kamenev, and Levitov (AGKL) \\cite{AGKL}, who emphasized that it is the spectrum of final states that should be continuous for the FGR picture to be applicable. In the problem of the hot-electron decay, final states are three-particle states with two electrons and one hole, and the corresponding mean level spacing can be estimated as\n\\begin{equation}\n  \\Delta_3(\\varepsilon) \\sim \\Delta^3\/\\varepsilon^2\n  .\n\\label{Delta3}\n\\end{equation}\nComparing $\\gamma_0(\\varepsilon)$ and $\\Delta_3(\\varepsilon)$, AGKL came to the conclusion that the FGR description should be valid for sufficiently large energies, $\\varepsilon>\\varepsilon_\\text{FGR}$, where\n\\begin{equation}\n\\varepsilon_\\text{FGR}=\\Delta\\sqrt g,\n\\label{eps*}\n\\end{equation}\nand $g=E_{\\text{Th}}\/\\lambda\\Delta\\gg1$ sets the scale, $\\Delta\/g$, of the interaction matrix elements (for the screened Coulomb interaction with $\\lambda=1$, $g$ coincides with the dimensionless conductance of the dot).\n\nIn order to study the spreading of a single-particle excitation over the space of many-particle states beyond the FGR approximation,\nAGKL proposed a remarkable mapping of the initial problem to a tight-binding Anderson model on a hierarchical lattice, treating each site as a basis vector in the many-particle Fock space. Motivated by the observation that the structure of this lattice for the quantum dot problem locally looks like a tree, AGKL approximated it by the Bethe lattice with a large branching number. Then using the known solution for the Anderson model on the Bethe lattice \\cite{Cayley}, AGKL predicted the localization transition in Fock space of an interacting quantum dot at the energy $\\varepsilon_\\text{MBL}\\sim\\Delta\\sqrt{g\/\\ln g}$, which appeared to be parametrically smaller than the FGR breaking energy scale $\\varepsilon_\\text{FGR}$. Thus, in the AGKL picture of many-body localization one should distinguish between three regimes \\cite{AGKL,MF97}.\nIn the localized regime realized at $\\varepsilon<\\varepsilon_\\text{MBL}$, the Slater determinants of one-particle states are very close to the exact many-body states, and the width of quasiparticle states is exactly zero.\nIn the intermediate region, $\\varepsilon_\\text{MBL}<\\varepsilon<\\varepsilon_\\text{FGR}$, quasiparticle states are delocalized, but they are strongly fractal and non-ergodic, with the spectral weight given by a number of slightly broadened lines.\nThe spectral weight acquires a Lorentzian form with a well defined width $\\gamma$ (corresponding to a simple exponential decay $e^{-\\gamma t}$ in the time domain) only in the regime $\\varepsilon\\gg\\varepsilon_\\text{FGR}$, where the FGR description finally sets in.\n\nThe AGKL paper has triggered a boost of activity in the field of many-body localization. The suggested mapping onto the lattice model has been proved to be extremely fruitful: instead of studying an interacting problem one now can deal with a non-interacting quantum mechanics, but on a very complicated lattice with an exponentially large number of sites.\n(Though all states in a finite system are localized by definition, even an extremely weak coupling to the external reservoir providing a level width larger than the exponentially small distance between the many-body states, which can be estimated as $\\exp(-\\alpha\\sqrt{\\varepsilon\/\\Delta})$ with $\\alpha\\sim1$ \\cite{Silv,Bethe1936,BohrMottelson},\nrenders the spectrum of delocalized states effectively continuous.)\nSince the complexity of the problem is encoded in the lattice topology, the crucial point is\nto identify the relevant features responsible for many-body localization.\n\nQuite soon it was recognized that the Bethe lattice with a constant branching number is an oversimplified model of Fock space.\nFirstly, the coordination number of the lattice decreases with the number of generations \\cite{Silv,Jacquod97}. Secondly, the actual lattice is not a tree, and the presence of loops essentially modifies combinatorics of the perturbative expansion in the localized region \\cite{Silv}, increasing the AGKL estimate for $\\varepsilon_\\text{MBL}$.\nNumerical studies \\cite{LTB98,Kamenev2002} demonstrated gradual delocalization with the growth of the quasiparticle energy.\nAnyway, despite the lack of a rigorous theory of many-body localization in a quantum dot, the general understanding achieved in the beginning of 2000s was that in finite systems one should expect a localization\/delocalization crossover, though its position was not firmly established.\n\n\nLater, the concept of many-body localization developed in the quantum dot problem\nwas applied to extended systems of interacting electrons with spatially localized single-particle states \\cite{BAA,GMP}, where the transition between the localized and delocalized many-body phases occurs at a finite temperature.\nThe related issues of ergodicity and thermalization are currently being actively investigated (for a review, see Ref.~\\cite{Polkovnikov}).\n\nVery recently, many-body localization in a quantum dot was reconsidered by Gornyi, Mirlin, and Polyakov \\cite{Mirlin2015}. Working in the framework of the lattice model, they found an additional factorial contribution in the perturbative series in the number of involved generations, which renders coupling to distant generations less efficient, thus acting in favor of localization.\nFor the hot-electron decay problem, this leads to the estimate for the threshold energy $\\varepsilon_\\text{MBL}\\sim g\\Delta\/\\ln g$, which is much larger than the original AGKL estimate.\nFor a thermal many-body state with the temperature $T$ and the total energy ${\\cal E}\\sim T^2\/\\Delta$, the FGR temperature $T_\\text{FGR}\\sim \\Delta\\sqrt{g}$ was shown to be logarithmically larger than the localization transition point, $T_\\text{MBL}\\sim\\Delta\\sqrt{g\/\\ln g}$. This result formally  coincides with the AGKL \\emph{energy}\\ threshold and is in full agreement with Ref.~\\cite{BAA}.\n\nSince, contrary to AGKL, Refs.~\\cite{Silv,Mirlin2015} place the many-body localization threshold $\\varepsilon_\\text{MBL}$ for the hot-electron problem above the FGR-breaking scale $\\varepsilon_\\text{FGR}$,\none can ask how to reconcile the FGR description with many-body localization. This question was answered by Silvestrov \\cite{Silvestrov2001} who studied the temporal decay of a quasiparticle with $\\varepsilon\\gg\\varepsilon_\\text{FGR}$. He showed that the FGR relaxation rate $\\gamma_0(\\varepsilon)$ describes the initial stage of exponential relaxation, that slows down at larger times due to smaller relaxation rate of descendant states, and eventually due to many-body localization.\n\nIn this paper, we address the initial temporal stage of energy relaxation in a quantum dot, when quantum many-body localization effects are not yet visible. Assuming the FGR approach is applicable, we calculate mesoscopic fluctuations of the energy relaxation rate. The smallness of fluctuations compared to the average relaxation rate provides an \\emph{a posteriori}\\ condition for the FGR applicability. In our analysis, we do not use the lattice model of Fock space and work in terms of Keldysh diagram technique in real space, where non-perturbative disorder averaging is performed by means of the non-linear supersymmetric sigma model. Such an approach allows us to derive the expression for the variance of the relaxation rate without any simplifications concerning the nature of the electron-electron interaction.\n\nThe paper is organized as follows. In Sec.~\\ref{S:Model} we introduce the model and summarize the results. Section~\\ref{S:General} describes the Keldysh kinetic approach\nalong with its modification simplifying the study of weak nonequilibrium.\nIn Sec.~\\ref{S:Average} we rederive the SIA result and generalize it to the case of an arbitrary temperature and interaction radius. Section~\\ref{S:MF} is devoted to the discussion of the general strategy for the calculation of mesoscopic fluctuations of the energy relaxation rate in terms of the exact (disorder-dependent) electron Green functions. The product of several Green functions is averaged non-perturbatively in Sec.~\\ref{S:nonpert}. The final step of the calculation of mesoscopic fluctuations is performed, for not very short-range interaction, in Sec.~\\ref{S:final} and, in the general case, in Sec.~\\ref{S:mesofluct-rs}. Our results are summarized in Sec.~\\ref{Discussion}. Numerous technical details are relegated to several Appendices.\n\nWe use the system of units with $\\hbar=k_B=1$.\n\n\n\n\n\n\n\n\\section{Model description and results}\n\\label{S:Model}\n\n\\subsection{Model}\n\nWe consider an isolated chaotic quantum dot with the broken time-reversal symmetry (unitary symmetry class).\nImpurity scattering in the dot is supposed to be strong enough to completely\nrandomize the electron trajectories establishing the diffusive regime with\n$l\\ll L$, where $l$ is the elastic mean free path,\nand $L$ is the characteristic size of the dot.\nFor generality, we consider the case of an arbitrary spin degeneracy, $N_s$, which plays the role of the number of independent fermionic flavors (the case $N_s=1$ corresponds to completely spin-polarized electrons).\n\nWe will discuss two models of electron-electron interactions: (i) the long-range Coulomb interaction with a small gas parameter $r_s \\equiv e^2\/\\epsilon v_F \\ll 1$ (where $\\epsilon$ is the dielectric constant of the medium, and $v_F$ is the Fermi velocity), and (ii) an arbitrary weak interaction. In the case of the Coulomb interaction, its screening should be taken into account, whereas for a weak interaction this procedure is unnecessary.\nIn both cases, the statically screened interaction in the real space can be written in the form\n\\begin{equation}\n\\label{V(r)}\n  V({\\bf r})\n  =\n  \\frac{\\lambda}{N_s\\nu} \\delta_\\kappa({\\bf r}) ,\n\\end{equation}\nwhere $\\lambda$ is the dimensionless interaction strength,\nand $\\delta_\\kappa({\\bf r})$ is the delta function smeared on the scale of the interaction radius $1\/\\kappa$\n[the latter is assumed to be much smaller than the size of the quantum dot,\njustifying the notion of the smeared delta function in Eq.~(\\ref{V(r)})].\nFor the long-range Coulomb interaction, $\\lambda=1$ and $1\/\\kappa$ is the Thomas-Fermi screening length,\nsee Eq.~(\\ref{kappa}).\nBesides the dimensionless strength $\\lambda$, the only characteristic of the potential (\\ref{V(r)}) that will be important in the following is the ratio, $F$, of the Hartree and Fock diagrams, which is a measure of the interaction range \\cite{Altshulerbook,Akkermans}. In the three- and two-dimensional cases it is given by (see \\ref{S:Non-RPA} for details)\n\\begin{equation}\n\\label{f-gen}\n  F\n  =\n  \\int_0^{1} v_\\kappa(2p_Fx) \\varphi(x) \\, dx ,\n\\qquad\n  \\varphi(x)\n  =\n  \\begin{cases}\n    2 x, & \\text{in 3D}, \\\\\n    (2\/\\pi) (1-x^2)^{-1\/2}, & \\text{in 2D}, \\\\\n  \\end{cases}\n\\end{equation}\nwhere $v_\\kappa({\\bf q})$ is the Fourier transform of $\\delta_\\kappa({\\bf r})$ in Eq.~(\\ref{V(r)}).\nThe limiting cases of $F=0$ and $F=1$ correspond to long-range ($\\kappa\\ll k_F$) and point-like screened interactions, respectively.\n\nWe will assume that interaction can be taken into account perturbatively, that can be justified in the two partially overlapping limits:\n\n\\begin{equation}\n\\label{lambda-f}\n  \\text{$\\lambda\\ll1$ or $F\\ll1$} ,\n\\end{equation}\ni.e., in the limit of weak interaction or for sufficiently large interaction radius ($\\kappa\\ll k_F$).\nThe two models of electron-electron interaction discussed above conform to the condition (\\ref{lambda-f}):\nFor the Coulomb interaction with $\\lambda=1$, its applicability is provided by the smallness of the gas parameter $r_s$ since $F\\sim r_s\\ln 1\/r_s$ [see Eq.~(\\ref{f-Coulomb})]; for weak interaction with an arbitrary radius $1\/\\kappa$, it is justified as long as $\\lambda\\ll1$.\n\n\n\n\\subsection{Average energy relaxation rate}\n\nWe start with generalizing the SIA result (\\ref{SIA}) to the case of a finite temperature $T$, arbitrary spin degeneracy $N_s$ and arbitrary interaction radius measured by the parameter $F$ [assuming that the restriction (\\ref{lambda-f}) holds].\nThe temperature and excitation energy are supposed to be smaller than the Thouless energy, $\\max \\{ \\varepsilon, T \\}\\ll E_{\\text{Th}}$.\nThe average energy relaxation rate calculated in the second order in the statically screened interaction (\\ref{V(r)}) is given by\n\\begin{equation}\n  \\gamma_0(\\varepsilon,T)\n  =\n  \\frac{\\lambda^2c(N_s,F)}{\\pi}\n  \\frac{\\Delta}{E_2^2} (\\varepsilon^2+\\pi^2 T^2) ,\n\\label{SIAT}\n\\end{equation}\nwhere $\\Delta=(\\nu V)^{-1}$ is the mean level spacing in the dot\n($\\nu$ is the single-particle density of states at the Fermi level per one spin projection, and $V$ is volume of the dot), and the energy scale $E_2$ is determined by the diffusion inside the dot:\n\\begin{equation}\n\\label{En}\n  E_n\\equiv\\biggl[\\mathop{{\\sum}'}_m \\frac{1}{(Dq^2_m)^n}\\biggr]^{-1\/n} \\sim E_{\\text{Th}}\n  .\n\\end{equation}\nHere $D$ is the diffusion coefficient, and $q^2_m$ are non-zero eigenvalues\nof the Laplace operator in the dot, $- \\nabla^2 \\psi_m({\\bf r}) = q_m^2 \\psi_m({\\bf r})$,\nwith the von Neumann boundary conditions.\nThe magnitude of $E_n$ defined by Eq.~(\\ref{En}) is set by the Thouless energy,\n$E_{\\text{Th}}=D\/L^2$, where $L$ is the typical size of the quantum dot.\n\nThe factor $c(N_s,F)$ in Eq.~(\\ref{SIAT}) takes into account spin degeneracy and the spatial structure of the interaction potential (\\ref{V(r)}):\n\\begin{equation}\n\\label{c-res}\n  c(N_s,F)\n  =\n  \\frac{1}{N_s^2} \\left(N_s - 2F + N_s F^2 \\right) .\n\\end{equation}\nThe function $c(N_s,F)$ has an important property $c(1,1)=0$, which ensures vanishing of interaction effects for polarized ($N_s=1$) fermions with a contact interaction,\nas a consequence of the Pauli exclusion principle. It should be noted\nthat the anticipated cancellation of $c(1,1)$ takes place only if the Hartree-type diagrams yielding the terms with $F$ in Eq.~(\\ref{c-res}) are taken into account (see Sec.~\\ref{SS:rs}). This issue is often overlooked in literature, leading to a variety of claimed prefactors in Eq.~(\\ref{SIA}).\n\n\n\n\\subsection{Mesoscopic fluctuations of the energy relaxation rate}\n\nIn the FGR description, the relaxation rate $\\gamma_i$ of a given single-particle state $|i\\rangle$ is a function of its energy only, implying that the states close in energy should have nearly the same inelastic width.\nThe validity of this approximation can be verified \\emph{a posteriori}\\\nby studying mesoscopic, i.~e., level-to-level fluctuations of $\\gamma_i$.\nWe calculate them under the same assumptions of the zero-dimensional geometry,\n$\\max \\{ \\varepsilon, T \\}\\ll E_{\\text{Th}}$, used for the determination of the average $\\gamma_0(\\varepsilon,T)$.\n\nWe describe the relaxation rate in the framework of the quantum kinetic equation for electrons in dirty metals~\\cite{RS}. This approach differs significantly from the lattice model proposed by AGKL \\cite{AGKL} and extensively used in subsequent publications \\cite{MF97,Silv,Jacquod97,LTB98,Mirlin2015}.\nThe method of kinetic equation is not suitable for studying many-body localization, but is sufficiently simple to unveil the limits of applicability of the FGR description.\nNevertheless, the kinetic approach, working well for interacting systems with continuous spectrum, meets significant difficulties in the considered region $\\max\\{\\varepsilon,T\\}\\llE_{\\text{Th}}$, where individual energy levels are well-resolved, $\\gamma_{0}(\\varepsilon,T)\\ll \\Delta$.\nDiscreteness of energy spectrum requires non-perturbative averaging over disorder, which is performed by means of the non-linear supersymmetric sigma model \\cite{Efetov-book}.\n\nThe main idea behind our calculations is the following. In the zeroth approximation (which is equivalent to the FGR), energy levels acquire an inelastic width $\\gamma_0(\\varepsilon,T)$. The inverse of this quantity yields the temporal scale\nat which single-particle coherence is maintained.\nThis corresponds to the appearance of the `mass' of the order $\\gamma_{0}(\\varepsilon,T)$ for the zero-dimensional diffusons (diffusons with zero momentum).\nAt the next stage, when loop corrections to the FGR result are considered, this `mass' will regularize the otherwise divergent contributions, producing $\\gamma_0(\\varepsilon,T)$ in the denominator, precisely in the way it occurs in the case of the non-interacting quantum dot in an external field \\cite{dyn-loc,Kubo,Kubo4} and in the high-temperature phase in the problem of many-body localization \\cite{BAA}. Thus, with decreasing the energy of excitations and\/or temperature, the loop corrections to the quasiclassical rate increase, and at some energy scale they become comparable, indicating the breakdown of the FGR description.\n\n\nFluctuations of the energy relaxation rate $\\gamma(\\varepsilon,T)$ around its FGR average value $\\gamma_0(\\varepsilon,T)$ given by Eq.~(\\ref{SIAT}) are characterized by the irreducible average:\n\\begin{equation}\n\\label{corr-irr-def}\n  \\corr{\\gamma^2(\\varepsilon,T)}=\\gamma_0^2(\\varepsilon,T) + \\ccorr{\\gamma^2(\\varepsilon,T)} .\n\\end{equation}\nWe calculate the leading contribution to $\\ccorr{\\gamma^2(\\varepsilon, T)}$ in the delocalized regime and obtain\n\\begin{equation}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}\n  =\n  \\frac{\\lambda^4\\Delta^5}{4\\pi^3}\n  \\left[\\frac{c_2(N_s,F)}{E_2^4}+\\frac{c_4(N_s,F)}{E_4^4}\\right]\n  \\int d\\varepsilon_1 d\\omega_1\n  \\frac{({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega_1})^2({\\cal B}_{\\omega_1}+{\\cal F}_{\\varepsilon-\\omega_1})^2}\n  {\\gamma_0(\\varepsilon-\\omega_1,T)+\\gamma_0(\\varepsilon_1,T)+\\gamma_0(\\varepsilon_1-\\omega_1,T)}\n  ,\n\\label{subfinal}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{FandB}\n  {\\cal F}_{\\varepsilon}=\\tanh \\frac{\\varepsilon}{2T},\n  \\qquad\n  {\\cal B}_{\\omega}=\\coth\\frac{\\omega}{2T}\n\\end{equation}\nare equilibrium fermionic and bosonic distribution functions, respectively, while the energies $E_2$ and $E_4$ (both of the order of $E_{\\text{Th}}$) are defined in Eq.~(\\ref{En}). The functions $c_2(N_s,F)$ and $c_4(N_s,F)$ in Eq.~(\\ref{subfinal}) are given by\n\\begin{subequations}\n\\label{c2c4}\n\\begin{gather}\n\\label{c2}\n  c_2(N_s,F)\n  =\n  \\frac{1}{N_s^4}\n  \\left[\n    (3N_s^2+1) - 16N_s F + 2(5N_s^2+7) F^2\n  - 16N_s F^3 + (3N_s^2+1) F^4\n  \\right]\n,\n\\\\\n\\label{c4}\n  c_4(N_s,F)\n  =\n  \\frac{1}{N_s^4}\n  \\left[\n    2N_s^2 - 8N_s F + 4(N_s^2+2) F^2\n  - 8N_s F^3 + 2N_s^2 F^4\n  \\right] .\n\\end{gather}\n\\end{subequations}\nThey obey an important relation $c_2(1,1)=c_4(1,1)=0$, which guarantees vanishing of mesoscopic fluctuations [as well as the inelastic width itself, see Eq.~(\\ref{c-res})] for spin-polarized fermions with a contact interaction.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=.55\\textwidth]{upsilon.pdf}}\n\\caption{The function $\\Upsilon(x)$ [Eq.~(\\ref{Upsilon-def})], which determines the energy and temperature dependence of $\\ccorr{\\gamma^2(\\varepsilon,T)}$, see Eq.~(\\ref{gamma-Upsilon}).}\n\\label{F:Upsilon}\n\\end{figure}\n\nTo single out the energy and temperature dependence of Eq.~(\\ref{subfinal}), we rewrite it in the form\n\\begin{equation}\n\\label{gamma-Upsilon}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}\n  =\n  \\frac{\\lambda^2 \\Delta^4}{4\\pi^2} \\frac{E_2^2}{c(N_s,F)} \\left[\\frac{c_2(N_s,F)}{E_2^4}+\\frac{c_4(N_s,F)}{E_4^4}\\right]\n  \\Upsilon \\left( \\frac{\\varepsilon}{2T} \\right),\n\\end{equation}\nwhere the dimensionless function $\\Upsilon(x)$ plotted in Fig.~\\ref{F:Upsilon} is defined as\n\\begin{equation}\n\\label{Upsilon-def}\n  \\Upsilon(x)\n  =\n  \\int dy \\, dz \\,\n  \\frac{[\\tanh(y)-\\tanh (y-z)]^2[\\coth(z)-\\tanh(z-x)]^2}{(x-z)^2+y^2+(y-z)^2 + 3\\pi^2\/4}\n  =\n  \\begin{cases}\n    0.248 , & x = 0 ; \\\\\n    16.95 , & x = \\infty .\n  \\end{cases}\n\\end{equation}\nHence, the magnitude of mesoscopic fluctuations of the relaxation rate can be roughly estimated as\n\\begin{equation}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}\n  \\sim\n  \\lambda^2\n  \\frac {\\Delta^4}{E_{\\text{Th}}^2} .\n\\label{qualresult1}\n\\end{equation}\nNote however that, due to a huge variation of the function $\\Upsilon(x)$, fluctuations at the Fermi energy ($\\varepsilon\\ll T$) are nearly 100 times smaller than the naive estimate (\\ref{qualresult1}).\n\n\nExpression (\\ref{subfinal}) for mesoscopic fluctuations of the energy relaxation rate in the FGR regime is the main result of our work.\nThe relative strength of mesoscopic fluctuations can be estimated as\n\\begin{equation}\n  \\frac{\\ccorr{ \\gamma^2(\\varepsilon,T)}}{\\gamma_{0}^2(\\varepsilon,T)}\n  \\sim\n  \\frac{\\Delta_3(\\max\\{\\varepsilon,T\\})}{\\gamma_{0}(\\varepsilon,T)}\n  \\sim\n  \\left( \\frac{\\varepsilon_\\text{FGR}}{\\max\\{ \\varepsilon, T\\}} \\right)^4 ,\n\\label{qualresult2}\n\\end{equation}\nwhere $\\Delta_3(\\varepsilon)$ is the mean three-particle level spacing defined in Eq.~(\\ref{Delta3}). The ratio (\\ref{qualresult2}) becomes of the order of unity as $\\max\\{\\varepsilon,T\\}$ approaches the FGR-breaking scale $\\varepsilon_\\text{FGR} = \\Delta\\sqrt{g}$. Hence for the validity of the FGR description of the initial stage of quasiparticle disintegration, the temperature $T$ of electrons in the dot and the excitation energy $\\varepsilon$ at which the decay rate is studied play nearly the same role. They will act differently at larger time scales, when many-body localization may have enough time to show up \\cite{Mirlin2015,Silvestrov2001}.\nPhysically, at $\\max\\{\\varepsilon,T\\}\\gg\\varepsilon_\\text{FGR}$ the state has many available routes to decay into three-particle excitations. Fluctuations of the number of such routes are small and are determined by Eq.~(\\ref{qualresult2}).\n\n\n\n\n\n\\section{General expression for the energy relaxation rate}\n\\label{S:General}\n\n\\subsection{Keldysh technique}\n\\label{SS:Keldysh}\n\nIn order to express the inelastic energy relaxation rate, we use the Keldysh technique \\cite{Keldysh} in the representation of the functional integral \\cite{KA99,KL09}. For an interacting system, the Keldysh action\nis a functional of the fermionic Grassmann fields $\\psi_\\sigma({\\bf r},t)$\n($\\sigma=1,\\dots,N_s$ counts spin projections)\nand the bosonic plasmon field $\\phi({\\bf r},t)$:\n\\begin{equation}\n\\label{action1}\n  S =\n  \\int dt\\left\\{\\sum_{\\sigma=1}^{N_s} \\int d{\\bf r} \\, \\psi^+_{\\sigma}({\\bf r},t) \\left( \\hat G_0^{-1} + \\phi_a({\\bf r},t) \\gamma^a \\right) \\psi_{\\sigma}({\\bf r},t)\n+ \\int(d{\\bf q}) V_0^{-1}({\\bf q}) \\, \\phi^T({\\bf q},t)\\gamma_2\\phi(-{\\bf q},t)\\right\\}.\n\\end{equation}\nThe fields $\\psi_\\sigma({\\bf r},t)$ and $\\phi({\\bf r},t)$ are two-component vectors in the Keldysh space.\nIn the Keldysh-rotated basis \\cite{KA99,KL09},\n\\begin{equation}\n\\hat G_0^{-1}=\\left(i\\frac{\\partial}{\\partial t}+\\frac{\\nabla^2}{2m}-U_{\\text{dis}}({\\bf r})\\right)\\gamma_1\\equiv G_0^{-1}\\gamma_1,\n\\qquad\n\\gamma^1=\\begin{pmatrix} 1&0\\\\0&1 \\end{pmatrix},\n\\qquad\n\\gamma^2=\\begin{pmatrix} 0&1\\\\1&0 \\end{pmatrix},\n\\end{equation}\nwhere $U_{\\text{dis}}({\\bf r})$ is the disorder potential\nwith the correlation function $\\corr{U_{\\text{dis}}({\\bf r})U_{\\text{dis}}({\\bf r}')} = \\delta({\\bf r}-{\\bf r}')\/(2\\pi\\nu\\tau)$\n($\\tau$~is the elastic mean free time, and $\\nu$ is the density of states per one spin projection\nat the Fermi energy). We assume no spin-dependent interactions so that all matrices\nact as a unit matrix in the spin space.\nThe second term in Eq.~(\\ref{action1}) is the action of the plasmon field,\nwith $V_0({\\bf q})$ being the bare interaction potential,\nand $(d{\\bf q})\\equiv d^dq\/(2\\pi)^d$ is the momentum integration measure in $d$ dimensions.\n\nThe Green functions are defined as ($\\xi_i$ denotes the pair ${\\bf r}_i,t_i$):\n\\begin{gather}\n\\hat G(\\xi_1,\\xi_2)=-i\\langle\\psi(\\xi_1)\\psi^+(\\xi_2)\\rangle\n=-i\\int D\\psi^{*}D\\psi D\\phi \\, e^{iS} \\, \\psi(\\xi_1)\\psi^+(\\xi_2),\n\\\\\n\\hat V(\\xi_1,\\xi_2)=-2i\\langle\\phi(\\xi_1)\\phi^T(\\xi_2)\\rangle\n=-2i\\int D\\psi^{*}D\\psi D\\phi \\, e^{iS} \\, \\phi(\\xi_1)\\phi^T(\\xi_2) .\n\\label{propV}\n\\end{gather}\nThe electron Green function has the triangular structure in the Keldysh space:\n\\begin{equation}\n\\hat G = \\begin{pmatrix} G^\\text{R} & G^\\text{K}\\\\ 0&G^\\text{A} \\end{pmatrix},\n\\qquad\nG^\\text{K} = G^\\text{R}\\circ {\\cal F}-{\\cal F}\\circ G^\\text{A},\n\\end{equation}\nwhere ${\\cal F}$ is the fermion distribution function, and the symbol ``$\\circ$''\ndenotes the convolution over intermediate spatial and time indices.\nAt the equilibrium with the temperature $T$, ${\\cal F}_\\varepsilon=\\tanh(\\varepsilon\/2T)$.\n\n\n\\subsection{Interaction propagator}\n\nIn the derivation below we will mainly assume that electrons in the dot interact via the Coulomb potential $V_0({\\bf r})=e^2\/\\epsilon r$, where $\\epsilon$ is the dielectric constant of the medium.\nDue to the long-range nature of the Coulomb interaction, $V_0^{-1}({\\bf q}\\to 0)=0$,\nthe propagator $\\hat V$ should be calculated taking screening into account\n(this procedure is not required for an arbitrary weak and not long-range potential).\nAs usual, we do this in the random phase approximation (RPA) \\cite{Mahan} justified in the case of a small gas parameter:\n\\begin{equation}\n\\label{rs}\n  r_s = \\frac{e^2}{\\epsilon v_F} \\ll 1 .\n\\end{equation}\nUnder this condition the ratio of the Hartree and Fock diagrams, $F$, introduced in Eq.~(\\ref{f-gen}) is small too, see Eq.~(\\ref{f-Coulomb}). Then one should sum only the bubble contributions to the effective propagator, with independent averaging over disorder in each bubble.\n\nThe RPA-screened\ninteraction propagator has the standard structure:\n\\begin{equation}\n  \\hat V = \\begin{pmatrix} V^\\text{K} & V^\\text{R} \\\\ V^\\text{A} & 0 \\end{pmatrix},\n\\qquad\n  V^\\text{K}=V^\\text{R}\\circ {\\cal B}-{\\cal B}\\circ V^\\text{A},\n\\label{CProp}\n\\end{equation}\nwhere ${\\cal B}$ is the boson distribution function defined as \\cite{KA99}\n\\begin{equation}\n{\\cal B}_{\\omega}=\\frac 1{2\\omega}\\int_{-\\infty}^{\\infty}(1-{\\cal F}_{\\varepsilon'}{\\cal F}_{\\varepsilon'-\\omega})d\\varepsilon' .\n\\end{equation}\nAt the thermal equilibrium, ${\\cal B}_\\omega=\\coth\\left(\\omega\/2T\\right)$.\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=120mm]{35textcropped.pdf}}\n\\caption{\nThe dynamically screened interaction $V(\\omega,{\\bf q})$ (wavy line) expressed in terms\nof the statically screened interaction (SSI) $V({\\bf q})$ (zigzag line)\nand the dynamic polarization operator $G^\\text{R}G^\\text{A}$.\nFor a quantum-dot at $(\\varepsilon,T)\\llE_{\\text{Th}}$, each dynamic polarization bubble contains a small factor of $\\omega\/E_{\\text{Th}}$.}\n\\label{F:SSI}\n\\end{figure}\n\nThe dynamically screened interaction can be written in the form (see Fig.~\\ref{F:SSI})\n\\begin{equation}\n  V^{\\text{R}(\\text{A})}(\\omega,{\\bf q})\n  =\n  \\bigl[\n    V^{-1}({\\bf q}) + \\Pi^{\\text{R}(\\text{A})}_\\text{dyn}(\\omega,{\\bf q})\n  \\bigr]^{-1} ,\n\\label{CProp10}\n\\end{equation}\nwhere the real function $V({\\bf q})$ is the statically screened interaction (SSI):\n\\begin{equation}\n  V({\\bf q})\n  =\n  \\bigl[\n    V_0^{-1}({\\bf q}) + N_s\\nu\n  \\bigr]^{-1} ,\n\\label{V-SSI}\n\\end{equation}\nand $\\Pi_\\text{dyn}$ is the dynamic part of the polarization operator originating from the bubble $G^\\text{R}G^\\text{A}$:\n\\begin{equation}\n  \\Pi^{\\text{R}(\\text{A})}_\\text{dyn}(\\omega,{\\bf q})\n  =\n  N_s\\nu \\frac{\\pm i\\omega}{Dq^2\\mp i\\omega}\n  .\n\\label{Pi10}\n\\end{equation}\nThe simple form of the polarization operator $\\Pi({\\bf q},\\omega)$ in Eqs.~(\\ref{V-SSI}) and (\\ref{Pi10}) corresponds to the limit $q\\ll p_F$ (and $\\omega\\ll E_F$). In the case of the Coulomb interaction with a small $r_s$, the spatial dispersion of $\\Pi({\\bf q},0)$ is irrelevant as the interaction in Eq.~(\\ref{f-gen}) is determined by $q\\sim\\kappa\\ll p_F$. In the case of a weak short-scale interaction, $\\lambda\\ll1$, static screening can be neglected and the form of $\\Pi({\\bf q},0)$ is not important.\n\nThe SSI defined in Eq.~(\\ref{V-SSI}) can be conveniently rewritten in the form [Fourier transform of Eq.~(\\ref{V(r)})]\n\\begin{equation}\n\\label{V-static}\n  V({\\bf q})\n  = \\frac{\\lambda}{N_s\\nu} v_\\kappa({\\bf q})\n ,\n\\end{equation}\nwhere\n$v_\\kappa({\\bf q})$\nis the momentum\nrepresentation of the $\\delta$ function smeared over the interaction radius, $\\delta_\\kappa({\\bf r})$.\nBy definition, $v_\\kappa(0)=1$.\nIn the important case of the Coulomb interaction, $\\lambda^\\text{Coul}=1$ and\n\\begin{equation}\n\\label{delta-kappa-Coul}\n  v_\\kappa^\\text{Coul}({\\bf q})\n  =\n  \\begin{cases}\n    \\kappa_\\text{3D}^2\/(q^2+\\kappa_\\text{3D}^2), & \\text{in 3D,} \\\\\n    \\kappa_\\text{2D}\/(q+\\kappa_\\text{2D}), & \\text{in 2D,}\n  \\end{cases}\n\\end{equation}\nwhere $1\/\\kappa$ is the Thomas-Fermi screening length given by\n\\begin{equation}\n\\label{kappa}\n  \\kappa^2_\\text{3D} = 4\\pi N_s\\nu_3 e^2\/\\epsilon,\n\\qquad\n  \\kappa_\\text{2D} = 2\\pi N_s\\nu_2 e^2\/\\epsilon .\n\\end{equation}\n\n\n\n\\subsection{Kinetic equation}\n\nIn this Section we sketch the main steps in the derivation of the quantum kinetic equation\nin the Keldysh formalism (for a detailed discussion see, e.g, Refs.~\\cite{RS,Altshuler}).\nFrom the Dyson equation for the exact (disorder-dependent)\nelectron Green function $\\hat G$ we obtain\n\\begin{equation}\n  \\bigl[\\hat G_0^{-1},\\hat G\\bigr]=\\bigl[\\hat{\\Sigma},\\hat G\\bigr],\n\\label{comm}\n\\end{equation}\nwhere $[A,B]=A\\circ B-B\\circ A$,\nand $\\hat{\\Sigma}$ is the irreducible self-energy due to interaction,\nbearing the same triangular structure as the electron Green function:\n\\begin{equation}\n\\label{Sigma-RAK}\n  \\hat{\\Sigma}=\\begin{pmatrix} \\Sigma^\\text{R}& \\Sigma^\\text{K} \\\\ 0& \\Sigma^\\text{A} \\end{pmatrix}.\n\\end{equation}\n\nAs we are interested in the slow dynamics of the system,\nit is convenient to switch to the mixed energy-time (Wigner) representation,\n\\begin{equation}\n  f_{\\varepsilon}(t) = \\int d\\tau \\, f\\left(t+\\frac {\\tau}2, t-\\frac{\\tau}2\\right)e^{i\\varepsilon \\tau} ,\n\\end{equation}\nwhich allows us to get rid of the convolution on the right-hand side\nof Eq.~(\\ref{comm}).\nThe kinetic equation is obtained from the Keldysh component of Eq.~(\\ref{comm})\nby tracing over the space.\nWe also assume that the distribution function does not depend\non the coordinates. Therefore the left-hand side of Eq.~(\\ref{comm})\nproduces only the time derivatives of ${\\cal F}$, and we arrive at\nthe standard form of the kinetic equation,\n\\begin{equation}\n\\label{F-St}\n  \\partial_t {\\cal F}_\\varepsilon = \\text{St}[{\\cal F}_\\varepsilon] .\n\\end{equation}\nThe collision integral $\\text{St}[{\\cal F}_\\varepsilon]$ is given by\n\\begin{equation}\n  \\text{St}[{\\cal F}_\\varepsilon]\n  \\int d{\\bf r} \\, \\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\n  =\n  -i\\int d{\\bf r} \\, d{\\bf r}'\\left\\{\n  \\Delta \\Sigma_{\\varepsilon}({\\bf r}, {\\bf r}')G^\\text{K}_{\\varepsilon}({\\bf r}',{\\bf r})\n- \\Sigma^\\text{K}_{\\varepsilon}({\\bf r},{\\bf r}')\\Delta G_{\\varepsilon}({\\bf r}',{\\bf r})\n  \\right\\},\n\\label{kinur0}\n\\end{equation}\nwhere all functions have an implicit central-time argument $t$, and we denote\n\\begin{equation}\n  \\Delta G= G^\\text{R}-G^\\text{A},\\qquad \\Delta \\Sigma=\\Sigma^\\text{R}-\\Sigma^\\text{A}.\n\\label{DeltaG}\n\\end{equation}\n\n\n\n\\subsection{Modification of the Keldysh technique}\n\\label{SS:Modified}\n\nAs our future analysis will involve diagrams with many interaction propagators,\nit is convenient to modify the standard Keldysh diagrammatic technique\ndiscussed above to make it suitable for routine calculations.\nWe find it appropriate to `eliminate' the Keldysh component\nof the electron Green function and to describe the system\nin terms of the retarded and advanced Green functions only.\nTo this end, we note that in the case of weak nonequilibrium\none has $G^\\text{K}_{\\varepsilon}(t)={\\cal F}_{\\varepsilon}(t)\\Delta G_{\\varepsilon}(t)$,\nwhich allows us to diagonalize the Green function as\n\\begin{equation}\n\\hat G=U^{-1}_{\\varepsilon} g_{\\varepsilon} U_{\\varepsilon},\n\\qquad\ng_{\\varepsilon}=\\begin{pmatrix} G^\\text{R}_{\\varepsilon}&0\\\\0&G^\\text{A}_{\\varepsilon} \\end{pmatrix},\n\\qquad\nU_{\\varepsilon}=\\begin{pmatrix} 1&{\\cal F}_{\\varepsilon}\\\\0&-1\\end{pmatrix} ,\n\\label{gtransform}\n\\end{equation}\nwhere the time argument is suppressed for brevity.\nIn order to construct the perturbation theory in terms of\nthe Green function $g$,\nit is convenient to include $U_{\\varepsilon}$ into the definition\nof the interaction vertex:\n\\begin{equation}\n\\Gamma^k(\\varepsilon_1,\\varepsilon_2)=U_{\\varepsilon_1}\\gamma^k U^{-1}_{\\varepsilon_2}.\n\\end{equation}\nThe resulting $\\Gamma^k(\\varepsilon_1,\\varepsilon_2)$ depend on two\nenergy indices owing to the energy dependence of the distribution function:\n\\begin{equation}\n\\Gamma^1(\\varepsilon_1,\\varepsilon_2)=\\begin{pmatrix} 1&{\\cal F}_{\\varepsilon_2}-{\\cal F}_{\\varepsilon_1}\\\\0&1\\end{pmatrix}, \\qquad \\Gamma^2(\\varepsilon_1,\\varepsilon_2)=\\begin{pmatrix} {\\cal F}_{\\varepsilon_1}&-1+{\\cal F}_{\\varepsilon_1}{\\cal F}_{\\varepsilon_2}\\\\-1&-{\\cal F}_{\\varepsilon_2}\\end{pmatrix}.\n\\end{equation}\n\nThis modification of the Keldysh technique that will be used below allows us\nto simplify calculations with many interaction lines significantly,\nsince now the electron Green function $g$ is diagonal in the Keldysh space\nand does not contain the distribution function. The interaction line is defined now as\n\\begin{equation}\n\\label{Y-def}\nY_{abcd}(\\varepsilon,\\varepsilon',\\omega,{\\bf q})=\n\\raisebox{-24pt}{\n  \\begin{picture}(88,53)(-10,0)\n    \\put(0,10){\\includegraphics[scale=0.15]{Yabcd1.jpg}}\n    \\put(2,2){$\\varepsilon$}\n    \\put(2,48){$\\varepsilon-\\omega$}\n    \\put(43,48){$\\varepsilon'-\\omega$}\n    \\put(58,2){$\\varepsilon'$}\n    \\put(-5,15){$a$}\n    \\put(-5,34){$b$}\n    \\put(68,34){$c$}\n    \\put(68,15){$d$}\n  \\end{picture}\n  }\n  =\n  \\Gamma^k_{ab}(\\varepsilon,\\varepsilon-\\omega)\\Gamma^l_{cd}\\left(\\varepsilon'-\\omega,\\varepsilon'\\right)V^{kl}_{\\omega}({\\bf q}),\n\\end{equation}\nwhere indices $a,b,c,d$ take the values $\\text{R}$ and $\\text{A}$,\nand we imply the correspondence $\\text{R}\\leftrightarrow 1$ and $\\text{A}\\leftrightarrow 2$.\nExplicit expressions for $Y_{abcd}$ are listed in \\ref{A:Y}.\nAccording to the general rules of the diagrammatic technique,\nthe element $Y_{abcd}(\\varepsilon,\\varepsilon',\\omega,{\\bf q})$ enters with the coefficient $i\/2$\n[the factor $2$ is inherited from Eq.~(\\ref{propV})].\n\nThe irreducible self-energy in the new basis, $\\Sigma^{ij}$,\nis related to the self-energy (\\ref{Sigma-RAK}) as\n\\begin{equation}\n  \\Sigma^\\text{R}=\\Sigma^{\\text{RR}},\n\\quad\n  \\Sigma^\\text{A}=\\Sigma^{\\text{AA}},\n\\quad\n  \\Sigma^\\text{K}=\\left(\\Sigma^{\\text{RR}}-\\Sigma^{\\text{AA}}\\right) {\\cal F}_{\\varepsilon}-\\Sigma^{\\text{RA}},\n\\quad\n  \\Sigma^{\\text{AR}}=0.\n\\end{equation}\nThus, in the modified Keldysh technique, Eq.~(\\ref{kinur0})\nfor the collision integral takes a compact form:\n\\begin{equation}\n  \\text{St}[{\\cal F}_\\varepsilon] \\int d{\\bf r}\\,\\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\n  =\n  -i\\int d{\\bf r}\\, d{\\bf r}'\\,\\Sigma^{\\text{RA}}_{\\varepsilon}({\\bf r},{\\bf r}') \\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}).\n\\label{kinur1}\n\\end{equation}\n\n\n\\subsection{Collision integral and the energy relaxation rate}\n\nThe inelastic energy relaxation rate, $\\gamma(\\varepsilon,T)$, can be obtained\nfrom the collision integral in the usual way~\\cite{Altshuler,Schmid-ep,Reizer}:\n\\begin{equation}\n  \\gamma(\\varepsilon,T) = -\\frac{\\delta \\, \\text{St}[{\\cal F}_{\\varepsilon}]}{\\delta {\\cal F}_{\\varepsilon}}.\n\\label{tau}\n\\end{equation}\nUsing $\\text{St}[{\\cal F}_{\\varepsilon}]$ from Eq.~(\\ref{kinur1}), we arrive at the following\nexpression for $\\gamma(\\varepsilon,T)$:\n\\begin{equation}\n  \\gamma(\\varepsilon,T) \\int d{\\bf r}\\,\\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\n  =\n  i\\frac{\\delta}{\\delta {\\cal F}_{\\varepsilon}}\n  \\int d{\\bf r} \\, d{\\bf r}'\\,\\Sigma^{\\text{RA}}_{\\varepsilon}({\\bf r},{\\bf r}') \\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}).\n\\label{gamma}\n\\end{equation}\n\nEquation (\\ref{gamma}) is the starting point for the calculation of the energy relaxation rate.\nThis equation contains exact disorder-dependent Green functions, and averaging its $n$'th power\nover the random potential generates the $n$'th moment of $\\gamma(\\varepsilon,T)$.\nThe simplest is the first moment, $\\corr{\\gamma(\\varepsilon,T)}$, related to the average\ncollision integral $\\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}$. In this case, the left-hand side of Eq.~(\\ref{kinur1})\ngives the average density of states, $\\corr{\\Delta G_{\\varepsilon}({\\bf r},{\\bf r})} = -2\\pi i \\nu$,\nand one arrives at\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}\n  =\n  \\frac{\\Delta}{2\\pi} \\int d{\\bf r}\\, d{\\bf r}'\n  \\langle\\Sigma^{\\text{RA}}_{\\varepsilon}({\\bf r},{\\bf r}') \\Delta G_{\\varepsilon}({\\bf r}',{\\bf r})\\rangle .\n\\label{kinur01}\n\\end{equation}\nIn the next Section we show how this equation reproduces Sivan, Imry and Aronov result (\\ref{SIAT}) for the average inelastic rate in the limit $F\\to0$, and generalize it to the case of an arbitatry parameter $F$.\nThe second moment of $\\gamma(\\varepsilon,T)$ will be considered in Secs.~\\ref{S:MF}--\\ref{S:mesofluct-rs}.\n\n\n\n\n\\section{Average energy relaxation rate}\n\\label{S:Average}\n\nIn this Section we rederive the result of Sivan, Imry and Aronov \\cite{SIA}\nand generalize it to the case\nof an arbitrary temperature $T$, spin degeneracy $N_s$ and interaction radius characterized by the parameter $F$.\nOur treatment closely follows the standard derivation\nof the kinetic equation in the RPA approximation \\cite{RS}, but within the modified\nKeldysh technique introduced in Sec.~\\ref{SS:Modified}.\nThe purpose of this Section is to illustrate the usage of this technique\nand to prepare the ingredients for the analysis of mesoscopic fluctuations\nof $\\gamma(\\varepsilon,T)$ in Sec.~\\ref{S:MF}.\n\n\n\n\\subsection{Derivation of the Sivan, Imry and Aronov result}\n\\label{SS:SIA}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=163mm]{30textcropped.pdf}}\n\\caption{(a) The simplest\n(and the most important for small $F$) contribution to the self-energy $\\Sigma^{\\text{RA}}_{\\varepsilon}$.\n(b) The graphic representation of the product $\\Sigma^{\\text{RA}}_{\\varepsilon} \\Delta G_{\\varepsilon}$\nwhich determines the collision integral (\\ref{kinur01}). The thick line\nstanding for $\\Delta G_{\\varepsilon} = G^\\text{R}_{\\varepsilon}-G^\\text{A}_{\\varepsilon}$ corresponds to\nthe electron state whose decay is studied.\n(c) The diagram (b) averaged over disorder, with the shaded block representing the diffuson.}\n\\label{F:SIA2}\n\\end{figure}\n\nThe simplest diagram for the self-energy $\\Sigma^{\\text{RA}}_{\\varepsilon}$\nis shown in Fig.~\\ref{F:SIA2}(a).\nIn the limit $F\\ll1$ it gives the leading contribution\nto the relaxation rate (a more general situation is discussed\nin Sec.~\\ref{SS:rs}).\nThe corresponding analytic expression\ncan be easily written in terms of the elements $Y$ introduced in Eq.~(\\ref{Y-def}):\n\\begin{equation}\n\\Sigma^{\\text{RA}}_{1\\varepsilon}({\\bf r},{\\bf r}')=\\frac i2\\int (d\\omega)\\left\\{G^\\text{R}_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')Y_{\\text{RR},\\text{RA}}(\\varepsilon,\\varepsilon,\\omega, {\\bf r}-{\\bf r}')+G^\\text{A}_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')Y_{\\text{RA},\\text{AA}}(\\varepsilon,\\varepsilon,\\omega,{\\bf r}-{\\bf r}')\\right\\},\n\\end{equation}\nwith $(d\\omega)\\equiv d\\omega\/2\\pi$.\nThe interaction propagators are assumed to be RPA screened\nand averaged over disorder (see Sec.~\\ref{SS:Keldysh}),\nwhile the electron Green functions are still exact in a given realization of disorder.\nUsing Eqs.~(\\ref{RRRA}) and (\\ref{RAAA}), and employing the analyticity properties\nwe obtain\n\\begin{equation}\n\\Sigma^{\\text{RA}}_{1\\varepsilon}({\\bf r},{\\bf r}')\n=\n\\frac i2 \\int (d\\omega) \\Phi(\\varepsilon,\\omega)\n\\Delta G_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')\n\\Delta V(\\omega, {\\bf r}-{\\bf r}'),\n\\label{Sigma1}\n\\end{equation}\nwhere $\\Delta V=V^\\text{R}-V^\\text{A}$, and\n\\begin{equation}\n  \\Phi(\\varepsilon,\\omega)=({\\cal F}_{\\varepsilon}-{\\cal F}_{\\varepsilon-\\omega}){\\cal B}_{\\omega}-(1-{\\cal F}_{\\varepsilon}{\\cal F}_{\\varepsilon-\\omega})\n\\end{equation}\nis a combination of the fermionic and bosonic distribution functions which vanishes at the equilibrium (detailed balance).\n\nThe collision integral (\\ref{kinur01}) involves the product $\\Sigma^{\\text{RA}}_{\\varepsilon} \\Delta G_{\\varepsilon}$,\nwhich is to be averaged over disorder. To make this calculation more intuitive,\nwe represent it diagrammatically on a separate graph in Fig.~\\ref{F:SIA2}(b),\nwhere the thick line stands for the Green function $\\Delta G_{\\varepsilon}$\nof the initial state. To avoid confusion we emphasize that this picture is just\na simple graphical notation for $\\Sigma^{\\text{RA}}_{\\varepsilon} \\Delta G_{\\varepsilon}$, since\nthe vertices with a thick line are not described by the rule of Eq.~(\\ref{Y-def}).\nSubstituting the self-energy (\\ref{Sigma1}) into Eq.~(\\ref{kinur01})\nwe obtain for the averaged collision integral:\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}\n  =\n  \\frac{i\\Delta}{4\\pi}\\int d{\\bf r} \\,d{\\bf r}'(d\\omega) \\Phi(\\varepsilon,\\omega)\n  \\langle\\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}) \\Delta G_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')\\rangle\n  \\Delta V(\\omega,{\\bf r}-{\\bf r}') .\n\\end{equation}\nThe averaged product of two Green functions,\n\\begin{equation}\n\\label{<GG>}\n  \\corr{\\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}) \\Delta G_{\\varepsilon-\\omega}({\\bf r},{\\bf r}')}\n  =\n  -4\\pi\\nu \\mathop{\\rm Re} D_0^\\text{R}(\\omega,{\\bf r}-{\\bf r}')\n\\end{equation}\nis expressed in terms\nof the particle-hole ladder (diffuson), $D_0^\\text{R}(\\omega,{\\bf q}) = 1\/(Dq^2-i\\omega)$,\nsee Fig.~\\ref{F:SIA2}(c). Then one arrives at the well-known result for the collision integral in dirty metals (see, for example, Refs.~\\cite{RS,AltshulerAronov,Schmid-ee}):\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_\\varepsilon]}\n  =\n  2\\int(d{\\bf q})(d\\omega)\\Phi(\\varepsilon,\\omega)\\mathop{\\rm Re} D_0^\\text{R}(\\omega,{\\bf q})\\mathop{\\rm Im} V^\\text{R}(\\omega,{\\bf q}) ,\n\\label{StF}\n\\end{equation}\nwhere $(d{\\bf q})\\equiv d^dq\/(2\\pi)^d$, and $d$ is the space dimensionality.\n\nAccording to Eq.~(\\ref{CProp10}), the imaginary part of the fluctuation propagator is determined by the dynamic polarization operator $\\Pi_\\text{dyn}$.\nFor a quantum dot in the zero-dimensional regime ($\\omega\\llE_{\\text{Th}}$),\n$\\Pi_\\text{dyn}$ is a small correction to $V^{-1}({\\bf q})$,\nand $\\mathop{\\rm Im} V^\\text{R}(\\omega,{\\bf q})$ can be written as\n\\begin{equation}\n\\label{ImV}\n  \\mathop{\\rm Im} V^\\text{R}(\\omega,{\\bf q})\n  \\approx\n  - V^2({\\bf q}) \\mathop{\\rm Im} \\Pi^{\\text{R}}_\\text{dyn}(\\omega,{\\bf q}) .\n\\end{equation}\nUsing Eq.~(\\ref{Pi10}) and assuming that the system size, $L$, is larger than the interaction radius, $1\/\\kappa$, we obtain in the zero-dimensional regime:\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}\n  =\n  -\\frac{2\\lambda^2}{N_s\\nu}\\int (d\\omega) (d{\\bf q})\\frac{\\omega \\, \\Phi(\\varepsilon,\\omega)}{(Dq^2)^2}\n  =\n  - \\frac{2\\lambda^2\\Delta}{N_s} \\mathop{{\\sum}'}_m\n  \\int (d\\omega)\\frac {\\omega\\,\\Phi(\\varepsilon,\\omega)}{(Dq_m^2)^2} ,\n\\label{StF1}\n\\end{equation}\nwhere $q_m^2$ are the eigenvalues of the operator $-\\nabla^2$ in the dot\nwith von Neumann boundary conditions, with the zero mode being excluded\ndue to electroneutrality \\cite{SIA,Blanter,AGKL,AG98,BlanterMirlin1997}.\nPerforming summation over discrete momenta, we get\n\\begin{equation}\n  \\corr{\\text{St}[{\\cal F}_{\\varepsilon}]}\n  =\n  - \\frac{2\\lambda^2\\Delta}{N_s E_2^2}\n  \\int (d\\omega) \\, \\omega \\, \\Phi(\\varepsilon,\\omega) ,\n\\end{equation}\nwhere the energy $E_2\\simE_{\\text{Th}}$ is defined in Eq.~(\\ref{En}).\nThe inelastic energy relaxation rate at the equilibrium\ncan be extracted with the help of Eq.~(\\ref{tau}):\n\\begin{equation}\n\\label{gamma0-integral}\n\\gamma_0^\\text{RPA}(\\varepsilon,T)=\\frac{2\\lambda^2\\Delta}{N_s E_2^2} \\int (d\\omega)\\,\\omega \\left\\{ \\coth\\left(\\frac{\\omega}{2T}\\right)+\\tanh\\left(\\frac{\\varepsilon-\\omega}{2T}\\right)\\right\\}.\n\\end{equation}\nIntegrating over $\\omega$, we arrive at Eq.~(\\ref{SIAT})\nfor the average energy relaxation rate $\\gamma_0(\\varepsilon,T)$\nin the limit $F\\to0$.\n\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=150mm]{401textcropped.pdf}}\n\\vspace{0mm}\n\\caption{\nThe diagrams for the collision integral with two SSI lines\nprior to disorder averaging.\n(a) The RPA diagram\n\\ref{F:SIA2}(b)\nredrawn in terms of the SSI lines (\\ref{V-static})\nand the dynamic polarization bubble $G^\\text{R}G^\\text{A}$. The corresponding contribution to the relaxation rate is given by the $X_4$ term in Eq.~(\\ref{gamma1}).\n(b) The other, non-RPA diagram discarded in Sec.~\\ref{SS:SIA}, leading to the $Y_4$ term in Eq.~(\\ref{gamma1}). Its contribution to the mean relaxation rate can be neglected\nat small $F$ [see Eq.~(\\ref{gamma1full})].}\n\\label{F:2Coulomb}\n\\end{figure}\n\n\n\n\n\n\\subsection{Physical interpretation and the other diagram}\n\\label{SS:Physical}\n\nIt is instructive to rederive the result for $\\gamma_0^\\text{RPA}(\\varepsilon,T)$\nin a slightly different manner to clarify the physics of the decay\nprocess responsible for the width of an one-electron level.\nReal decay processes are described by $\\mathop{\\rm Im} V^\\text{R}(\\omega,{\\bf q})$ [see Eq.~(\\ref{StF})],\nwhich is determined by the dynamic screening of the bare interaction.\nIn the zero-dimensional limit, $(E,T,\\omega)\\llE_{\\text{Th}}$,\nthe dynamic part of the polarization bubble,\n$\\Pi_\\text{dyn}(\\omega,{\\bf q})$, contains\na small factor $\\omega\/E_{\\text{Th}}$ [see Eq.~(\\ref{Pi10})].\nThis means that with the accuracy of order $\\omega\/E_{\\text{Th}}$ we may\nconsider $\\Pi_\\text{dyn}(\\omega,{\\bf q})$ as a small perturbation\nand retain only\nthe first dynamic bubble in the expansion of $V(\\omega,{\\bf q})$\nnear the statically screened interaction (SSI)\n$V({\\bf q})$, see Eq.~(\\ref{ImV}).\n\nTherefore with the accuracy of order $\\omega\/E_{\\text{Th}}$ the diagram in Fig.~\\ref{F:SIA2}(a)\nfor the average collision integral with one dynamically screened interaction\ncan be redrawn as the diagram in Fig.~\\ref{F:2Coulomb}(a)\nwith two SSI lines.\nThe advantage of the diagrammatic representation of Fig.~\\ref{F:2Coulomb}(a)\nis that it elucidates the physics of the inelastic collision:\nThe cross-section of this diagram corresponds to the decay of an electron with\nenergy $\\varepsilon$ into an electron with energy $\\varepsilon-\\omega$ and an electron-hole\npair with energies $\\varepsilon_1$ and $\\varepsilon_1-\\omega$.\nThe corresponding contribution to the self-energy\n(before disorder averaging)\ncan be easily read off from Eq.~(\\ref{Y-def}):\n\\begin{equation}\n\\label{SigmaRAa}\n\\Sigma^{\\text{RA}}_{a}=-N_s\\left(\\frac i2\\right)^2\\sum_{abc}G^a_{\\varepsilon-\\omega}({\\bf r}_1,{\\bf r}_2)G^b_{\\varepsilon_1-\\omega}({\\bf r}_4,{\\bf r}_3)G^c_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4) Y_{\\text{R}abc}(\\varepsilon,\\varepsilon_1,\\omega,{\\bf r}_1,{\\bf r}_3)Y_{cba\\text{A}}(\\varepsilon_1,\\varepsilon,\\omega,{\\bf r}_2,{\\bf r}_4),\n\\end{equation}\nwhere here and in what follows the interaction line $Y_{abcd}$ corresponds to the SSI $V({\\bf q})$.\nThe factor $-N_s$ in Eq.~(\\ref{SigmaRAa}) comes from the upper closed electron loop [the bottom bubble containing a highlighted Green function at the external energy $\\Delta G_{\\varepsilon}$\nis not a closed loop in the diagrammatic sense (see Fig.~\\ref{F:SIA2}) and does not contribute an extra factor $-N_s$].\n\nThe diagram \\ref{F:2Coulomb}(a) is not the only one with two SSI lines\nthat describes quasiparticle decay process. The other diagram is shown in Fig.~\\ref{F:2Coulomb}(b). Though it is not as intuitive as the diagram \\ref{F:2Coulomb}(a), it also makes a contribution to the relaxation rate. That contribution is usually discarded\nsince it vanishes in the limit $F\\to0$ (see below).\nHowever as we show in Sec.~\\ref{Seconddiagram},\nthe diagram \\ref{F:2Coulomb}(b) should be taken into account\nin calculating mesoscopic fluctuations of the relaxation rate,\nsince its contribution is comparable to that of the diagram \\ref{F:2Coulomb}(a)\neven in the limit $F\\to0$.\nThe self-energy for the diagram \\ref{F:2Coulomb}(b) is given by\n\\begin{equation}\n\\Sigma^{\\text{RA}}_{b}=\\left(\\frac i2\\right)^2\\sum_{abc}G^a_{\\varepsilon_1}({\\bf r}_1,{\\bf r}_2)G^b_{\\varepsilon_1-\\omega}({\\bf r}_2,{\\bf r}_3)G^c_{\\varepsilon-\\omega}({\\bf r}_3,{\\bf r}_4) Y_{\\text{R}abc}(\\varepsilon,\\varepsilon-\\omega,\\varepsilon-\\varepsilon_1,{\\bf r}_1,{\\bf r}_3)Y_{abc\\text{A}}(\\varepsilon_1,\\varepsilon,\\omega,{\\bf r}_2,{\\bf r}_4).\n\\end{equation}\nDue to the absence of the closed electron loop in the diagram \\ref{F:2Coulomb}(b), $\\Sigma^{\\text{RA}}_{b}$ does not contain an extra factor $-N_s$ compared to $\\Sigma^{\\text{RA}}_{a}$.\n\nThe energy relaxation rate $\\gamma(\\varepsilon, T)$ (in a given realization of disorder) due to processes with two SSI\nlines can be obtained from Eq.~(\\ref{gamma}) with $\\Sigma^{\\text{RA}}=\\Sigma^{\\text{RA}}_{a}+\\Sigma^{\\text{RA}}_{b}$.\nAfter some algebra involving analyticity properties we obtain\n\\begin{equation}\n  \\gamma(\\varepsilon,T)\\int d{\\bf r} \\, \\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\n  =\n  - \\frac{i\\lambda^2}{4N_s^2\\nu^2}\\int d{\\bf r}_1\\ldots d{\\bf r}_4\\int (d\\varepsilon_1)(d\\omega)({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega})({\\cal F}_{\\varepsilon-\\omega}+{\\cal B}_{\\omega})\n  \\left[ N_s X_4-Y_4 \\right],\n\\label{gamma1}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{X4}\n  X_4\n  =\n  \\delta_\\kappa({\\bf r}_1-{\\bf r}_3)\\delta_\\kappa({\\bf r}_2-{\\bf r}_4) \\,\n  \\Delta G_{\\varepsilon}({\\bf r}_2,{\\bf r}_1)\\Delta G_{\\varepsilon-\\omega}({\\bf r}_1,{\\bf r}_2)\n  \\Delta G_{\\varepsilon_1-\\omega}({\\bf r}_4,{\\bf r}_3)\\Delta G_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4)\n\\end{equation}\nand\n\\begin{equation}\n\\label{Y4}\nY_4=\\delta_{\\kappa}({\\bf r}_1-{\\bf r}_3)\\delta_{\\kappa}({\\bf r}_2-{\\bf r}_4)\\Delta G_{\\varepsilon}({\\bf r}_4,{\\bf r}_1)\\Delta G_{\\varepsilon_1}({\\bf r}_1,{\\bf r}_2)\\Delta G_{\\varepsilon_1-\\omega}({\\bf r}_2,{\\bf r}_3)\\Delta G_{\\varepsilon-\\omega}({\\bf r}_3,{\\bf r}_4).\n\\end{equation}\nIn Eq.~(\\ref{gamma1}), the term $N_s X_4$ comes from $\\Sigma^{\\text{RA}}_{a}$ [diagram \\ref{F:2Coulomb}(a)], and the term $-Y_4$ comes from $\\Sigma^{\\text{RA}}_{b}$ [diagram \\ref{F:2Coulomb}(b)].\nAn expression similar to Eq.~(\\ref{gamma1}) has been derived in Ref.~\\cite{Basko}, where only the contribution from the\ndiagram \\ref{F:2Coulomb}(a) has been considered.\nIn the limit when the interaction radius $1\/\\kappa$ exceeds the Fermi wavelength, i.e. at $F\\ll1$, disorder averaged $\\langle \\gamma(\\varepsilon,T)\\rangle$\nreproduces the result (\\ref{gamma0-integral}) for $\\gamma_0^\\text{RPA}(\\varepsilon, T)$ [see Eq.~(\\ref{gamma1full})].\n\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=150mm]{505textcropped.pdf}}\n\\caption{Possible ways to average the diagrams for the relaxation rate shown in Fig.~\\ref{F:2Coulomb} over disorder (other types of averaging with two diffusons exist but are suppressed at least by the factor $l\/L\\ll1$, where $l$ is the mean free path).\n(a) RPA averaging of the diagram \\ref{F:2Coulomb}(a)\nreproducing the SIA result for $\\gamma_0^\\text{RPA}(\\varepsilon,T)$ [Eq.~(\\ref{gamma0-integral})].\n(b) Non-RPA averaging of the diagram \\ref{F:2Coulomb}(a)\nwith the contribution $F^2\\gamma_0^\\text{RPA}(\\varepsilon,T)$.\n(c) and (d) Two ways to average the diagram \\ref{F:2Coulomb}(b), each producing the contribution $-(F\/N_s)\\gamma_0^\\text{RPA}(\\varepsilon,T)$ to the relaxation rate.}\n\\label{F:2Coulomb-av}\n\\end{figure}\n\n\n\\subsection{Disorder averaging and the role of the parameter $F$}\n\\label{SS:rs}\n\nEquation (\\ref{gamma1}) is the starting point for evaluating moments of the relaxation rate. In order to find its mean\nvalue one has to average the combinations of four Green functions in Eqs.~(\\ref{X4}) and (\\ref{Y4}).\n\nThe RPA result of Sec.~\\ref{SS:SIA} is reproduced if one takes only the diagram \\ref{F:2Coulomb}(a) (the term $X_4$) into account and average each bubble independently [see Fig.~\\ref{F:2Coulomb-av}(a)]:\n\\begin{equation}\n\\label{X4RPA}\n  \\corr{X_4}_\\text{RPA}\n  =\n  \\delta_\\kappa({\\bf r}_1-{\\bf r}_3)\\delta_\\kappa({\\bf r}_2-{\\bf r}_4)\n  \\corr{\\Delta G_{\\varepsilon}({\\bf r}_2,{\\bf r}_1)\\Delta G_{\\varepsilon-\\omega}({\\bf r}_1,{\\bf r}_2)}\n  \\corr{\\Delta G_{\\varepsilon_1-\\omega}({\\bf r}_4,{\\bf r}_3)\\Delta G_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4)} .\n\\end{equation}\nAveraging with the help of Eq.~(\\ref{<GG>}) and integrating\nover $\\varepsilon_1$, one arrives at Eq.~(\\ref{gamma0-integral}) for $\\gamma_0^\\text{RPA}(\\varepsilon,T)$.\n\nAlong with the RPA averaging (\\ref{X4RPA}), there exists another way to average $X_4$ over disorder shown in Fig.~\\ref{F:2Coulomb-av}(b). It has the same structure of diffusons as the RPA diagram \\ref{F:2Coulomb-av}(a),\nbut with the interaction lines taken at fast momenta $q\\sim\\min(\\kappa,p_F)$.\nThe corresponding contribution to the relaxation rate, $F^2\\gamma_0^\\text{RPA}(\\varepsilon,T)$,\ndiffers from the RPA-contribution by a factor $F^2$, where $F$ is the standard notation for the ratio of the Hartree and Fock diagrams \\cite{Altshulerbook,Akkermans,AAL} discussed in \\ref{S:Non-RPA}.\n\nFinally, consider disorder averaging of the diagram \\ref{F:2Coulomb}(b) corresponding to the term $Y_4$ in Eq.~(\\ref{gamma1}). For this diagram, there are also two ways to draw two-diffuson configurations shown in Figs.~\\ref{F:2Coulomb-av}(c) and (d).\nTheir contributions are equal and can be calculated similar to that of the\ndiagram \\ref{F:2Coulomb-av}(b), but now only one of the two SSI lines are taken at fast momentum making the result proportional to the first power of $F$.\nThe overall correction of the diagram \\ref{F:2Coulomb}(b) to the average relaxation rate\nis given by $-(2F\/N_s)\\gamma_0^\\text{RPA}(\\varepsilon, T)$.\n\nThus we see that among four possible contributions to the\naverage relaxation rate shown in Fig.~\\ref{F:2Coulomb-av},\nall non-RPA diagrams [(b), (c) and (d)] are proportional to some\npower of $F$ and hence are suppressed\nif the interaction radius is larger than the Fermi wavelength.\nThe smallness of the non-RPA diagrams\nin this limit should be attributed to the Friedel oscillations\nwhich suppress the contribution of the corresponding process \\cite{Mahan}.\nSuch a situation was considered, e.g., in Refs.~\\cite{RS,AltshulerAronov,Schmid-ee}.\nOn the other hand, for the point-like interaction (corresponding to $F=1$),\nthe diagrams \\ref{F:2Coulomb-av}(a) and \\ref{F:2Coulomb-av}(b) give the same\ncontribution. This short-range limit was considered in\nRefs.~\\cite{Blanter,AGKL,AG98,ABG}.\n\nCollecting the contributions of all diagrams with two SSI lines\nand two diffusons (Fig.~\\ref{F:2Coulomb-av}), we obtain the final result for the average energy relaxation rate $\\gamma_0(\\varepsilon,T)=\\corr{\\gamma(\\varepsilon,T)}$:\n\\begin{equation}\n\\label{gamma1full}\n  \\gamma_0(\\varepsilon, T)\n  =\n  \\left( 1 - 2F\/N_s + F^2 \\right) \\gamma_0^\\text{RPA}(\\varepsilon,T) ,\n\\end{equation}\nleading to Eq.~(\\ref{SIAT}).\nNote that for spinless electrons with point-like interaction,\nthe inelastic relaxation rate vanishes, $\\gamma=0$,\nwhich is a consequence of the Pauli exclusion principle.\nIt is essential that such a cancellation takes place only\nif the diagram \\ref{F:2Coulomb}(b) is taken into account.\n\n\n\n\n\n\n\n\n\\section{Mesoscopic fluctuations of the energy relaxation rate: general consideration}\n\\label{S:MF}\n\nIn this Section we discuss the general approach to the calculation of mesoscopic fluctuations of the energy relaxation rate $\\gamma(\\varepsilon,T)$.\nThe starting point is the diagrams for the collision integral shown\nin Figs.~\\ref{F:2Coulomb}(a) and \\ref{F:2Coulomb}(b).\nThe corresponding expression for $\\gamma(\\varepsilon,T)$ is given by\nEq.~(\\ref{gamma1}) which is written for a particular realization of impurities. Mesoscopic fluctuations of the relaxation rate are determined by the square of $\\gamma(\\varepsilon,T)$ averaged over disorder:\n\\begin{multline}\n  \\langle \\gamma^2(\\varepsilon,T)\\rangle\n  \\int d{\\bf r}\\, d{\\bf r}' \\,\n  \\langle \\Delta G_{\\varepsilon}({\\bf r},{\\bf r})\\Delta G_{\\varepsilon}({\\bf r}',{\\bf r}')\\rangle\n  =\n- \\frac{\\lambda^4}{16N_s^4\\nu^4}\\int d{\\bf r}_1\\ldots d{\\bf r}_8\n  \\int (d\\varepsilon_1)(d\\varepsilon_2)(d\\omega_1)(d\\omega_2)\n\\\\{}\n  \\times\n  ({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega_1})({\\cal F}_{\\varepsilon-\\omega_1}+{\\cal B}_{\\omega_1})\n  ({\\cal F}_{\\varepsilon_2}-{\\cal F}_{\\varepsilon_2-\\omega_2})({\\cal F}_{\\varepsilon-\\omega_2}+{\\cal B}_{\\omega_2})\n  \\corr{(N_s X_4-Y_4)(N_s X_4'-Y_4')} ,\n\\label{corr}\n\\end{multline}\nwhere the objects $X_4$ and $Y_4$ containing different products of four Green functions and two SSI lines are defined in Eqs.~(\\ref{X4}) and (\\ref{Y4}), while\n$X_4'$ and $Y_4'$ are obtained from them through the replacement\n\\begin{equation}\n  \\{{\\bf r}_i\\} \\to \\{{\\bf r}_{i+4}\\} ,\n\\quad\n  \\varepsilon_1 \\to \\varepsilon_2 ,\n\\quad\n  \\omega_1 \\to \\omega_2 .\n\\end{equation}\nEquation (\\ref{corr}) contains three different products of eight Green functions, $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$, which need to be separately averaged over disorder (the term $Y_4X_4'$ reduces to $X_4Y_4'$ after an obvious change of variables). For example, the explicit form of $X_4X_4'$ is given by\n\\begin{multline}\n  X_4X_4'\n  =\n  \\delta_\\kappa({\\bf r}_1-{\\bf r}_3)\\delta_\\kappa({\\bf r}_2-{\\bf r}_4)\n  \\delta_\\kappa({\\bf r}_5-{\\bf r}_7)\\delta_\\kappa({\\bf r}_6-{\\bf r}_8)\n  \\,\n  \\Delta G_{\\varepsilon}({\\bf r}_2,{\\bf r}_1) \\Delta G_{\\varepsilon-\\omega_1}({\\bf r}_1,{\\bf r}_2)\n\\\\{}\n  \\times\n  \\Delta G_{\\varepsilon_1-\\omega_1}({\\bf r}_4,{\\bf r}_3) \\Delta G_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4)\n  \\Delta G_{\\varepsilon}({\\bf r}_6,{\\bf r}_5) \\Delta G_{\\varepsilon-\\omega_2}({\\bf r}_5,{\\bf r}_6)\n  \\Delta G_{\\varepsilon_2-\\omega_2}({\\bf r}_8,{\\bf r}_7)\\Delta G_{\\varepsilon_2}({\\bf r}_7,{\\bf r}_8)\n  .\n\\label{X8}\n\\end{multline}\nSince $\\Delta G = G^\\text{R}-G^\\text{A}$, each of the products $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$ contains $2^8=256$ different elementary contributions in terms of $G^\\text{R}$ and $G^\\text{A}$.\n\n\nOur task is to calculate the irreducible part $\\ccorr{\\gamma^2(\\varepsilon,T)}$ defined in Eq.~(\\ref{corr-irr-def}).\nIn a usual situation, mesoscopic fluctuations of a random quantity $x$ are determined by irreducible averaging over disorder, when two copies of $x$ are connected by at least one impurity line. In the present case the situation is different as the left-hand side of the basic Eq.~(\\ref{corr}) contains the pairwise correlator $\\corr{ \\Delta G_{\\varepsilon} \\Delta G_{\\varepsilon} }$, which deviates significantly from its reducible part $\\corr{\\Delta G_{\\varepsilon}}^2$ (see Sec.~\\ref{paircorr}).\nTherefore, even extracting the square of the average, $\\corr{\\gamma(\\varepsilon,T)}^2$, from Eq.~(\\ref{corr}) should be done with care as it involves irreducible disorder averaging of\n$\\corr{(N_s X_4-Y_4)(N_s X_4'-Y_4')}$\nneeded for non-perturbative account for correlations between\ntwo Green functions with the energy $\\varepsilon$. The details of this calculation are presented in \\ref{reducible part}.\nSuch a complication is the price one has to pay for extracting the property of a single discrete level with the technique well suited for describing continuous spectra.\nKeeping that in mind we proceed with evaluation of the irreducible part of $\\corr{\\gamma^2(\\varepsilon,T)}$.\n\n\nIn order to determine $\\corr{\\gamma^2(\\varepsilon,T)}$ one has to compute\n$\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$,\nand evaluate four remaining energy integrals in Eq.~(\\ref{corr}).\nThis calculation is rather nontrivial and will be performed in\nSecs.~\\ref{S:nonpert}, \\ref{S:final} and \\ref{S:mesofluct-rs}.\nMeanwhile we discuss the main idea and outline the principal technical steps of the derivation.\n\nThe most complicated task of this procedure is to perform disorder averaging which becomes rather involved in the low-energy limit, $(\\varepsilon,T)\\llE_{\\text{Th}}$, we are interested in.\nIndeed, the single-particle spectrum is well resolved in this case, $\\corr{\\gamma(\\varepsilon,T)}\\ll\\Delta$, indicating that the averaging is to be performed non-perturbatively.\nThat can be achieved with the help of the Efetov's nonlinear supersymmetric sigma model technique \\cite{Efetov-book}, properly generalized to the case of several Green functions with different energies.\n\nThe averaging of two Green functions on the left-hand side of Eq.~(\\ref{corr}) is straightforward and can be done exactly in the zero-dimensional geometry, see Sec.~\\ref{paircorr}.\n\nThe calculation of\n$\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$,\ncontaining eight Green functions is much more complicated and cannot be performed exactly in the general case.\nFortunately, the exact expression is not required for determination of the leading contribution to mesoscopic fluctuations of the relaxation rate.\nA significant simplification is suggested by analysing the physics of the decay process.\nAccording to the FGR, $\\gamma(\\varepsilon, T)$ can be considered as a sum of the decay processes of a single-particle excitation into all possible final three-particle states allowed by the energy conservation:\n$\\varepsilon\\to(\\varepsilon-\\omega,\\varepsilon',-\\varepsilon'+\\omega)$. Consequently, $\\gamma^2(\\varepsilon, T)$ given by Eq.~(\\ref{corr}) contains,\nin principle, six different final states: $(\\varepsilon-\\omega_1,\\varepsilon_1,-\\varepsilon_1+\\omega_1, \\varepsilon-\\omega_2,\\varepsilon_2,-\\varepsilon_2+\\omega_2)$.\n\nHowever, as it was first demonstrated in Ref.~\\cite{BAA}, the leading contribution to mesoscopic fluctuations of the relaxation rate comes from those configurations that describe \\emph{the square of the same decay process}, i.e., from the terms with the identical set of the final states.\nSuch a situation can be realized with two choices:\n\\begin{equation}\n\\label{energy-matching}\n  \\text{(a) \\: $\\varepsilon_1 \\approx \\varepsilon_2$ and $\\omega_1 \\approx \\omega_2$},\n\\qquad\n  \\text{(b) \\: $\\varepsilon_1 \\approx \\varepsilon-\\omega_2$ and $\\varepsilon_2 \\approx \\varepsilon-\\omega_1$} ,\n\\end{equation}\nwhere the energies should coincide with the accuracy of the single level width $\\gamma$.\nTechnically, the importance of such configurations is related to the appearance of the formally divergent delta function of zero frequency, $\\delta(0)$, if conditions (\\ref{energy-matching}) are considered as strict equalities, corresponding to the use of non-interacting Green functions in\n$\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$.\nIn order to take into account the single-particle level broadening due to interaction one should add an imaginary part to the energy argument $E$ of the Green function, replacing it by $E_+$ (for $G^\\text{R}$) and $E_-$ (for $G^\\text{A}$):\n\\begin{equation}\n\\label{width}\n  E_\\pm = E \\pm i\\gamma(E)\/2 ,\n\\end{equation}\nwhere $\\gamma(E) \\equiv \\gamma_0(E,T)$ is the average value of the relaxation rate obtained in the FGR approximation [Eq.~(\\ref{SIAT})].\nThe substitution (\\ref{width}) is equivalent to including the elastic part of the electron-electron interaction in the zero-dimensional diffuson (see Sec.~\\ref{slowaveraging} and \\ref{eldiffuson}).\nA finite imaginary part cuts the singularity in $\\delta(0)$ which should be replaced roughly by $1\/\\gamma$, making fluctuations\nfinite but divergent as $\\varepsilon$ and $T$ are decreased.\nThis enhancement of fluctuations at low energies and temperatures is precisely the effect we are looking for.\n\nSuch a mechanism of enhancement of mesoscopic fluctuations of the inelastic width with decreasing temperature was suggested in Ref.~\\cite{BAA} for a model when the matrix elements of the interaction $V_{\\alpha\\beta\\gamma\\delta}$ in the basis of exact single-particle states were assumed to be independently distributed Gaussian variables.\nOur task is more complicated as we do not make any assumptions about the structure of the matrix elements in Fock space and calculate them for the real Coulomb interaction. Though we do not use the language of the matrix elements $V_{\\alpha\\beta\\gamma\\delta}$, they are effectively generated after proper averaging over fast diffusive modes\n\\cite{Blanter,AGKL}.\n\nHaving discussed the main idea, we now outline the crucial steps in the calculation of\n$\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$:\n\\begin{itemize}\n\\item\nEach of $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$ contains in general $2^8=256$ elementary products of $G^\\text{R}$ and $G^\\text{A}$.\nAs we discuss in Secs.~\\ref{slowaveraging} and \\ref{qual},\nthe principal contribution to $\\ccorr{\\gamma^2(\\varepsilon,T)}$ originates from the terms with four $G^\\text{R}$ and four $G^\\text{A}$. Each of these  $C_8^4=70$ terms\nmay be presented as a functional integral over the 16-component superfields which is then averaged over disorder following the standard technique \\cite{Efetov-book}.\nThe resulting supersymmetric nonlinear sigma model is formulated in terms of the functional integral over the $16\\times16$ superfield $Q({\\bf r})$, see Sec.~\\ref{SS:SUSY}.\n\n\\item\nAs the SSI lines carry nonzero momenta (otherwise the diagram is zero due to electroneutrality), momentum conservation requires the presence of some number of fast ($q\\neq0$) diffusive modes.\nSince each fast diffuson contributes a small factor of $\\Delta\/E_{\\text{Th}}$, their number should be minimized.\nFor $\\corr{\\gamma(\\varepsilon,T)}$ the minimal number was two [see Fig.~\\ref{F:2Coulomb-av}], and for $\\corr{\\corr{\\gamma^2(\\varepsilon,T)}}$ it is four, producing the\nfactors $1\/E_2^4$ and $1\/E_4^4$.\nFollowing the method developed in Ref.~\\cite{KM}, we integrate over these fast modes perturbatively and derive an effective action for the zero-mode (spatially-uniform supermatrix $Q$). This procedure is described in details in Sec.~\\ref{fastaveraging}, with the total number of emergent contributions $7!!=105$\nfor every given elementary term from $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$ with four $G^\\text{R}$ and four $G^\\text{A}$.\n\n\\item\nThe resulting zero-dimensional sigma model is still too complicated\nbecause of the large size of the $Q$ matrix.\nHowever, as described above, the leading contribution comes from configurations\nwhere among energy arguments of eight Green functions, four are pairwise equal\n(with an uncertainty of $\\gamma$).\nSince the typical energy difference between pairs is $(\\varepsilon,T)\\gg\\Delta$,\nour extended sigma model splits into four blocks, each corresponding to the standard Efetov sigma model for $\\corr{G^\\text{R}G^\\text{A}}$.\nThe integrals over these sigma models are then evaluated non-perturbatively using the standard technique \\cite{Efetov-book}.\n\nThe detailed discussion of this step in connection with the choice of pairs is presented in Secs.~\\ref{slowaveraging} and \\ref{SS:further}.\n\n\\item\nAs a result of the described procedure, quite a few terms will be generated.\nFortunately, most of them will be either zero or less singular than expected.\nOnly a small number of terms will effectively contribute to fluctuations of\nthe relaxation rate.\nThis selection is discussed in Sec.~\\ref{S:final} in the simplest case of $F\\to0$.\nThe final evaluation of $\\ccorr{\\gamma^2(\\varepsilon,T)}$ in the general case of arbitrary parameter $F$ is performed in Sec.~\\ref{S:mesofluct-rs}.\n\n\\end{itemize}\nThe announced program will be realized step by step in Secs.~\\ref{S:nonpert}, \\ref{S:final} and \\ref{S:mesofluct-rs}.\n\n\n\n\n\n\\section{Nonperturbative averaging of eight Green functions}\n\\label{S:nonpert}\n\n\\subsection{Pairwise correlator $\\corr{ \\Delta G_{\\varepsilon} \\Delta G_{\\varepsilon}}$}\n\\label{paircorr}\n\nWe start with discussing the pair correlation function on the left-hand side of Eq.~(\\ref{corr}). It is instructive to introduce a small energy mismatch $\\omega$ between the energies of two Green functions and to consider a more general correlation function\n\\begin{equation}\n\\label{R-def}\n  \\int d{\\bf r} \\, d{\\bf r}' \\,\n  \\corr{ \\Delta G_{\\varepsilon}({\\bf r},{\\bf r}) \\Delta G_{\\varepsilon-\\omega}({\\bf r}',{\\bf r}')}\n  =\n  -4\\pi^2\\nu^2 R_\\gamma(\\omega) .\n\\end{equation}\nAs usual, the terms $G^\\text{R}G^\\text{R}$ and $G^\\text{A}G^\\text{A}$ are averaged trivially, each contributing 1\/4 to $R_\\gamma(\\omega)$. Averaging of the cross term $G^\\text{R}G^\\text{A}$ is performed with the help of Efetov's zero-dimensional supersymmetric sigma model \\cite{Efetov-book}, where in accordance with Eq.~(\\ref{width}) one has to introduce a complex frequency\n\\begin{equation}\n  \\Omega = \\varepsilon_+ - (\\varepsilon-\\omega)_-\n  =\n  \\omega + i[\\gamma(\\varepsilon)+\\gamma(\\varepsilon-\\omega)]\/2 .\n\\end{equation}\nEvaluating the standard integrals with the help of the machinery developed in \\ref{slowcontrrules}, we obtain\n\\begin{equation}\n\\label{R-general}\n  R_\\gamma(\\omega)\n  =\n  1-\\mathop{\\rm Re} X(-i\\pi\\Omega\/\\Delta) ,\n\\end{equation}\nwhere the function $X(a)$ is given by Eq.~(\\ref{X}):\n\\begin{equation}\n  X(a)=\\frac{1-e^{-2a}}{2a^2} .\n\\end{equation}\n\nIn the limit of vanishing width ($\\gamma\\to0$), $R_0(\\omega)$ reproduces the pair correlation\nfunction for the Gaussian unitary ensemble in the random matrix theory \\cite{Mehta}:\n$\n  R_0^\\text{RMT}(\\omega) = 1-(\\sin x\/x)^2+\\pi \\delta(x)\n$,\nwhere $x=\\pi\\omega\/\\Delta$.\nA finite but small width acts as a regularizer of the $\\delta$ function, accounting for the contribution of the same broadened level into the correlation function. In the relevant limit of a resolved discrete spectrum, $(\\omega,\\gamma)\\ll\\Delta$, $R_\\gamma(\\omega)$ can be written as\n\\begin{equation}\n\\label{R-delta}\n  R_\\gamma(\\omega)\n  \\approx\n  \\Delta \\delta_\\gamma(\\omega) ,\n\\end{equation}\nwhere $\\delta_\\gamma(\\omega)$ is a Lorentzian approximation of the delta function:\n\\begin{equation}\n\\label{delta_gamma}\n  \\delta_\\gamma(\\omega)\n  =\n  \\frac{1}{\\pi} \\frac{\\gamma(\\varepsilon)}{\\gamma^2(\\varepsilon)+\\omega^2} .\n\\end{equation}\nIt is worth noting that Eq.~(\\ref{R-delta}) is non-perturbative. Though it looks just like a one-diffuson contribution, $R_\\gamma(\\omega) \\approx -\\mathop{\\rm Im} 1\/(\\omega+i\\gamma)$, it is an asymptotic expansion of the exact non-perturbative expression (\\ref{R-general}).\n\nThe left-hand side of Eq.~(\\ref{corr}) can be expressed with the help of Eq.~(\\ref{R-def}), with $R_\\gamma(0)=\\Delta\/\\pi\\gamma(\\varepsilon)$.\n\n\n\n\n\\subsection{Supersymmetric sigma model for eight Green functions (the case $F=0$)}\n\\label{SS:SUSY}\n\nThe objects $X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$\ncontain the products of eight $\\Delta G = G^\\text{R}-G^\\text{A}$ which should be averaged over disorder.\nTo introduce the method we start with\nthe simplest case of $F\\to0$. Generalization to the case of an arbitrary $F$ will be performed in Sec.~\\ref{S:mesofluct-rs}. To be specific, we focus on calculating $\\corr{X_4X_4'}$ (disorder averaging of other products is performed analogously and the result is presented in Sec.~\\ref{Seconddiagram}).\nAs we will see in Secs.~\\ref{slowaveraging} and \\ref{qual}, only terms with the equal number of retarded and advanced Green functions make the leading contribution to mesoscopic fluctuations.\nConsider, e.~g., a particular choice of $G^\\text{R}$ and $G^\\text{A}$ from $X_4X_4'$ and calculate\n\\begin{equation}\nK=\\langle G^\\text{R}_{\\varepsilon}({\\bf r}_2,{\\bf r}_1) G^\\text{A}_{\\varepsilon}({\\bf r}_6,{\\bf r}_5) G^\\text{R}_{\\varepsilon-\\omega_2}({\\bf r}_5,{\\bf r}_6) G^\\text{A}_{\\varepsilon-\\omega_1}({\\bf r}_1,{\\bf r}_2)\nG^\\text{R}_{\\varepsilon_1-\\omega_1}({\\bf r}_4,{\\bf r}_3)\nG^\\text{A}_{\\varepsilon_2-\\omega_2}({\\bf r}_8,{\\bf r}_7)  G^\\text{R}_{\\varepsilon_2}({\\bf r}_7,{\\bf r}_8) G^\\text{A}_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4) \\rangle .\n\\label{GGGG}\n\\end{equation}\nDisorder averaging of other relevant terms from $X_4X_4'$ [listed in Eq.~(\\ref{RA-4ways})] can be performed analogously (see Secs.~\\ref{SS:X8a} and \\ref{SS:X8b}).\n\nIn order to calculate $K$ we follow the standard line of Efetov's supersymmetric sigma model \\cite{Efetov-book}, generalizing it to the case of eight Green functions. We group the Green functions into four RA pairs in the sequence they appear in Eq.~(\\ref{GGGG}):\n\\begin{equation}\n\\label{1234}\n  \\begin{tabular}{|c|cccc|}\n  \\hline\n      & 1 & 2 & 3 & 4 \\\\\n  \\hline\n    R & $\\varepsilon$ & $\\varepsilon-\\omega_2$ & $\\varepsilon_1-\\omega_1$ & $\\varepsilon_2$ \\\\\n  \\hline\n    A & $\\varepsilon$ & $\\varepsilon-\\omega_1$ & $\\varepsilon_2-\\omega_2$ & $\\varepsilon_1$ \\\\\n  \\hline\n  \\end{tabular}\n\\end{equation}\nthereby introducing a new space of pairs, that will be referred to as 1234.\nThe way this space is introduced is consistent with the pairing (a) in Eq.~(\\ref{energy-matching}) shown in Fig.~\\ref{F:AGKL31}(a). The leading contribution to mesoscopic fluctuations from this pairing will come from configurations when the energies in each pair nearly coincide, corresponding to $Q$ matrices which are block-diagonal in the space 1234 [see Eq.~(\\ref{Q-blocks}) below].\nThe other relevant contribution originates from the pairing (b) in Eq.~(\\ref{energy-matching}) shown in Fig.~\\ref{F:AGKL31}(b).\nIn principle, it can be also handled using the 1234 structure of Eq.~(\\ref{1234}), however the corresponding $Q$ matrices will have a cumbersome structure in this basis.\nTherefore in the study of the pairing (b) in Sec.~\\ref{SS:X8b} we will introduce 1234 space in a different way consistent with that pairing.\n\nThe resulting sigma model is formulated in terms of the $16\\times 16$ supermatrix field $Q({\\bf r})$ acting in the tensor product of the spaces $\\text{FB}\\otimes\\text{RA}\\otimes 1234$, where FB stands for the superspace. The matrix $Q$ can be written as $Q=T^{-1}\\Lambda T$, where $T$ spans the supersymmetric coset\n$\\text{U}(8|4,4)\/\\text{U}(4|4)\\times\\text{U}(4|4)$, and $\\Lambda=\\sigma_3^\\text{RA}$. The sigma-model action is given by (we adopt the fermion-dominated notation of Ref.~\\cite{Efetov-book})\n\\begin{equation}\n  S[Q]\n  =\n  \\frac{\\pi\\nu}4\\int d{\\bf r}\n  \\mathop{\\rm str} \\bigl[ D\\left( \\nabla Q({\\bf r})\\right)^2+4i\\hat E Q ({\\bf r}) \\bigr],\n\\label{action}\n\\end{equation}\nwhere $\\hat E$ is the diagonal matrix made of the energy arguments [with widths, according to Eq.~(\\ref{width})] of the corresponding Green functions:\n\\begin{equation}\n\\label{E8-def}\n  \\hat E\n  =\n  \\mathop{\\rm diag}\n\\bigl\\{\n\\varepsilon_+,\n(\\varepsilon-\\omega_2)_+, (\\varepsilon_1-\\omega_1)_+, (\\varepsilon_2)_+,\n\\varepsilon_-,\n(\\varepsilon-\\omega_1)_-, (\\varepsilon_2-\\omega_2)_-,(\\varepsilon_1)_-\n\\bigr\\}\\otimes 1_{\\text{FB}}.\n\\end{equation}\n\nIn the sigma-model language, the average product of eight Green functions in Eq.~(\\ref{GGGG}) transforms to\n\\begin{equation}\n  K=(\\pi \\nu)^8\\int Q^{\\text{AR}}_{21}({\\bf r}_1)Q^{\\text{RA}}_{12}({\\bf r}_2)Q^{\\text{RA}}_{21}({\\bf r}_5)Q^{\\text{AR}}_{12}({\\bf r}_6) Q^{\\text{AR}}_{43}({\\bf r}_3)Q^{\\text{RA}}_{34}({\\bf r}_4)Q^{\\text{RA}}_{43}({\\bf r}_7)Q^{\\text{AR}}_{34}({\\bf r}_8) e^{-S[Q]}DQ,\n\\label{Qcorr}\n\\end{equation}\nwhere all the elements of $Q$ matrices in the preexponent are taken from the BB sector (omitted for brevity). Equation (\\ref{Qcorr}) holds only in the limit $F\\to0$, when the interaction range is much larger than the Fermi wave length, and additional terms in the preexponent are suppressed by Friedel oscillations. In the general case discussed in Sec.~\\ref{S:mesofluct-rs}, Eq.~(\\ref{Qcorr}) should be replaced by Eq.~(\\ref{Qcorr-rs}).\n\n\n\n\\subsection{Integration over fast modes}\n\\label{fastaveraging}\n\n\nBesides eight Green functions, the block $X_4X_4'$ given by Eq.~(\\ref{X8}) contains four SSI lines, each carrying a non-zero momentum. Therefore in calculating $K$ in Eq.~(\\ref{Qcorr}) we should (i) allow fast ($q\\neq0$) diffuson modes to make $\\corr{X_4X_4'}$ nonzero, (ii) minimize their number as every fast mode brings a small factor of $\\Delta\/E_{\\text{Th}}$, and (iii) be able to handle the resulting zero-dimensional integral non-perturbatively in order to resolve discrete levels. This task can be accomplished following the strategy of Ref.~\\cite{KM}, where non-universal corrections to the random-matrix level statistics were calculated beyond the zero-dimensional limit.\nFor this purpose we write\n\\begin{equation}\n  Q({\\bf r}) = T^{-1} Q'({\\bf r}) T,\n\\label{param}\n\\end{equation}\nwhere the matrix $T$ is spatially uniform, and $Q'({\\bf r})$ describes all fast modes with non-zero momenta.\nSince the latter are to be accounted perturbatively, we expand $Q'({\\bf r})$ near the origin:\n\\begin{equation}\n  Q'({\\bf r})=\\Lambda\\left[1+W({\\bf r})+W^2({\\bf r})\/2+\\ldots\\right], \\qquad \\{W({\\bf r}),\\Lambda\\}=0.\n\\label{Q'-W}\n\\end{equation}\n\nTo get the leading contribution to $\\corr{X_4X_4'}$ we extract one fast $W$ from each of the eight $Q$ matrices in the preexponent of Eq.~(\\ref{Qcorr}), and average it with the Gaussian action\n\\begin{equation}\n  S^{(2)}[W] = - \\frac{\\pi\\nu D}4 \\int d{\\bf r}\\, \\mathop{\\rm str}[\\nabla W({\\bf r})]^2 ,\n\\end{equation}\ncoming from the gradient term of Eq.~(\\ref{action}).\nUsing Wick's theorem, the correlator of eight $W$ fields can be expressed as the sum of all possible products of four pairwise correlators. The latter can be easily calculated with the help of the following contraction rule valid for arbitrary matrices $P$ and $R$:\n\\begin{equation}\n  \\langle \\mathop{\\rm str} PW({\\bf r}) \\mathop{\\rm str} RW({\\bf r}') \\rangle_W\n  =\n  D({\\bf r},{\\bf r}')\\mathop{\\rm str}(P\\Lambda R\\Lambda -PR),\n\\qquad\n  D({\\bf r},{\\bf r}')=\\frac{\\corr{{\\bf r}|(-\\nabla^2)^{-1}|{\\bf r}'}}{\\pi\\nu D} .\n\\label{contruction rules}\n\\end{equation}\nAccording to Wick's theorem, averaging over fast modes generates $7!!=105$ different terms in the expression for $K$.\nNot all of them are equally important. To pick up the relevant terms one should understand how to perform further integration over the zero mode. This procedure will be discussed in Sec.~\\ref{slowaveraging}, and in Sec.~\\ref{SS:further} we will proceed with the derivation based on Eq.~(\\ref{contruction rules}).\n\n\n\n\n\\subsection{Block structure of the zero-dimensional sigma model and energy pairs}\n\\label{slowaveraging}\n\nHaving integrated out fast diffusive modes, we end up with the effective zero-dimensional sigma model for the $16 \\times 16$ supermatrix $Q=T^{-1}\\Lambda T$ with the action\n\\begin{equation}\n\\label{action0}\n  S_0[Q]=\\frac{\\pi i }{\\Delta}\\mathop{\\rm str} \\hat E Q.\n\\end{equation}\nOwing to a large size of the matrix $Q$, this is still a complicated theory. Fortunately, we do not need its exact solution since, as discussed in Sec.~\\ref{S:MF}, the most singular contribution to $\\ccorr{\\gamma^2(\\varepsilon,T)}$ comes from pairwise coinciding energies, see Eq.~(\\ref{energy-matching}). Since we are interested in the limit $(\\varepsilon,T)\\gg\\Delta$, energy difference between pairs is typically large, $|\\varepsilon_i-\\varepsilon_j|\\gg\\Delta$, and hence correlations between different pairs can be neglected (configurations with $|\\varepsilon_i-\\varepsilon_j|\\lesssim\\Delta$ which require non-perturbative treatment exist but their weight is small). On the other hand, energies from the same pair match with the accuracy of $\\gamma$ and should be treated non-perturbatively like in Sec.~\\ref{paircorr}. In the language of the zero-dimensional sigma model, large energy difference between pairs suppresses degrees of freedom in the supermatrix $Q$ which are off-diagonal in the space of pairs. As a result, $Q$ becomes block-diagonal in the space of pairs, and the full theory splits into a product of four standard Efetov's supermatrix sigma models for $\\corr{G^\\text{R}G^\\text{A}}$, see Eq.~(\\ref{S-splitting}) below.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=1.0\\textwidth]{65textcropped.pdf}}\n\\caption{\nTwo types of arranging eight Green functions in\nthe product\n$X_4X_4'$ [Eq.~(\\ref{X8})] into four pairs that produce the most singular contributions $\\corr{X_4X_4'}_\\text{sing}^{(a)}$ and $\\corr{X_4X_4'}_\\text{sing}^{(b)}$ in Eq.~(\\ref{X8-sing-reg}).\nGreen functions from the same pair are marked by a dotted line. The pairings (a) and (b) correspond to the two ways to match energies in Eq.~(\\ref{energy-matching}). All energy and coordinate indices in (b) are the same as in (a).}\n\\label{F:AGKL31}\n\\end{figure}\n\nThe next step is to understand what are the possible ways to group eight Green functions into four pairs to maximize their contribution to fluctuations of $\\gamma(\\varepsilon,T)$. First of all, it is clear that the Green functions with the energies $\\varepsilon$ always need to be in a pair, as they immediately produce the smeared delta function of zero argument, $\\delta_\\gamma(0)=1\/\\pi\\gamma$ [cf.\\ Eq.~(\\ref{delta_gamma})]. It is this large factor that compensates the analogous contribution from $\\corr{G^\\text{R}_\\varepsilon G^\\text{A}_\\varepsilon}$ on the left-hand side of Eq.~(\\ref{corr}). Our aim then is to identify those pairings that may introduce an additional large factor of $\\delta_\\gamma(0)$ after integration over all intermediate energies. Such a situation can be realized only if coincidence of the energies in two pairs automatically implies coincidence of the energies in the third pair. That can be achieved only in two physically relevant cases, (a) and (b), listed in Eq.~(\\ref{energy-matching}) and shown diagrammatically in Fig.~\\ref{F:AGKL31}\n(there exist three other unphysical pairings which should be discarded as demonstrated in \\ref{A:spurious}). We emphasize that possible choices of pairing are dictated by the energy arguments of electron Green functions and are not specific for the particular arrangement of $G^\\text{R}$ and $G^\\text{A}$. The only  natural requirement is that each pair contains one retarded and one advanced Green function (otherwise, there is no correlations within a pair at all).\n\n\n\nAccording to this general logic, each diagram for $\\corr{X_4X_4'}$, $\\corr{X_4Y_4'}$ and $\\corr{Y_4Y_4'}$ can be written in the form\n\\begin{equation}\n  \\corr{A} = \\corr{A}_\\text{sing}^{(a)} + \\corr{A}_\\text{sing}^{(b)} + \\corr{A}_\\text{reg} ,\n\\label{X8-sing-reg}\n\\end{equation}\nwhere $\\corr{\\dots}_\\text{sing}^{(a)}$ and $\\corr{\\dots}_\\text{sing}^{(b)}$ denote singular (in the limit $\\gamma\\to0$) contributions from the pairings (a) and (b).\nWe proceed with calculating these singular contributions for the term $K$ [a particular element of $\\corr{X_4X_4'}$ defined in Eq.~(\\ref{GGGG})] from the zero-dimensional sigma model (\\ref{action0}).\nThe introduction of the 1234 space in Eq.~(\\ref{1234}) is consistent with the pairing (a): in the 1234 basis the corresponding matrix $Q$ becomes diagonal. On the contrary, the pairing (b) is described by non-diagonal matrices in the 1234 space. To calculate $K_\\text{sing}^{(b)}$ we find it more convenient to change the basis and introduce a new $1234'$ space consistent with the pairing (b), in which the matrix $Q$ is diagonal.\nThen the calculation of $K_\\text{sing}^{(b)}$ becomes completely analogous to the calculation of $K_\\text{sing}^{(a)}$.\nTo illustrate the technique, we focus on the contribution of $K$ due to the pairing (a) [the contribution of the pairing (b) is discussed in Sec.~\\ref{SS:X8b}].\n\n\n\n\n\n\n\n\n\n\\subsection{Singular contribution of the pairing (a)}\n\\label{SS:further}\n\nNow we are ready to proceed with the calculation started in Sec.~\\ref{fastaveraging}.\nIn order to apply the contraction rules (\\ref{contruction rules}) for averaging the preexponent in Eq.~(\\ref{Qcorr}) over fast modes, it is convenient to introduce projectors onto different sectors of the $Q$ manifold.\nBy definition, we write $Q^{ab}_{ij}({\\bf r})=\\mathop{\\rm str} P^{ab}_{ij}Q({\\bf r})$, where $a$ and $b$ refer to the RA space, whereas $i$ and $j$ refer to the 1234 space (BB sector is also implied). For example, the projector $P^{\\text{AR}}_{21}$ is given by\n\\begin{equation}\n  P^{\\text{AR}}_{21} = \\begin{pmatrix} 0&P^{\\text{AR}}&0&0\\\\0&0&0&0\\\\0&0&0&0\\\\0&0&0&0 \\end{pmatrix}_{1234},\n\\quad\n  P^{\\text{AR}} = \\begin{pmatrix} 0&P_\\text{BB}\\\\0&0 \\end{pmatrix}_{\\text{RA}},\n\\quad\n  P_\\text{BB} = \\begin{pmatrix} 0&0\\\\0&1 \\end{pmatrix}_{\\text{FB}}.\n\\end{equation}\n\nAs we discussed in Sec.~\\ref{slowaveraging}, the singular contribution $K_\\text{sing}^{(a)}$ originates from the matrices $Q$ which are diagonal in $1234$ space. Such a structure of the $Q$ matrix guarantees that only four out of $7!!=105$ terms appearing after averaging over fast modes [see Eq.~(\\ref{param})] turn out to be non-zero:\n\\begin{align}\n  K_\\text{sing}^{(a)}\n  =\n  (\\pi\\nu)^8\n  \\bigl\\langle\\bigl\\{\n    D({\\bf r}_1,{\\bf r}_2) & D({\\bf r}_5,{\\bf r}_6)\\mathop{\\rm str} \\left(P^{\\text{AR}}_{21}QP^{\\text{RA}}_{12}Q-P^{\\text{AR}}_{21} P^{\\text{RA}}_{12}\\right) \\mathop{\\rm str} \\left(P^{\\text{RA}}_{21}QP^{\\text{AR}}_{12}Q-P^{\\text{RA}}_{21} P^{\\text{AR}}_{12}\\right)\n\\nonumber\n\\\\{}\n  + D({\\bf r}_1,{\\bf r}_6) & D({\\bf r}_2,{\\bf r}_5) \\mathop{\\rm str} \\left(P^{\\text{AR}}_{21}QP^{\\text{AR}}_{12}Q-P^{\\text{AR}}_{21} P^{\\text{AR}}_{12}\\right) \\mathop{\\rm str} \\left(P^{\\text{RA}}_{21}QP^{\\text{RA}}_{12}Q- P^{\\text{RA}}_{21} P^{\\text{RA}}_{12}\\right)\\bigr\\}\n\\nonumber\n\\\\{}\n\\times \\bigl\\{ D({\\bf r}_3,{\\bf r}_4) & D({\\bf r}_7,{\\bf r}_8)\\mathop{\\rm str} \\left(P^{\\text{AR}}_{43}QP^{\\text{RA}}_{34}Q-P^{\\text{AR}}_{43} P^{\\text{RA}}_{34}\\right) \\mathop{\\rm str} \\left(P^{\\text{RA}}_{43}QP^{\\text{AR}}_{34}Q-P^{\\text{RA}}_{43} P^{\\text{AR}}_{34}\\right)\n\\nonumber\n\\\\{}\n+ D({\\bf r}_3,{\\bf r}_8) & D({\\bf r}_4,{\\bf r}_7) \\mathop{\\rm str} \\left(P^{\\text{AR}}_{43}QP^{\\text{AR}}_{34}Q-P^{\\text{AR}}_{43} P^{\\text{AR}}_{34}\\right) \\mathop{\\rm str} \\left(P^{\\text{RA}}_{43}QP^{\\text{RA}}_{34}Q- P^{\\text{RA}}_{43} P^{\\text{RA}}_{34}\\right)\\bigr\\}\\bigr\\rangle\n,\n\\label{Ka}\n\\end{align}\nwhere $\\langle \\ldots \\rangle$ denotes averaging over the $16\\times16$ zero-dimensional sigma-model manifold with the action (\\ref{action0}).\n\nSince $Q$ is diagonal in the 1234 space, it can be represented as\n\\begin{equation}\n\\label{Q-blocks}\n  Q = \\mathop{\\rm diag} \\{Q_1, Q_2, Q_3, Q_4\\} ,\n\\end{equation}\nwhere $Q_j$ are independent $4\\times4$ supermatrices in the $\\text{RA}\\otimes\\text{FB}$ space spanning the standard supermanifold of the sigma model for $\\corr{G^\\text{R}G^\\text{A}}$.\nThe action (\\ref{action0}) then splits into a sum of four separate Efetov's sigma-model actions:\n\\begin{equation}\n\\label{S-splitting}\n  S_0[Q]\n  =\n  \\sum_{j=1}^4 S_{0j}[Q_j]\n  =\n  \\frac{\\pi i}{\\Delta} \\sum_{j=1}^4 \\mathop{\\rm str} \\hat E_j Q_j,\n\\end{equation}\nwhere the elements of the diagonal matrix $\\hat E_j$ are taken from the corresponding sector of $\\hat E$ defined in Eq.~(\\ref{E8-def}). Now averaging over all four $Q_j$ is performed independently and can be done exactly in the standard way \\cite{Efetov-book}. To handle the supertrace structure in Eq.~(\\ref{Ka}), it is convenient to use the zero-dimensional contraction rules (\\ref{contr-rules-0D}) derived in \\ref{slowcontrrules}. The result can be expressed in a compact form in terms of the quantities\n\\begin{equation}\n  X_j \\equiv X\\left(-i\\pi\\Omega_j\/\\Delta \\right) ,\n\\qquad\n  Z_j \\equiv Z\\left(-i\\pi\\Omega_j\/\\Delta \\right) , \\label{XjZj}\n\\end{equation}\nwhere the functions $X(a)$ and $Y(a)$ are given by Eqs.~(\\ref{X}) and (\\ref{Z}):\n\\begin{equation}\n\\label{XZ}\n  X(a)=\\frac{1-e^{-2a}}{2a^2} ,\n\\qquad\n  Z(a)=\\frac 1a ,\n\\end{equation}\nand $\\Omega_j$ is the energy difference between the arguments of $G^\\text{R}$ and $G^\\text{A}$ in the corresponding pair [cf.\\ Eq.~(\\ref{1234})]:\n\\begin{gather}\n\\label{Omega1234}\n  \\Omega_1 = \\varepsilon_+ - \\varepsilon_- ,\n  \\!\\!\\qquad\n  \\Omega_2 = (\\varepsilon-\\omega_2)_+ - (\\varepsilon-\\omega_1)_- ,\n  \\!\\!\\qquad\n  \\Omega_3 = (\\varepsilon_1-\\omega_1)_+ - (\\varepsilon_2-\\omega_2)_-  ,\n  \\!\\!\\qquad\n  \\Omega_4 = (\\varepsilon_2)_+ - (\\varepsilon_1)_- .\n\\end{gather}\n\nTo demonstrate the technique, consider for example one of the four terms in Eq.~(\\ref{Ka}):\n\\begin{equation}\n  K_\\text{sing}^{(a)}\n  =\n  (\\pi\\nu)^8 D({\\bf r}_1,{\\bf r}_2)D({\\bf r}_5,{\\bf r}_6) D({\\bf r}_3,{\\bf r}_4)D({\\bf r}_7,{\\bf r}_8) L_1 + \\dots\n\\label{Ka2}\n\\end{equation}\nSubstituting $Q$ in the form (\\ref{Q-blocks}) and tracing over the 1234 space we obtain\n\\begin{align}\n  L_1\n  =\n  \\bigl\\langle & \\mathop{\\rm str} (P^{\\text{AR}}Q_2P^{\\text{RA}}Q_1-P^{\\text{AR}}_{21} P^{\\text{RA}}_{12}) \\mathop{\\rm str} (P^{\\text{RA}}Q_2P^{\\text{AR}}Q_1-P^{\\text{RA}}_{21} P^{\\text{AR}}_{12})\n\\nonumber\n\\\\{} \\times\n  & \\mathop{\\rm str} (P^{\\text{AR}}Q_4P^{\\text{RA}}Q_3-P^{\\text{AR}}_{43} P^{\\text{RA}}_{34}) \\mathop{\\rm str} (P^{\\text{RA}}Q_4P^{\\text{AR}}Q_3-P^{\\text{RA}}_{43} P^{\\text{AR}}_{34})\\bigr\\rangle.\n\\label{K11}\n\\end{align}\nSequentially applying the contraction rules (\\ref{contr-rules-0D}) for averaging over all $Q_j$, we arrive at the exact non-perturbative expression\n\\begin{equation}\n  L_1=(4+2X_1+2X_2+4X_1X_2)(4+2X_3+2X_4+4X_3X_4) .\n\\end{equation}\nContributions of the three other terms in Eq.~(\\ref{Ka2}) are calculated analogously, and we obtain finally the singular part of $K$ due to pairing (a):\n\\begin{align}\n  K_\\text{sing}^{(a)}\n  =(\\pi\\nu)^8\n  & \\left\\{D({\\bf r}_1,{\\bf r}_2)D({\\bf r}_5,{\\bf r}_6)(4+2X_1+2X_2+4X_1X_2)+ 4D({\\bf r}_1,{\\bf r}_6)D({\\bf r}_2,{\\bf r}_5)Z_1Z_2\\right\\}\n\\nonumber\n\\\\{}\n  \\times\n  & \\left\\{ D({\\bf r}_3,{\\bf r}_4)D({\\bf r}_7,{\\bf r}_8)(4+2X_3+2X_4+4X_3X_4)+4D({\\bf r}_3,{\\bf r}_8)D({\\bf r}_4,{\\bf r}_7)Z_3Z_4\\right\\}.\n\\label{K1}\n\\end{align}\n\nThough the singular contribution due to the energy pairing (a) may originate only from Eq.~(\\ref{K1}), not all terms in this equation do actually produce the singular contribution. We analyze this expression and perform the final step of calculation in the next Section.\n\n\n\n\n\\section{Mesoscopic fluctuations at $F=0$}\n\\label{S:final}\n\n\\subsection{General recipe}\n\\label{qual}\n\nThe most singular contribution from Eq.~(\\ref{K1}) originates from the terms which contain the maximal number (four) of $X_j$ and $Z_j$. We have already explained that this choice is dictated by the necessity to obtain two smeared delta-functions (\\ref{delta_gamma}) with zero argument, contributing a large factor $\\Delta\/\\gamma$ each.\nThis is also the reason why we have to choose each pair in the $1234$ space to consist of one $G^R$ and one $G^\\text{A}$: large factors $X_j$ and $Z_j$ appear as a result of averaging $\\langle G^\\text{R} G^\\text{A} \\rangle$ over the zero-dimensional diffusive model.\n\nIn calculating the energy integrals in Eq.~(\\ref{corr}), the important contribution comes from the vicinity of the poles of $X_j$ and $Z_j$. This observation allows us to work in the limit $\\Omega_j \\sim \\gamma \\ll\\Delta$ [with $\\Omega_j$ introduced in Eq.~(\\ref{Omega1234}) being the energy mismatch within a pair] and use the leading-order asymptotics of Eqs.~(\\ref{XZ}):\n\\begin{equation}\n  X_j \\approx Z_j = \\frac{i\\Delta}{\\pi\\Omega_j} .\n\\end{equation}\nThe chosen sequence of $G^\\text{R}$ and $G^\\text{A}$ in the correlator $K$ [Eq.~(\\ref{GGGG})] also ensures that the product $1\/\\Omega_2\\Omega_3\\Omega_4$ has poles both in the upper and lower half-planes of the energy variables $\\varepsilon_1$, $\\varepsilon_2$, $\\omega_1$ and $\\omega_2$. Hence integration over intermediate energies does not vanish and indeed produces the desired delta-function-like contribution.\n\nNow the recipe for extracting the leading singular term from Eq.~(\\ref{K1}) and  similar expressions for other diagrams can be formulated as follows.\nOne should substitute all $X_j$ by $Z_j$ and take the term proportional to ${\\cal Z} = Z_1Z_2Z_3Z_4$.\nThen the coefficient ${\\cal Z}$ should be replaced by\n\\begin{equation}\n  {\\cal Z}\n  \\longrightarrow\n  \\frac{4\\Delta^4}{\\pi^2 \\gamma(\\varepsilon)}\n  \\frac{\\delta(\\varepsilon_1-\\varepsilon_2)\\delta(\\omega_1-\\omega_2)}{\\gamma(\\varepsilon-\\omega_1)+\\gamma(\\varepsilon_1)+\\gamma(\\varepsilon_1-\\omega_1)} ,\n\\label{XXX}\n\\end{equation}\nwhere the factor $\\Delta\/\\pi\\gamma(\\varepsilon)$ originates from $Z_1$ and will be canceled by a similar factor from the left-hand side of Eq.~(\\ref{corr}) (see Sec.~\\ref{paircorr}), while the coefficient in front of the delta functions can be easily obtained by calculating, e.g., $\\int Z_2Z_3Z_4\\,d\\varepsilon_2d\\omega_2$.\nThe last step is to integrate four diffuson propagators in Eq.~(\\ref{K1}) over ${\\bf r}_i$ [see Eq.~(\\ref{corr})].\nIn doing that, $\\delta_\\kappa({\\bf r}-{\\bf r}')$ in the expressions for\n$X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$\ncan be considered as the usual zero-range delta function.\nThen depending on the term under consideration, diffusons combine either into $(\\mathop{\\rm tr} D^2)^2$ or  $\\mathop{\\rm tr} D^4$. These traces evaluated in the Fourier space produce either $1\/E_2^4$ or $1\/E_4^4$, where $E_n\\simE_{\\text{Th}}$ are defined in Eq.~(\\ref{En}).\nAs a result, the contribution to mesoscopic fluctuations of the inelastic rate can be written in the form\n\\begin{equation}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}_{A}^{(i)}\n  =\n  \\frac{{\\lambda^4}\\Delta^5}{4\\pi^3}\n  c_{A}^{(i)}\n  \\int d\\varepsilon_1 d\\omega_1\n  \\frac{({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega_1})^2({\\cal B}_{\\omega_1}+{\\cal F}_{\\varepsilon-\\omega_1})^2}\n  {\\gamma_0(\\varepsilon-\\omega_1,T)+\\gamma_0(\\varepsilon_1,T)+\\gamma_0(\\varepsilon_1-\\omega_1,T)}\n  ,\n\\label{gamma-c}\n\\end{equation}\nwhere $A$ is a particular diagram or a set of diagrams considered, and $i$ labels the type of the energy pairing (a or b).\n\n\n\\subsection{Contribution from $X_4X_4'$ due to the energy pairing (a)}\n\\label{SS:X8a}\n\nApplying this general recipe to expression (\\ref{K1}) for $K_\\text{sing}^{(a)}$, we first reduce it to the form\n\\begin{equation}\n  K_\\text{sing}^{(a)}\n  =16(\\pi\\nu)^8 \\, {\\cal Z}\n  \\left\\{D({\\bf r}_1,{\\bf r}_2)D({\\bf r}_5,{\\bf r}_6) + D({\\bf r}_1,{\\bf r}_6)D({\\bf r}_2,{\\bf r}_5)\\right\\}\n  \\left\\{ D({\\bf r}_3,{\\bf r}_4)D({\\bf r}_7,{\\bf r}_8) + D({\\bf r}_3,{\\bf r}_8)D({\\bf r}_4,{\\bf r}_7)\\right\\} ,\n\\label{K1-Z}\n\\end{equation}\nand then, tracing the diffuson propagators, obtain the coefficient $c_{K}^{(a)}$ in Eq.~(\\ref{gamma-c}):\n\\begin{equation}\n  c_{K}^{(a)}\n  =\n  \\frac{1}{2N_s^2}\\left(\\frac 1{E_2^4}+\\frac 1{E_4^4}\\right) .\n\\label{gamma1'}\n\\end{equation}\n\nTo complete the analysis of the contribution from $X_4X_4'$ due type-(a) energy pairing, one has to consider other arrangements of $G^\\text{R}$ and $G^\\text{A}$. The requirement of having one $G^\\text{R}$ and one $G^\\text{A}$ in each pair limits the number of various possibilities to $2^4=16$. However, not all of them should be taken into account. Consider, for example, the correlator\n$$\nK'=\\langle G^\\text{R}_{\\varepsilon}({\\bf r}_2,{\\bf r}_1) G^\\text{A}_{\\varepsilon}({\\bf r}_6,{\\bf r}_5)  G^\\text{R}_{\\varepsilon-\\omega_2}({\\bf r}_5,{\\bf r}_6) G^\\text{A}_{\\varepsilon-\\omega_1}({\\bf r}_1,{\\bf r}_2)\nG^\\text{R}_{\\varepsilon_1-\\omega_1}({\\bf r}_4,{\\bf r}_3) G^\\text{A}_{\\varepsilon_2-\\omega_2}({\\bf r}_8,{\\bf r}_7) G^\\text{A}_{\\varepsilon_2}({\\bf r}_7,{\\bf r}_8) G^\\text{R}_{\\varepsilon_1}({\\bf r}_3,{\\bf r}_4)\\rangle ,\n$$\nwhich differs from $K$ [Eq.~(\\ref{GGGG})] by changing R${}\\leftrightarrow{}$A in the last two Green functions.\nThe expression analogous to Eq.~(\\ref{K1}) will contain  $X_4^*$ instead of $X_4$ and $Z_4^*$ instead of $Z_4$, which renders the poles of both $X_3X_4^*$ and $Z_3Z_4^*$  lying in the upper half-plane of $\\varepsilon_2$. Therefore, the contribution of this term is non-singular and should be disregarded.\nA simple analysis demonstrates that there are only four  possibilities to arrange $G^\\text{R}$ and $G^\\text{A}$\nin $X_4X_4'$:\n\\begin{equation}\n\\label{RA-4ways}\n  \\text{RARARARA},\n\\quad\n  \\text{ARRARARA},\n\\quad\n  \\text{RAARARAR},\n\\quad\n  \\text{ARARARAR},\n\\end{equation}\nwhere all energy and coordinate indices follow Eq.~(\\ref{GGGG}). Each choice gives the same contribution given by Eq.~(\\ref{gamma1'}). As a result, the total contribution of the term $X_4X_4'$ due to type-(a) energy pairing (shown in Fig.~\\ref{F:AGKL31}a) is described by the coefficient\n\\begin{equation}\n  c_{XX}^{(a)}\n  =\n  \\frac{2}{N_s^2}\\left(\\frac 1{E_2^4}+\\frac 1{E_4^4}\\right) .\n\\label{gamma11}\n\\end{equation}\n\n\n\\subsection{Contribution from $X_4X_4'$ due to the energy pairing (b)}\n\\label{SS:X8b}\n\nFollowing the same line we can analyze the singular part of $K$ due to pairing (b) shown in Fig.~\\ref{F:AGKL31}(b). Instead of Eq.~(\\ref{K1-Z}) we obtain:\n\\begin{equation}\n  K_\\text{sing}^{(b)}\n  =\n  16 (\\pi\\nu)^8 \\, {\\cal Z} \\,\n  D({\\bf r}_1,{\\bf r}_2) D({\\bf r}_3,{\\bf r}_4)\n  D({\\bf r}_5,{\\bf r}_6) D({\\bf r}_7,{\\bf r}_8) .\n\\label{K2-Z}\n\\end{equation}\nFollowing the procedure described in Sec.~\\ref{qual} and utilizing the same four possibilities (\\ref{RA-4ways}) to arrange $G^\\text{R}$ and $G^\\text{A}$, we arrive at\n\\begin{equation}\n  c_{XX}^{(b)}\n  =\n  \\frac{1}{N_s^2E_2^4} .\n\\label{gamma12}\n\\end{equation}\n\n\n\n\n\\subsection{Contributions from $X_4Y_4'$ and $Y_4Y_4'$}\n\\label{Seconddiagram}\n\n\nIt is left to discuss the other two terms, $X_4Y_4'$ and $Y_4Y_4'$, in Eq.~(\\ref{corr}). Following the same steps we obtain that the contributions of the cross term $\\corr{X_4Y_4'}$ from both the pairings (a) and (b) vanish after integration over fast diffusive modes due to the diagonal structure of the $Q$ matrix in the 1234 space, while in calculating $\\corr{Y_4Y_4'}$ only the pairing (b) should be taken into account for the same reason.\n\nConsider, e.~g., a particular realization of $G^\\text{R}$ and $G^\\text{A}$ from $\\corr{Y_4Y_4'}$ [compare with $K$ defined in Eq.~(\\ref{GGGG})]:%\n\\begin{equation}\n\\tilde K=\\langle G^\\text{R}_{\\varepsilon}({\\bf r}_4,{\\bf r}_1) G^\\text{A}_{\\varepsilon_1}({\\bf r}_1,{\\bf r}_2) G^\\text{R}_{\\varepsilon_1-\\omega_1}({\\bf r}_2,{\\bf r}_3) G^\\text{A}_{\\varepsilon-\\omega_1}({\\bf r}_3,{\\bf r}_4) G^\\text{A}_{\\varepsilon}({\\bf r}_8,{\\bf r}_5) G^\\text{R}_{\\varepsilon_2}({\\bf r}_5,{\\bf r}_6) G^\\text{A}_{\\varepsilon_2-\\omega_2}({\\bf r}_6,{\\bf r}_7) G^\\text{R}_{\\varepsilon-\\omega_2}({\\bf r}_7,{\\bf r}_8) \\rangle .\n\\end{equation}\nIts singular contribution due to the pairing (b) is given by\n\\begin{equation}\n  \\corr{\\tilde K}_\\text{sing}^{(b)}\n  =\n  16 (\\pi\\nu)^8 \\, {\\cal Z} \\,\n  D({\\bf r}_1,{\\bf r}_2) D({\\bf r}_3,{\\bf r}_4)\n  D({\\bf r}_5,{\\bf r}_6) D({\\bf r}_7,{\\bf r}_8) ,\n\\end{equation}\nTracing the diffusons and taking into account four possibilities (\\ref{RA-4ways}), we get\n\\begin{equation}\n  c_{YY}^{\\text{(b)}}\n  =\n  \\frac{1}{N_s^4E_2^4} .\n\\label{gamma2}\n\\end{equation}\n\nFinally, adding the contributions (\\ref{gamma11}), (\\ref{gamma12}) and (\\ref{gamma2}) we obtain the final expression for mesoscopic fluctuations of the inelastic rate in the limit $F\\to0$:\n\\begin{equation}\n  \\ccorr{\\gamma^2(\\varepsilon,T)}\n  =\n  \\frac{\\lambda^4\\Delta^5}{4\\pi^3N_s^4}\n  \\left(\\frac{3N_s^2+1}{E_2^4}+\\frac{2N_s^2}{E_4^4}\\right)\n  \\int d\\varepsilon_1 d\\omega_1\n  \\frac{({\\cal F}_{\\varepsilon_1}-{\\cal F}_{\\varepsilon_1-\\omega_1})^2({\\cal B}_{\\omega_1}+{\\cal F}_{\\varepsilon-\\omega_1})^2}\n  {\\gamma_0(\\varepsilon-\\omega_1,T)+\\gamma_0(\\varepsilon_1,T)+\\gamma_0(\\varepsilon_1-\\omega_1,T)}\n  .\n\\label{gamma-total-f0}\n\\end{equation}\n\n\n\n\\section{Mesoscopic fluctuations at arbitrary $F$}\n\\label{S:mesofluct-rs}\n\n\\subsection{Account for a finite $F$ in the sigma-model language}\n\nIn this Section we generalize the result (\\ref{gamma-total-f0}) to the case of an arbitrary interaction parameter $F$ and derive the general expression (\\ref{subfinal}).\nFirst, we explain the main idea with an example of the type-(a) energy pairing in the diagram for $X_4X_4'$, and then apply it to the other diagrams ($X_4Y_4'$ and $Y_4Y_4'$) and energy pairing (b).\n\nWhen the range of the SSI (\\ref{V(r)}) characterized by the function $\\delta_\\kappa({\\bf r})$ becomes comparable to the Fermi wave length (i.~e., $F$ becomes non-negligible), a simple sigma-model expression (\\ref{Qcorr}) for the correlator $K$ breaks down. To modify it one has to turn back to the intermediate step in the derivation of the sigma model, prior to the final integration over the 16-component supervector $\\psi$ used to represent Green functions in the functional form.\nAt this stage, the correlator $K$ [Eq.~(\\ref{GGGG})] is written as\n\\begin{align}\n  K\n  =\n  \\int\n  \\langle\n& \\psi_\\text{R1}({\\bf r}_2) \\psi^*_\\text{R1}({\\bf r}_1)\n  \\psi_\\text{A1}({\\bf r}_6) \\psi^*_\\text{A1}({\\bf r}_5)\n  \\psi_\\text{R2}({\\bf r}_5) \\psi^*_\\text{R2}({\\bf r}_6)\n  \\psi_\\text{A2}({\\bf r}_1) \\psi^*_\\text{A2}({\\bf r}_2)\n\\nonumber\n\\\\{} \\times{}\n& \\psi_\\text{R3}({\\bf r}_4) \\psi^*_\\text{R3}({\\bf r}_3)\n  \\psi_\\text{A3}({\\bf r}_8) \\psi^*_\\text{A4}({\\bf r}_7)\n  \\psi_\\text{R4}({\\bf r}_7) \\psi^*_\\text{R4}({\\bf r}_8)\n  \\psi_\\text{A4}({\\bf r}_3) \\psi^*_\\text{A4}({\\bf r}_4)\n  \\rangle_{\\psi}\n  \\,\n  e^{-S[Q]} DQ ,\n\\label{psi16}\n\\end{align}\nwhere the bosonic components of the superfield $\\psi$ are implied.\nAveraging over $\\psi$ should be performed with the help of Wick's theorem with the correlation function\n\\begin{equation}\n  \\corr{\\psi({\\bf r})\\psi^*({\\bf r}')} = -i g({\\bf r},{\\bf r}') \\Lambda ,\n\\end{equation}\nwhere $g$ is the supermatrix Green function defined as\n\\begin{equation}\n  g({\\bf r},{\\bf r}')\n  =\n  \\langle\n    {\\bf r} | ( \\hat E - H_0 + iQ\/2\\tau )^{-1} | {\\bf r}'\n  \\rangle .\n\\end{equation}\n\nThe result of the $\\psi$-averaging is sensitive to the distance between the points ${\\bf r}_i$ controlled by the spatial range of the SSI propagators through the factors $\\delta_\\kappa({\\bf r}_i-{\\bf r}_j)$ in Eq.~(\\ref{X8}).\nSince the interaction is assumed to be short-range on the scale of the system size $L$, only correlations between $\\psi$'s coupled to the same SSI propagator should be taken into account. Therefore the $\\psi$-averaging in Eq.~(\\ref{psi16}) factorizes into four contributions corresponding to four SSI propagators in Eq.~(\\ref{X8}):\n\\begin{equation}\n\\label{K-KKKK}\n  K\n  =\n  \\int K^{(1,3)} K^{(2,4)} K^{(5,7)} K^{(6,8)} e^{-S[Q]} DQ .\n\\end{equation}\nConsider for example the group of four $\\psi$'s coupled by the interaction with $\\delta_\\kappa({\\bf r}_1-{\\bf r}_3)$:\n\\begin{equation}\n  K^{(1,3)}\n  =\n  \\langle\n  \\psi^*_\\text{R1}({\\bf r}_1)\n  \\psi_\\text{A2}({\\bf r}_1)\n  \\psi^*_\\text{R3}({\\bf r}_3)\n  \\psi_\\text{A4}({\\bf r}_3)\n  \\rangle_{\\psi}\n.\n\\label{psi4}\n\\end{equation}\nApplication of Wick's theorem generates two terms:\n\\begin{equation}\n  K^{(1,3)}\n  =\n  -\n  g_\\text{A2,R1}({\\bf r}_1,{\\bf r}_1)\n  g_\\text{A4,R3}({\\bf r}_3,{\\bf r}_3)\n  -\n  g_\\text{A2,R3}({\\bf r}_1,{\\bf r}_3)\n  g_\\text{A4,R1}({\\bf r}_3,{\\bf r}_1)\n.\n\\label{K13-gg}\n\\end{equation}\n\nThe first (diagonal) term in Eq.~(\\ref{K13-gg}) reduces to the $Q$ matrix due to the self-consistency equation $g({\\bf r},{\\bf r})=-i\\pi\\nu Q({\\bf r})$ \\cite{Efetov-book}, and then one immediately recovers Eq.~(\\ref{Qcorr}) used previously in the analysis of the $F=0$ case. In contrast, the second (off-diagonal) term in Eq.~(\\ref{K13-gg}) cannot be so easily expressed in terms of the $Q$ matrix. But here we can use the knowledge that at the later stage, in the process of averaging over fast modes (see Sec.~\\ref{fastaveraging}), each $Q$ will be expanded to the first power of $W$ [see Eqs.~(\\ref{param}), (\\ref{Q'-W})]. The linear-in-$W$ contribution to $g$ is given by:\n\\begin{equation}\n\\label{dg1}\n  \\delta g({\\bf r}_1,{\\bf r}_3)\n  =\n  - \\frac{i}{2\\tau}\n  \\int d{\\bf r}'\n  \\langle\n    {\\bf r}_1 | ( - H_0 + iQ\/2\\tau )^{-1} | {\\bf r}'\n  \\rangle\n  \\,\n  T^{-1} \\Lambda W({\\bf r}') T\n  \\,\n  \\langle\n    {\\bf r}' | ( - H_0 + iQ\/2\\tau )^{-1} | {\\bf r}_3\n  \\rangle ,\n\\end{equation}\nwhere we write $Q=T^{-1}\\Lambda T$, and neglect the term $\\hat E$ as we are working in the diffusive limit $\\max\\{\\varepsilon, T\\} \\tau \\ll 1$.\nSince $W$ anticommutes with $\\Lambda$, we can rewrite Eq.~(\\ref{dg1}) in the form\n\\begin{equation}\n\\label{dg2}\n  \\delta g({\\bf r}_1,{\\bf r}_3)\n  =\n  - \\frac{i}{2\\tau}\n  \\int d{\\bf r}'\n  \\,\n  T^{-1}\n  \\langle\n    {\\bf r}_1 | ( - H_0 + i\\Lambda\/2\\tau )^{-1} | {\\bf r}'\n  \\rangle\n  \\langle\n    {\\bf r}' | ( - H_0 - i\\Lambda\/2\\tau )^{-1} | {\\bf r}_3\n  \\rangle\n  \\Lambda W\n  T\n  =\n  - \\frac{i \\delta Q}{2\\tau}\n  (G^\\text{R}G^\\text{A})({\\bf r}_1-{\\bf r}_3)\n  ,\n\\end{equation}\nwhere $\\delta Q=T^{-1} \\Lambda W T$,\nand $G^\\text{R}G^\\text{A}$ is a scalar function given by Eq.~(\\ref{GRGA}).\n\nMultiplying the second term of Eq.~(\\ref{K13-gg}) by $\\delta_\\kappa({\\bf r}_1-{\\bf r}_3)$ and integrating over the difference ${\\bf r}_1-{\\bf r}_3$, we immediately recover the factor $F$ introduced in Eq.~(\\ref{f-I}).\nHence, Eq.~(\\ref{K13-gg}) can be written as\n\\begin{equation}\n  K^{(1,3)}\n  =\n  (\\pi\\nu)^2\n  \\left[\n  Q^{\\text{AR}}_{21}({\\bf r}_1) Q^{\\text{AR}}_{43}({\\bf r}_3)\n  + F\\,\n  Q^{\\text{AR}}_{23}({\\bf r}_1) Q^{\\text{AR}}_{41}({\\bf r}_3)\n  \\right] ,\n\\label{K13-QQ}\n\\end{equation}\nwhere with our accuracy we do not distinguish between ${\\bf r}_1$ and ${\\bf r}_3$.\nGenerally speaking, Eq.~(\\ref{K13-QQ}) is incorrect.\nWe write it in such a form for brevity, assuming that it will be further subject to the procedure of fast mode extraction described in Sec.~\\ref{fastaveraging}.\n\nPerforming the same analysis for other $K^{(i,j)}$ from Eq.~(\\ref{K-KKKK}), one concludes that for the purpose of calculation of mesoscopic fluctuations of the relaxation rate at $F\\neq0$, Eq.~(\\ref{Qcorr}) should be modified as\n\\begin{multline}\n  K\n  =\n  (\\pi \\nu)^8\\int\n  \\left[\n  Q^{\\text{AR}}_{21}({\\bf r}_1) Q^{\\text{AR}}_{43}({\\bf r}_3)\n  + F\\,\n  Q^{\\text{AR}}_{23}({\\bf r}_1) Q^{\\text{AR}}_{41}({\\bf r}_3)\n  \\right]\n  \\left[\n  Q^{\\text{RA}}_{12}({\\bf r}_2) Q^{\\text{RA}}_{34}({\\bf r}_4)\n  + F\\,\n  Q^{\\text{RA}}_{14}({\\bf r}_2) Q^{\\text{RA}}_{32}({\\bf r}_4)\n  \\right]\n\\\\{}\n  \\times\n  \\left[\n  Q^{\\text{RA}}_{21}({\\bf r}_5) Q^{\\text{RA}}_{43}({\\bf r}_7)\n  + F\\,\n  Q^{\\text{RA}}_{23}({\\bf r}_5) Q^{\\text{RA}}_{41}({\\bf r}_7)\n  \\right]\n  \\left[\n  Q^{\\text{AR}}_{12}({\\bf r}_6) Q^{\\text{AR}}_{34}({\\bf r}_8)\n  + F\\,\n  Q^{\\text{AR}}_{14}({\\bf r}_6) Q^{\\text{AR}}_{32}({\\bf r}_8)\n  \\right]\n  e^{-S[Q]}DQ,\n\\label{Qcorr-rs}\n\\end{multline}\nwhere the difference between the coordinates within the same bracket can be neglected.\nSimilar expressions originate in the analysis of other contributions from\n$X_4X_4'$, $X_4Y_4'$ and $Y_4Y_4'$.\n\n\n\n\\subsection{Contributions of different diagrams and energy pairings}\n\nTo find mesoscopic fluctuations at a finite $F$, one should use Eq.~(\\ref{Qcorr-rs}) as a starting point and perform all the steps outlined in Secs.~\\ref{S:nonpert} and \\ref{S:final}.\nThis is a routine procedure leading to the following results.\nIn the case $F\\neq0$, there are finite contributions from all the terms $X_4X_4'$, $X_4Y_4'$, and $Y_4Y_4'$, and from both energy pairings (a) and (b).\nThese six contributions are characterized by six coefficients $c_A^{(i)}$ in Eq.~(\\ref{gamma-c}). After some algebra, we obtain (the coefficients $c_{XY}^{(i)}$ already contain the factor 2 accounting for the mirror term $Y_4X_4'$):\n\\begin{gather}\n  c_{XX}^{(a)} =\n  \\frac{2(1+F^2)^2}{N_s^2 E_2^4} + \\frac{2(1+F^4)}{N_s^2 E_4^4} ,\n\\qquad\n  c_{XX}^{(b)} =\n  \\frac{1+6F^2+F^4}{N_s^2 E_2^4} + \\frac{4F^2}{N_s^2 E_4^4} ,\n\\label{gX}\n\\\\\n  c_{XY}^{(a)} = c_{XY}^{(b)} =\n  - \\frac{8(F+F^3)}{N_s^3 E_2^4} - \\frac{4(F+F^3)}{N_s^3 E_4^4} ,\n\\label{gXY}\n\\\\\n  c_{YY}^{(a)} = \\frac{8 F^2}{N_s^4 E_2^4} + \\frac{4 F^2}{N_s^4 E_4^4} ,\n\\qquad\n  c_{YY}^{(b)} = \\frac{1+6F^2+F^4}{N_s^4 E_2^4} + \\frac{4F^2}{N_s^4 E_4^4}.\n\\label{gY}\n\\end{gather}\nSumming the contributions (\\ref{gX})--(\\ref{gY}), we come to the final result (\\ref{subfinal}) valid for an arbitrary $F$.\n\n\n\n\n\\section{Discussion and conclusion}\n\\label{Discussion}\n\nIn this work we have analyzed the applicability of the Fermi-golden-rule description of the initial stage of quasiparticle decay in diffusive quantum dots in the regime when the single-particle levels are already resolved. Approaching the problem from the high energy\/temperature side, where each energy level can be characterized by a Lorentzian width $\\gamma_0(\\varepsilon,T)$, we have calculated mesoscopic fluctuations of the energy relaxation rate.\nThe leading contribution to fluctuations comes from the diagrams which describe the square of the same decay process, i.e.\\ have the same set of final states.\nThe resulting expression is non-perturbative in $\\gamma_0$, which appears in the denominator of Eq.~(\\ref{subfinal}), ensuring the growth of fluctuations with the decrease of the excitation energy and\/or temperature.\n\nQuantum relaxation of the initial state $|i\\rangle$ can be described by the return probability\n\\begin{equation}\n  P(t) = \\overline{\\bigl| \\corr{i|e^{-iHt}|i} \\bigr|^2} ,\n\\end{equation}\nwhere $H$ is the Hamiltonian of the interacting quantum dot, and the bar stands for the thermal average. In the semiclassical FGR picture this is just a pure exponential decay:\n\\begin{equation}\n  P_\\text{FGR}(t) = e^{-\\gamma(\\varepsilon,T)t} ,\n\\end{equation}\nwhere the rate $\\gamma(\\varepsilon,T)$ depends on a particular disorder realization.\nIts average value is given by Eq.~(\\ref{SIAT}), and we focused on its mesoscopic fluctuations.\nWe found that the FGR description of the initial stage of quasiparicle decay is applicable as long as $\\max\\{\\varepsilon,T\\}\\gg\\varepsilon_\\text{FGR}$. In this limit, each level is characterized by a well-defined energy width $\\gamma(\\varepsilon,T)$, which weakly fluctuates near the FGR mean:\n\\begin{equation}\n\\label{qualresult3}\n  \\frac{\\ccorr{ \\gamma^2(\\varepsilon,T)}}{\\gamma_{0}^2(\\varepsilon,T)}\n  \\sim\n  \\left( \\frac{\\varepsilon_\\text{FGR}}{\\max\\{ \\varepsilon, T\\}} \\right)^4 .\n\\end{equation}\nThe temperature and the excitation energy enter the result in a similar way [note, however, the presence of the factor $\\Upsilon(\\varepsilon\/2T)$ in Eq.~(\\ref{gamma-Upsilon}), that can change by two orders of magnitude]. This fact is a consequence of the lowest-order approximation. In higher orders their role is expected to be different, in accordance with the difference between the relaxation dynamics in the hot-electron and thermal problems \\cite{Mirlin2015}.\n\nIt is important that in the range of applicability of the FGR description, $\\max\\{\\varepsilon,T\\}\\gg\\varepsilon_\\text{FGR}$, the hybridiation with distant generations and eventually many-body localization effects (if any) become relevant at sufficiently large time scales. The characteristic time $t_*$ when $P_\\text{FGR}(t)$ crosses over to a weaker dependence is determined by the initial state.\nIn the limit $\\max\\{\\varepsilon,T\\}\\gg\\varepsilon_\\text{FGR}$, the scale $t_*$ satisfies $\\gamma_0(\\varepsilon,T)t_*\\gg1$, indicating that almost all the quasiparticle weight is lost during the FGR exponential relaxation\n(for the hot-electron problem, $\\gamma_0(\\varepsilon,0)t_* \\sim \\ln(\\varepsilon\/\\varepsilon_\\text{FGR})$ \\cite{Silvestrov2001}).\n\nOur approach is conceptually similar to the one used by Basko, Aleiner and Altshuler (BAA)~\\cite{BAA}, and hence it is instructive to compare the two results.\nBAA considered inelastic relaxation in a chaotic quantum dot (in Ref.~\\cite{BAA}, referred to as the localization cell), working in the basis of exact one-particle states $|\\alpha\\rangle$ and treating electron-electron interaction phenomenologically. Interaction matrix elements, $\\corr{\\alpha\\beta|V|\\gamma\\delta}$, were assumed to be independent normally distributed random variables with the standard deviation $\\lambda_\\text{BAA}\\Delta$ for energy difference smaller than the ultraviolet cutoff $M\\Delta$, and zero otherwise.\nThe FGR relaxation rate is then given by $\\gamma_{\\text{BAA}}(T)\\sim \\lambda_\\text{BAA}^2 MT$, and BAA obtained the following expression for  mesoscopic fluctuations:\n\\begin{equation}\n\\label{mesofluct-BAA}\n  \\frac{\\ccorr{\\gamma^2(T)}_{\\text{BAA}}}{\\gamma^2_{\\text{BAA}}(T)}\n  \\sim\n  \\frac{\\lambda_\\text{BAA}^4M\\Delta^2T}{\\gamma_{\\text{BAA}}^3(T)}\n  \\sim\n  \\frac{\\Delta^2}{\\lambda_\\text{BAA}^2M^2T^2}.\n\\end{equation}\nIn order to apply these results to the case of a diffusive quantum dot, one should put $\\lambda_\\text{BAA} \\sim \\lambda \\Delta\/E_{\\text{Th}}$~\\cite{AGKL} and $M\\sim T\/\\Delta$.\nThen $\\gamma_{\\text{BAA}}(T)$ coincides with the Sivan-Imry-Aronov relaxation rate (\\ref{SIAT}), while the estimate (\\ref{mesofluct-BAA}) reproduces our result (\\ref{qualresult3}) for mesoscopic fluctuations.\n\nFinally, we emphasize that our result (\\ref{subfinal}) for the leading contribution to mesoscopic fluctuations provides an exact account for the (screened) electron-electron interaction in a diffusive quantum dot. It has an important implementation for the statistics of the interaction matrix elements $\\corr{\\alpha\\beta|V|\\gamma\\delta}$ in the lattice-model language. Since Ref.~\\cite{AGKL}, it is usually assumed for simplicity that this statistics is Gaussian. However, our result (\\ref{subfinal}) demonstrates that for a real diffusive quantum dot such an assumption is generally incorrect. Indeed, for the Gaussian statistics all correlators of the four matrix elements in $\\ccorr{\\gamma^2(\\varepsilon,T)}$ can be expressed through pairwise correlators (the Wick theorem), which are known to be determined by $E_2$ only \\cite{Blanter,AGKL}.\nTherefore the presence of the quantity $E_4$ in Eq.~(\\ref{subfinal}) indicates that the statistics of the interaction matrix elements in a quantum dot is essentially non-Gaussian. This fact should be taken into account in constructing the theory of many-body localization in quantum dots.\n\nWe thank\nI. Aleiner,\nYa.~M. Blanter,\nM. V. Feigel'man,\nI. V. Gornyi,\nV. E. Kravtsov,\nA. D. Mirlin,\nand A. Silva\nfor useful discussions.\nThis work was supported by the Russian Science Foundation under Grant\nNo.\\ 14-42-00044 (M.A.S.).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nLet $\\left(X,Y\\right)$ be a random couple with joint cumulative distribution function $H$ and marginal distribution functions $F$ and $G$. The Sklar's theorem (see \\cite{skl}) says that there exists a bivariate distribution function $C$ on $[0,1]^2$  with uniform margins such that\n$$H(x,y)=C(F(x),G(y)).$$\nThe function $C$ is called the copula associated with $(X,Y)$ and couples the joint distribution $H$ with its marginals. If the marginal distribution functions $F$ and $G$ are continuous, then the copula $C$ is unique and we have  for all $(u,v)\\in [0,1]^2$,\n$$C(u,v)=H(F^{-1}(u),G^{-1}(v)),$$ \nwhere $F^{-1}(u)=\\inf\\{x: F(x)\\geq u\\}$ and $G^{-1}(v)=\\inf\\{y: G(y)\\geq v\\}$ are the generalized inverse functions of $F$ and $G$, respectively. \\\\\n\nThere are three main approaches for copula estimation : parametric, semiparametric and nonparametric. The parametric approach assumes parametric models for both the copula and the marginals and then deals with maximum likelihood or moment method estimation Oakes \\cite{oak}(1982). Semiparametric estimation specifies a parametric copula while leaving the marginals nonparametric (see, e.g. Genest \\textit{et al.}\\cite{ggr}(1995)). The nonparametric approch offers the greatest generality and was initiated by Deheuvels \\cite{deh} (1979), who proposed an estimator based on a multivariate empirical distribution function and its marginals. Afterward, some kernel smoothed estimators have been proposed in the literature (see for instance \\cite{r4},\\cite{r8},\\cite{r3},\\cite{r2},\\cite{r7}).\\\\\n\nIn this paper we are interested with the kernel estimators proposed in Chen and Huang \\cite{r2}(2007), Gijbels and Mielniczuk \\cite{r4}(1990) and Fermanian \\textit{et al.} \\cite{r3}(2004), res pectively called the local linear, the mirror-reflection and the transformation estimators. We will establish for each of them a uniform in bandwidth law of the iterated logarithm for its deviation. These results allows to study the uniform consistency of kernel copula estimators over compact sets $[a,b]^2$, with $0<a<b<1.$\\\\\n\nLet $(X_1,Y_1),...,(X_n,Y_n)$ be an independent and identically distributed sample of the bivariate random vector $(X,Y)$ with joint cumulative distribution function $H$ and marginal distribution functions $F$ and $G$. Denote by $F_n$ and $G_n$  the marginal empirical cumulative distribution functions. Let $\\phi:[0,1]\\mapsto[0,1]$ be an increasing transformation and  $K(\\cdot,\\cdot)$ a bivariate kernel function. For all $0\\leq u,v\\leq 1$ and $0<h<1$, define the general estimator\n\\begin{equation}\\label{ge}\n\\hat{C}_{n,h}^{(\\cdot)}(u,v)=\\frac{1}{n}\\sum_{i=1}^{n}K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{U}_i)}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{V}_i)}{h} \\right)\n\\end{equation}\nIf $K(x,y)$ is a multiplicative kernel, i.e. $K(x,y)=K(x)K(y)$ and the pseudo-observations are $\\hat{U}_i=\\frac{n}{n+1}F_n(X_i)$ and $\\hat{V}_i=\\frac{n}{n+1}G_n(Y_i)$, \\eqref{ge} is exactly the transformation estimator which is defined by\n\\begin{equation}\\label{tra}\n\\hat{C}_{n,h}^{(T)}(u,v)=\\frac{1}{n}\\sum_{i=1}^{n}K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{U}_i)}{h} \\right)K\\left(\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{V}_i)}{h} \\right).\n\\end{equation}\nFor the local linear estimator,  suppose first that the marginals $F$ and $G$ are estimated by\n\\begin{center} \n$\\displaystyle \\hat{F}_n(x)=\\frac{1}{n}\\sum_{i=1}^n K\\left(\\frac{x-X_i}{b_{n1}}\\right),\\ \\ \\ \\ \\ \\hat{G}_n(y)=\\frac{1}{n}\\sum_{i=1}^n K\\left(\\frac{y-Y_i}{b_{n2}}\\right),$\n\\end{center}  where $b_{n1}$ and $b_{n2}$ are  bandwidths converging to 0, as $n\\rightarrow\\infty$ and $K$ is the integral of a symmetric bounded kernel function $k$ supported on $[-1,1]$. \nNext, consider the pseudo-observations $\\hat{U}_i = \\hat{F}_n(X_i)$ and $\\hat{V}_i =\\hat{G}_n(Y_i)$ and choose a multiplicative $K(x,y)=K_{u,h}(x)K_{v,h}(y)$, for $u,v\\in[0,1]$, where $K_{w,h}(x)=\\int_{-\\infty}^{x}k_{w,h}(t)dt,\\; w=u,v$\n with $k_{w,h}$ a local linear version of the kernel $k$ given by\n$$\n k_{w,h}(t)=\\frac{k(t)\\{a_2(w,h)-a_1(w,h)t\\}}{a_0(w,h)a_2(w,h)-a_1^2(w,h)}\\mathbb{I}\\left\\{\\frac{w-1}{h}<t<\\frac{w}{h}\\right\\},$$\nwhere $ a_j(w,h)=\\int_{(w-1)\/h}^{w\/h} t^j k(t)dt$ for $j=0,1,2$ ; $w\\in[0,1]$ and $0<h<1$ is a bandwidth. Finally taking $\\phi(s)=t$, the identity function in \\eqref{ge}, we obtain the local linear kernel estimator  defined as\n\\begin{equation}\\label{lli}\n\\hat{C}_{n,h}^{(LL)}(u,v)=\\frac{1}{n}\\sum_{i=1}^{n}K_{u,h}\\left(\\frac{u-\\hat{U_i}}{h} \\right)K_{v,h}\\left(\\frac{v-\\hat{V_i}}{h} \\right).\n\\end{equation}\n  According to Omelka \\textit{et al} (2009), the mirror-reflection estimator can be represented as\n \\begin{equation}\\label{mrf}\n\\hat{C}_{n}^{(MR)}(u,v)=\\sum_{l=1}^9\\left[Z_n(l,u,v)-Z_n(l,u,0)-Z_n(l,0,v)+Z_n(l,0,0)\\right],\n\\end{equation}\nwhere\n\\begin{equation}\\label{bigz}\nZ_n(l,u,v)=\\frac{1}{n}\\sum_{i=1}^nK\\left(\\frac{u-\\hat{U}_i^{(l)}}{h_n}\\right)K\\left(\\frac{v-\\hat{V}_i^{(l)}}{h_n}\\right)\n\\end{equation}\nand \n\\begin{multline}\n\\left\\{\\left(\\hat{U}_i^{(l)},\\hat{V}_i^{(l)}\\right),i=1,...,n,l=1,...,9\\right\\}\\\\\n=\\left\\{\\left(\\pm\\hat{U}_i,\\pm\\hat{V}_i\\right),\\left(\\pm\\hat{U}_i,2-\\hat{V}_i\\right),\\left(2-\\hat{U}_i,\\pm\\hat{V}_i\\right),\\left(2-\\hat{U}_i,2-\\hat{V}_i\\right),i=1,...,n\\right\\}\\nonumber\n\\end{multline}\nSetting $\\phi(t)=t$ and using a multiplicative kernel $K(x,y)=K(x)K(y)$, one can see that each quantity $Z_n(l,u,v)$ may be put in the form \\eqref{ge}.\\\\\n\nThe remainder of the paper is organized as follows. In Section 2, we state our main results which consist of laws of the iterated logarithm for the deviations of the estimators \\ref{tra},\\ref{lli} and \\ref{mrf} from their means.  Section 3 is devoted to the proofs of the results. Finally, in Section 4 we give an appendix, where we show a key result for the proofs inspired by a general theorem of Mason and Swanpoel (2010), concerning the uniform in bandwidth consistency of kernel-type function estimators. \n\n\\section{Main results }\\label{ssec2}\nWe state our theoretical results in Theorems \\ref{t1}, \\ref{t2} and \\ref{t3}, which give  uniform in bandwidth laws of the iterated logarithm (LIL) for the maximal deviation of the estimators \\eqref{tra},\\eqref{lli} and \\eqref{mrf} from their expectations.\\\\\n Let $R_n =\\left(\\frac{n}{2\\log\\log n}\\right)^{1\/2}$. We have\n \\begin{theorem}\\label{t1}\nFor any sequence of positive constants $(b_n)_{n\\geq 1}$ satisfying $0<b_n<1, b_n\\rightarrow 0$, $b_n\\geq (\\log n)^{-1}$, and for some $c>0$, we have almost surely\n\\begin{equation}\\label{e2}\n\\limsup_{n\\rightarrow\\infty}\\left\\{R_n\\sup_{\\frac{c\\log n}{n}\\le h\\le b_n}\\sup_{(u,v)\\in(0,1)^2}\\left|\\hat{C}_{n,h}^{(LL)}(u,v)-\\mathbb{E}\\hat{C}_{n,h}^{(LL)}(u,v)\\right|\\right\\}\\leq 3.\n\\end{equation}\n\\end{theorem} \n\n\\begin{theorem}\\label{t2}\nSuppose that the increasing transformation $\\phi$ admits a bounded derivative $\\phi'$. Then, for any sequence of positive constants $(b_n)_{n\\geq 1}$ satisfying $0<b_n<1, b_n\\rightarrow 0$, $b_n\\geq (\\log n)^{-1}$, and for some $c>0$, we have almost surely \n\\begin{equation}\\label{ee2}\n\\limsup_{n\\rightarrow\\infty}\\left\\{R_n\\sup_{\\frac{c\\log n}{n}\\le h\\le b_n}\\sup_{(u,v)\\in(0,1)^2}\\left|\\hat{C}_{n,h}^{(T)}(u,v)-\\mathbb{E}\\hat{C}_{n,h}^{(T)}(u,v)\\right|\\right\\}\\leq 3.\n\\end{equation}\n\\end{theorem}\n\\begin{theorem}\\label{t3}\nFor any sequence of positive constants $(b_n)_{n\\geq 1}$ satisfying $0<b_n<1, b_n\\rightarrow 0$, $b_n\\geq (\\log n)^{-1}$, and for some $c>0$, we have almost surely \n\\begin{equation}\\label{eeq2}\n\\limsup_{n\\rightarrow\\infty}\\left\\{R_n\\sup_{\\frac{c\\log n}{n}\\le h\\le b_n}\\sup_{(u,v)\\in(0,1)^2}\\left|\\hat{C}_{n,h}^{(MR)}(u,v)-\\mathbb{E}\\hat{C}_{n,h}^{(MR)}(u,v)\\right|\\right\\}\\leq 3.\n\\end{equation}\n\\end{theorem}\n\nThe proofs of Theorems \\ref{t1},\\ref{t2},\\ref{t3} are similar and are postponded until Section 3. They are obtained by combining a general theorem of Mason and Swanpoel \\cite{r5}(2010) for proving the uniform in bandwidth consistency of kernel-type function estimators, and a law of the iterated logarithm for Kiefer processes due to Wichura \\cite{r10}(1973).\\\\\n\n\\textbf{Remark}.\n{\\rm \\begin{itemize}\n\\item[1)] Under smoothness conditions on the copula $C$, namely the existence bounded second-order partial derivatives, ensuring the uniform almost sure convergence of the bias of the estimators  to zero, we obtain the strong uniform in bandwidth consistency of the kernel copula estimators $\\hat{C}_{n,h}^{(LL)},\\hat{C}_{n,h}^{(MR)}$ and $\\hat{C}_{n,h}^{(T)}$.\n\\item[2)] The uniformity on the bandwidth allows the use of a large bandwidth selection methods including the method of shrinkage proposed by Omelka \\textit{et al.}\\cite{r7}(2009).\n\\item[3)] As the boundary bias is present, the uniform consistency of these kernel estimators is not valid over the entire $[0,1]^2$, but it is true for all  $(u,v)\\in[h,1-h]^2,\\; 0<h<1$.\n\\item[4)] If we consider a data-driven bandwidth ; that is $h=\\hat{h}_n$, we have the probability versions of Theorems \\ref{t1},\\ref{t2} and \\ref{t3}, stated in  Corollary \\ref{cr1} below.\\\\\n\\end{itemize} }\n\nLet $\\hat{C}_{n,h}^{(\\cdot)}$ denote a kernel estimator of the copula $C$, where $(\\cdot)$ stands for $(LL), (MR)$ and $(T)$, respectively. One has\n\\begin{corollary}\\label{cr1}\nUnder the assumptions of Theorems \\ref{t1},\\ref{t2} or \\ref{t3}, we have for all $0<\\epsilon <1$ and for any sequence of data-driven bandwidth $\\hat{h}_n$ satisfying \\begin{equation} \\label{ee4'}\n \\mathbb{P}(\\frac{c\\log n}{n}\\le \\hat{h}_n\\le b_n)\\rightarrow 1,\\; n\\rightarrow\\infty,\n \\end{equation}\n\\begin{equation}\\label{ee2}\n\\mathbb{P}\\left\\{R_n\\sup_{(u,v)\\in(0,1)^2}\\left|\\hat{C}_{n,\\hat{h}_n}^{(\\cdot)}(u,v)-\\mathbb{E}\\hat{C}_{n,\\hat{h}_n}^{(\\cdot)}(u,v)\\right|>  3(1+\\epsilon)\\right\\}=o(1).\n\\end{equation}\n\\end{corollary}\n\\textbf{Remark}. This result enables us to construct simultaneous confidence bands for the copula curve $C(u,v), 0<u,v<1$, with  asymptotic confidence level $100\\%$, under the condition that the bias of the estimator $\\mathbb{E}\\hat{C}_{n,\\hat{h}_n}^{(\\cdot)}-C(u,v)$ converges uniformly to zero with the same rate $R_n$, as $n\\rightarrow\\infty$, i.e.  for all $0<\\epsilon <1$,\n\\begin{equation}\\label{e2}\n\\mathbb{P}\\left\\{R_n\\sup_{(u,v)\\in(0,1)^2}\\left|\\mathbb{E}\\hat{C}_{n,\\hat{h}_n}^{(\\cdot)}(u,v)-C(u,v)\\right|>  \\epsilon\\right\\}=o(1).\n\\end{equation}\nAn example of such confidence bands is provided in \\cite{r1} for the local linear estimator.  \n\\section{Proofs}\nThe proofs of the theorems are similar. To simplify we will establish the results for a generalized estimator $\\hat{C}_{n,h}^{(\\cdot)}$ defined as follows. Let $\\phi:[0,1]\\mapsto[0,1]$ be an increasing transformation. For a bivariate kernel $K(\\cdot,\\cdot)$ and $0<u,v<1$, define the estimator\n\\begin{equation}\\label{ge}\n\\hat{C}_{n,h}^{(\\cdot)}(u,v)=\\frac{1}{n}\\sum_{i=1}^{n}K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{U}_i)}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{V}_i)}{h} \\right)\n\\end{equation}\nand set\n$$\\hat{D}_{n,h}^{(\\cdot)}(u,v)=:\\hat{C}_{n,h}^{(\\cdot)}(u,v)-\\mathbb{E}\\hat{C}_{n,h}^{(\\cdot)}(u,v).$$\nWe call $\\hat{D}_{n,h}^{(\\cdot)}(u,v)$ the deviation of the estimator $\\hat{C}_{n,h}^{(\\cdot)}(u,v)$ from its expectation and we will study its behavior by using general empirical process theory.\\\\\n\\indent We remark that if the kernel $K(\\cdot,\\cdot)$ is multiplicative, i.e. $K(x,y)=K(x)K(y)$, we obtain directly the transformation estimator. If $\\phi(t)=t$, the identity function and the local linear kernels $K_{u,h}(\\cdot)$ and $K_{v,h}(\\cdot)$ are employed, then we obtain the local linear kernel estimator. Note that for the mirror-reflection estimator, we consider the following decomposition due to Omelka \\textit{et al} (2009),\n \\begin{equation}\\label{mirror-local}\n\\hat{C}_{n}^{(MR)}(u,v)=\\sum_{l=1}^9\\left[Z_n(l,u,v)-Z_n(l,u,0)-Z_n(l,0,v)+Z_n(l,0,0)\\right],\n\\end{equation}\nwhere\n\\begin{equation}\\label{bigz}\nZ_n(l,u,v)=\\frac{1}{n}\\sum_{i=1}^nK\\left(\\frac{u-\\hat{U}_i^{(l)}}{h_n}\\right)K\\left(\\frac{v-\\hat{V}_i^{(l)}}{h_n}\\right).\n\\end{equation}\nSetting $\\phi(t)=t$ and using a multiplicative kernel, one can see that each quantity $Z_n(l,u,v)$ may be put in the form \\eqref{ge}.\\\\\n\nLet $H_n$, $F_n$ and $G_n$ be the empirical cumulative distribution functions of $H$, $F$ and $G$, respectively. Then  the estimator based directly on Sklar's Theorem is given by\n$$ C_n(u,v)= H_n(F_n^{-1}(u),G_n^{-1}(v)),$$\nwith $F_n^{-1}(u)=\\inf\\{x:F_n(x)\\geq u\\}$ and $G_n^{-1}(v)=\\inf\\{x:F_n(x)\\geq v\\}$. The bivariate empirical copula process is defined as\n$$\\mathbb{C}_n(u,v)=\\sqrt{n}[C_n(u,v)- C(u,v)],\\qquad (u,v)\\in [0,1]^2.$$\nIntroduce the following quantity.\n$$ \\widetilde{C}_n(u,v)=\\frac{1}{n}\\sum_{i=1}^n\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}$$\nwhich represents the uniform bivariate empirical distribution function based on a sample $(U_1,V_1),\\cdots,(U_n,V_n)$ of i.i.d random  variables uniformly distributed on $[0,1]^2$. \nDefine the following empirical process\n $$\\mathbb{\\widetilde{C}}_n(u,v)=\\sqrt{n}[\\widetilde{C}_n(u,v)- C(u,v)],\\quad (u,v)\\in[0,1]^2.$$\nThen, wecan easily prove that that \n\\begin{equation}\\label{c3}\n\\mathbb{\\widetilde{C}}_n(u,v)=\\mathbb{C}_n(u,v)+ \\frac{1}{\\sqrt{n}}.\n\\end{equation}\nFor any given increasing transformation $\\phi$ and $(u,v)\\in [0,1]^2$, define \n\\begin{eqnarray*}\n&g_{n,h}=\\hat{C}_{n,h}^{(\\cdot)}(u,v)-\\widetilde{C}_n(u,v)&\\\\\n&  = \\frac{1}{n}\\sum_{i=1}^{n}\\left [K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{U}_i)}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{V}_i)}{h} \\right)- \\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\right ]&\\\\\n& =\\frac{1}{n}\\sum_{i=1}^{n}\\left [K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\hat{F}_n oF^{-1}(U_i))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\hat{G}_n oG^{-1}(V_i))}{h} \\right)-\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\right]& \\\\\n& =: \\frac{1}{n}\\sum_{i=1}^{n}g(U_i,V_i,h),&\n\\end{eqnarray*}\nwhere $g$ belongs to the class of measurable functions $\\mathcal{G}$ defined as \n$$\n\\mathcal{G}=\\left\\{\\begin{array}{c} \ng:(s,t,h)\\mapsto g(s,t,h)=K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(s))}{h}, \\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(t))}{h}\\right)- \\mathbb{I}\\{s\\leq u,t\\leq v\\},\\\\\n u,v\\in[0,1], 0<h<1 \\,\\text{and }\\, \\zeta_{1,n}\\zeta_{2,n}:[0,1]\\mapsto [0,1] \\,\\text{nondecreasing.}\n\\end{array} \\right\\}\n$$\nSince $\\mathbb{E}\\widetilde{C}_n(u,v)=C(u,v)$, one can observe that \n$$\n\\sqrt{n}\\vert g_{n,h}- \\mathbb{E}g_{n,h}\\vert= \\vert\\sqrt{n}\\hat{D}_{n,h}^{(\\cdot)}(u,v)-\\mathbb{\\widetilde{C}}_n(u,v)\\vert.\n$$\n\nNow, we want to apply the main Theorem in \\cite{r5} due to Mason and Swanpoel. Towards this end, the class of functions $\\mathcal{G}$ must verify the following  four conditions: \n\\begin{itemize}\n\\item[(G.i)]There exists a finite constant $\\kappa>0$ such that\n $$ \\sup_{0\\leq h\\leq 1}\\sup_{g\\in \\mathcal{G}}\\left\\|g\\left(\\cdot,\\cdot,h\\right)\\right\\|_\\infty=\\kappa <\\infty. $$\n\\item[(G.ii)]\\ \\ \\ There exists a constant $C'>0$ such that for all $h\\in [0,1]$,\n$$ \\sup_{g\\in \\mathcal{G}}\\mathbb{E}\\left[g^2\\left(U,V,h\\right)\\right]\\leq C'h. $$\n\\item[(F.i)]\\ \\ \\ \n$\\mathcal{G}$ satisfies the uniform entropy condition, i.e., \n$$\\exists \\, C_0>0, \\nu_0>0\\ :\\ N\\left(\\epsilon,\\mathcal{G}\\right)\\leq C_0\\epsilon^{-\\nu_0}.$$\n\\item[(F.ii)]\\ \\ \\ $\\mathcal{G}$ is a pointwise measurable class, i.e there exists a countable sub-class $\\mathcal{G}_0$ of $\\mathcal{G}$ such that for all $g\\in \\mathcal{G}$, there exits $\\left(g_m\\right)_m\\subset \\mathcal{G}_0$ such that $g_m\\longrightarrow g.$\\\\\n\\end{itemize}\n\nThe checking of these conditions will be done in Appendix and constitutes the proof of the following proposition.\n\\begin{proposition}\\label{p1}\nSuppose that the copula function $C$ has bounded first order partial derivatives on $(0, 1)^2$ and the transformation $\\phi$ admits a bounded derivative $\\phi'$. Then assuming (G.i), (G.ii), (F.i) and (F.ii), we have for some $c > 0,\\ 0 < h_0 < 1,$ with probability one, \n$$\n\\limsup_{n\\rightarrow\\infty}\\sup_{\\frac{c\\log n}{n}\\leq h \\leq h_0}\\sup_{(u,v)\\in(0,1)^2}\n\\frac{|\\sqrt{n}\\hat{D}_{n,h}^{(\\cdot)}(u,v)-\\tilde{\\mathbb{C}}_n(u,v)|}{\\sqrt{h(|\\log h\\vert\\vee\\log\\log n)}}=A(c),\n$$\nwhere $A(c)$ is a positive constant.\n\\end{proposition}\n\n\\begin{corollary}\\label{crl1}\n Under the assumptions of Proposition \\ref{p1}, one has for any sequence of constants $0<b_n<1,$ satisfying $\\ b_n\\rightarrow 0,\\ b_n\\geq (\\log n)^{-1}$, with probability one, \n $$\n \\sup_{\\frac{c\\log n}{n}\\leq h \\leq b_n}\\sup_{(u,v)\\in(0,1)^2}\n\\frac{|\\sqrt{n}\\hat{D}_{n,h}^{(\\cdot)}(u,v)-\\tilde{\\mathbb{C}}_n(u,v)|}{\\sqrt{\\log\\log n}}=O(\\sqrt{b_n}).\n $$\n \\end{corollary}\n\\begin{proof}{( \\textbf{Corollary \\ref{crl1})}}\\\\\n{\\rm First, observe that the condition $b_n\\geq (\\log n)^{-1}$ implies\n\\begin{equation}\\label{dc}\n\\frac{\\vert \\log b_n\\vert}{\\log\\log n}\\leq 1.\n\\end{equation}\nNext, by the monotonicity of the function $x\\mapsto x\\vert\\log x \\vert$ on $[0,1\/e]$, one can write for $n$ large enough, $h\\vert\\log h \\vert\\leq b_n\\vert\\log b_n \\vert$ and hence,\n\\begin{equation}\nh(\\vert\\log h \\vert  \\vee\\log\\log n)\\leq b_n(\\vert\\log b_n \\vert \\vee\\log\\log n).\n\\end{equation}\nCombining this and Proposition \\ref{p1}, we obtain\n$$\n \\sup_{\\frac{c\\log n}{n}\\leq h \\leq b_n}\\sup_{(u,v)\\in(0,1)^2}\n\\frac{|\\sqrt{n}\\hat{D}_{n,h}(u,v)-\\tilde{\\mathbb{C}}_n(u,v)|}{\\sqrt{b_n\\log\\log n\\left(\\frac{\\vert \\log b_n\\vert}{\\log\\log n}\\vee 1\\right)}}=O(1).\n $$\nThus the Corollary \\ref{crl1} follows from  \\eqref{dc}.}\n\\end{proof}\nComing back to the proof of our theorems, we have to show that\n\\begin{equation}\\label{lil}\n\\limsup_{n\\rightarrow\\infty}\\sup_{\\frac{c\\log n}{n}\\leq h \\leq b_n}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\sqrt{n}\\hat{D}_{n,h}^{(\\cdot)}(u,v)\\right|}{\\sqrt{2\\log\\log n}}\\leq 3.\n\\end{equation}\nTowards this end, we make use of an approximation of the empirical copula process $\\mathbb{C}_n$ by a Kiefer process (see e.g., Zari\\cite{r11}, page 100). Let $\\mathbb{W}(u,v,t)$ be a $3$-parameters  Wiener process defined on $[0,1]^2\\times[0,\\infty)$. Then the Gaussian process $\\mathbb{K}(u,v,t)=\\mathbb{W}(u,v,t)-\\mathbb{W}(1,1,t).uv$ is called a $3$-parameters  Kiefer process defined on $[0,1]^2\\times[0,\\infty)$.\\\\\n\\indent By Theorem 3.2 in Zari\\cite{r11}, for $d=2$, there exists a sequence of Gaussian processes $\\left\\{\\mathbb{K}_{C}(u,v,n), u,v\\in[0,1], n>0\\right\\}$ such that\n$$ \\sup_{(u,v)\\in[0,1]^2}\\left|\\sqrt{n}\\mathbb{C}_n(u,v)-\\mathbb{K}_{C}^\\ast(u,v,n)\\right|=O\\left(n^{3\/8}(\\log n)^{3\/2}\\right),$$ where \n$$\\mathbb{K}_{C}^\\ast(u,v,n)=\\mathbb{K}_{C}(u,v,n)-\\mathbb{K}_{C}(u,1,n)\\frac{\\partial C(u,v)}{\\partial u}-\\mathbb{K}_\\mathbb{C}(1,v,n)\\frac{\\partial C(u,v)}{\\partial v}.$$\nThis yields\n\\begin{equation}\\label{c1}\n\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\mathbb{C}_n(u,v)\\right|}{\\sqrt{2\\log\\log n}}=\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\mathbb{K}_{C}^\\ast(u,v,n)\\right|}{\\sqrt{2n\\log\\log n}}.\n\\end{equation}\nBy the works of Wichura\\cite{r10} on the  law of the iterated logarithm , for $d=2$, one has almost surely\n\\begin{equation}\\label{c2}\n\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\mathbb{K}_\\mathbb{C}^\\ast(u,v,n)\\right|}{\\sqrt{2n\\log\\log n}}\\leq 3,\n\\end{equation}\n\n\\noindent which entails  \n$$\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\mathbb{C}_n(u,v)\\right|}{\\sqrt{2\\log\\log n}}\\leq 3.$$\nSince ${\\mathbb{C}}_n(u,v)$ and $\\tilde{\\mathbb{C}}_n(u,v)$ are asymptotically equivalent in view of (\\ref{c3}), one obtains\n$$\\limsup_{n\\rightarrow\\infty}\\sup_{(u,v)\\in[0,1]^2}\\frac{\\left|\\tilde{\\mathbb{C}}_n(u,v)\\right|}{\\sqrt{2\\log\\log n}}\\leq 3.$$\nApplying Corollary \\ref{crl1} and the fact that $\\sqrt{b_n}\\rightarrow 0$, we obtain \\eqref{lil} which proves the theorems.\n\n\\section*{Appendix}\n\\begin{proof}(\\textbf{Proposition \\ref{p1}})\\\\\nAssume that the function $K(\\cdot,\\cdot)$ is the integral of a symmetric bounded kernel $k(\\cdot,\\cdot)$, supported on $[-1,1]^2$, i.e. $K(x,y)=\\int_0^x\\int_0^y k(s,t)dsdt$. We have to check (G.i), (G.ii), (F.i) and (F.ii).\\\\\n\n\\noindent \\textbf{Checking for (G.i):}\nRecall that $(U_i,V_i), i\\geq 1$ are iid random variables uniformly distributed on $[0,1]^2$, $\\zeta_{1,n}(U_i)=\\hat{F}_n o F^{-1}(X_i)$ and $\\zeta_{2,n}(U_i)=\\hat{G}_n o G^{-1}(Y_i)$.\nFor any function $g\\in\\mathcal{G}$ and $0<h<1$, we can write \n\\begin{eqnarray*}\ng\\left(U_i,V_i,h\\right) & =&K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U_i))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V_i))}{h}\\right)- \\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\\\\n& =&\\int_{-\\infty}^{\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U_i))}{h}} \\int_{-\\infty}^{\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V_i))}{h}}k(s,t)dsdt-\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\\\\n& =&\\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U_i\\leq \\zeta_{1,n}^{-1}\\circ\\phi(\\phi^{-1}(u)-th) ,V_i\\leq \\zeta_{2,n}^{-1}\\circ\\phi(\\phi^{-1}(v)-sh)\\right\\}k(s,t)dsdt \\\\\n & & -\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\\\\n&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}k(t)k(s,t)dsdt-\\mathbb{I}\\{U_i\\leq u,V_i\\leq v\\}\\ \\leq 4 \\|k\\|^2+1,\\\\\n\\end{eqnarray*}\nwhere $\\displaystyle \\|k\\|=\\sup_{(s,t)\\in[-1,1]^2}|k(s,t)|$ represents the supremum norm on $[-1,1]^2$. Thus (G.i) holds by taking $\\kappa:=4\\|k\\|^2+1.$\\\\\n\n\\noindent \\textbf{Checking for (G.ii).} We have to show that $\\displaystyle \\sup_{g\\in \\mathcal{G}}\\mathbb{E}g^2(U,V,h)\\leq C_0 h$, where $C_0$ is a positive  constant. One can write\n\\begin{eqnarray*}\\small\n&\\mathbb{E}g^2(U,V,h) =  \\mathbb{E}\\left[K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)- \\mathbb{I}\\{U\\leq u,V\\leq v\\}\\right]^2 &\\\\\n&=  \\mathbb{E}\\left[K^2\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\right] &\\\\\n&  - 2\\mathbb{E}\\left[K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\mathbb{I}\\{U\\leq u,V\\leq v\\}\\right] + C(u,v)&\\\\\n& =:  A -2B + C(u,v)&.\n\\end{eqnarray*}\n Since the function $K(\\cdot,\\cdot)$ is a kernel of a distribution function, we may assume without loss of generality that it takes its values in $[0,1]$. Then, we can use the inequality $K^2(x,y)\\leq K(x,y)$ to bound up the term $A$ in the right hand side of the previous egality.\n\\begin{eqnarray*}\n&A  =   \\mathbb{E}\\left[K^2\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\right]&\\\\\n&\\leq  \\mathbb{E}\\left[K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\right]&\\\\\n& \\leq \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq \\zeta_{1,n}^{-1}\\circ\\phi(\\phi^{-1}(u)-sh),V\\leq \\zeta_{2,n}^{-1}\\circ\\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right].&\n\\end{eqnarray*}\nThe other term $B$ is written into\n\\begin{eqnarray*}\n&B  = \\mathbb{E}\\left[K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right)\\mathbb{I}\\{U\\leq u,V\\leq v\\}\\right]&\\\\\n&= \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{s\\leq \\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(U))}{h},t\\leq \\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(V))}{h}\\right\\} \\mathbb{I}\\{U\\leq u,V\\leq v\\}k(s,t)dsdt\\right]&\\\\\n&= \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq u\\wedge \\zeta_1^{-1}\\circ\\phi(\\phi^{-1}(u)-sh),V\\leq v\\wedge \\zeta_2^{-1}\\circ\\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right],&\n\\end{eqnarray*} where $x\\wedge y=\\min (x,y)$.\nNote that $$ C(u,v)=\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt,$$\nas the kernel $k(\\cdot,\\cdot)$ satisfies  $\\int_{-1}^{1}\\int_{-1}^{1}k(s,t)dsdt=1$. Thus\n\\begin{eqnarray*}\n&\\mathbb{E}g^2(U,V,h)\\leq \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq \\zeta_{1,n}^{-1}\\circ\\phi(\\phi^{-1}(u)-sh),V\\leq \\zeta_{2,n}^{-1}\\circ\\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right]&\\\\\n &  -2\\mathbb{E}\\bigg[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq u\\wedge \\zeta_{1,n}^{-1}\\circ\\phi(\\phi^{-1}(u)-sh),V\\leq v\\wedge \\zeta_{2,n}^{-1}\\circ\\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\bigg]& \\\\\n & + \\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt.&\n\n\\end{eqnarray*}\nWe shall suppose that the empirical kernel distributions $\\hat{F}_n$ and $\\hat{G}_n$ are asymptotically equivalent the classical empirical distribution function $F_n$ $G_n$. From the Chung's (1949) LIL, we can infer that for all $u\\in [0,1]$, as $n\\rightarrow\\infty$,\n$$\\zeta_{1,n}^{-1}(u)-u = F\\circ\\hat{F}_n^{-1}(u)-F\\circ F^{-1}(u)=O(n^{-1}\\log\\log n).$$\nThat is $\\zeta_{1,n}^{-1}(u)$ is asymptotically equivalent to $u$. Same for $\\zeta_{2,n}^{-1}(u)=G\\circ\\hat{G}_n^{-1}(u)$. Thus, for all large $n$, we can write\n\\begin{eqnarray*}\n&\\mathbb{E}g^2(U,V,h)\\leq \\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq \\phi(\\phi^{-1}(u)-sh),V\\leq \\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right]&\\\\\n &  -2\\mathbb{E}\\left[ \\int_{-1}^{1}\\int_{-1}^{1}\\mathbb{I}\\left\\{U\\leq u\\wedge \\phi(\\phi^{-1}(u)-sh),V\\leq v\\wedge \\phi(\\phi^{-1}(v)-th)\\right\\}k(s,t)dsdt \\right]&\\\\\n &  +\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt.&\n\\end{eqnarray*}\nThat is,\n\\begin{eqnarray}\\label{eqcas}\n\\mathbb{E}g^2(U,V,h)&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)k(s,t)dsdt \\nonumber\\\\\n & & -2\\int_{-1}^{1}\\int_{-1}^{1}C\\left(u\\wedge\\phi(\\phi^{-1}(u)-sh),v\\wedge \\phi(\\phi^{-1}(v)-th)\\right)k(s,t)dsdt \\\\\n &  & +\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt.\\nonumber\n\\end{eqnarray}\nNow, we have to discuss condition (G.ii) in the four following cases:\\\\\n\n\\textbf{Case 1.} $u\\wedge\\phi(\\phi^{-1}(u)-sh)=\\phi(\\phi^{-1}(u)-sh)\\ \\text{et}\\ v\\wedge \\phi(\\phi^{-1}(v)-th)=\\phi(\\phi^{-1}(v)-th)$.\\\\\nIn this case the second member of inequality \\eqref{eqcas} is reduced and we have \n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)&\\leq &-\\int_{-1}^{1}\\int_{-1}^{1}C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)k(s,t)dsdt \\\\\n  &  & +\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt\\\\\n\t&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}\\left[C(u,v)-C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)\\right]k(s,t)dsdt.\n\\end{eqnarray*}\nBy a Taylor expansion for the copula function $C$, we have\n\\begin{eqnarray*}\nC(u,v)-C(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th))&=&[u-\\phi(\\phi^{-1}(u)-sh)]C_u(u,v)\\\\\n &+ & [v-\\phi(\\phi^{-1}(v)-th)]C_v(u,v)+ o(h). \n\\end{eqnarray*}\nApplying again a Taylor-Young expansion for the function $\\phi$, we obtain\n$$ u-\\phi(\\phi^{-1}(u)-sh)=\\phi(\\phi^{-1}(u)) -\\phi(\\phi^{-1}(u)-sh)=\\phi'(u)sh+o(h)$$\nand\n$$ v-\\phi(\\phi^{-1}(v)-sh)=\\phi(\\phi^{-1}(v)) -\\phi(\\phi^{-1}(v)-th)=\\phi'(u)th+o(h).$$\nThus\n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}\\left[\\phi'(u)C_u(u,v)h+\\phi'(v)C_v(u,v)h\\right]k(s,t)dsdt \\\\\n\t&\\leq & 4h\\left[\\|C_u\\|+\\|C_v\\|\\right]\\sup_{x\\in[0,1]}|\\phi'(x)|\\|k\\|.\n\\end{eqnarray*}\nTaking $C_0=4\\left[\\left\\|C_u\\right\\|+\\left\\|C_v\\right\\|\\right]\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|$ gives condition (G.ii).\\\\\n\n\\textbf{Case 2.} $u\\wedge\\phi(\\phi^{-1}(u)-sh)=u\\ \\text{et}\\ v\\wedge \\phi(\\phi^{-1}(v)-th)=v$. \\\\\nHere the inequality \\eqref{eqcas} is reduced to\n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)\\leq \\int_{-1}^{1}\\int_{-1}^{1}\\left[C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)-C(u,v)\\right]k(s,t)dsdt.\n\\end{eqnarray*}\nUsing the same arguments as in Case 1, we obtain condition (G.ii):\n$\\displaystyle \\sup_{g\\in \\mathcal{G}}\\mathbb{E}g^2(U,V,h)\\leq C_0h$, with $C_0=4\\left[\\left\\|C_u\\right\\|+\\left\\|C_v\\right\\|\\right]\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|$.\\\\\n\n\\textbf{Case 3.} $u\\wedge\\phi(\\phi^{-1}(u)-sh)=\\phi(\\phi^{-1}(u)-sh)\\ \\text{et}\\ v\\wedge \\phi(\\phi^{-1}(v)-th)=v$.\\\\\nHere, inequality \\eqref{eqcas} is rewritten into\n\\begin{eqnarray*}\n&\\mathbb{E}g^2(U,V,h)\\leq \\int_{-1}^{1}\\int_{-1}^{1}C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)k(s,t)dsdt & \\\\\n & -2\\int_{-1}^{1}\\int_{-1}^{1}C\\left(\\phi(\\phi^{-1}(u)-sh),v\\right)k(s,t)dsdt  +\\int_{-1}^{1}\\int_{-1}^{1}C(u,v)k(s,t)dsdt &\\\\\n&\\leq \\int_{-1}^{1}\\int_{-1}^{1}\\left[C\\left(\\phi(\\phi^{-1}(u)-sh),\\phi(\\phi^{-1}(v)-th)\\right)-C\\left(\\phi(\\phi^{-1}(u)-sh),v\\right)\\right]k(s,t)dsdt &\\\\\n& -\\int_{-1}^{1}\\int_{-1}^{1}\\left[C\\left(\\phi(\\phi^{-1}(u)-sh),v\\right)-C(u,v)\\right]k(s,t)dsdt. &\n\\end{eqnarray*}\nBy applying successively a Taylor expansion for $C$ and for $\\phi$, we get\n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)&\\leq &\\int_{-1}^{1}\\int_{-1}^{1}C_v\\left(\\phi(\\phi^{-1}(u)-sh),\\theta_1\\right)\\left[\\phi(\\phi^{-1}(v)-th)-\\phi(\\phi^{-1}(v))\\right]k(s,t)dsdt\\\\\n& & -\\int_{-1}^{1}\\int_{-1}^{1}C_u\\left(\\theta_2,v\\right)\\left[\\phi(\\phi^{-1}(u)-sh)-\\phi(\\phi^{-1}(u))\\right]k(s,t)dsdt\\\\\n&\\leq & \\int_{-1}^{1}\\int_{-1}^{1}C_v\\left(\\phi(\\phi^{-1}(u)-sh),\\theta_1\\right)\\phi'(\\gamma_1).(-th)k(s,t)dsdt\\\\\n& & -\\int_{-1}^{1}\\int_{-1}^{1}C_u\\left(\\theta_2,v\\right)\\phi'(\\gamma_2).(-sh)k(s,t)dsdt,\n\\end{eqnarray*}\nwhere $\\theta_1\\in \\left(\\phi(\\phi^{-1}(v)-th),v\\right)\\ ;\\ \\theta_2\\in \\left(\\phi(\\phi^{-1}(u)-sh),u\\right)\\ ;\\ \\gamma_1\\in\\left(\\phi^{-1}(v)-th,\\phi^{-1}(v)\\right)\\ ;\\ \\gamma_2\\in\\left(\\phi^{-1}(u)-sh,\\phi^{-1}(u)\\right).$\\\\\nThis implies \n\\begin{eqnarray*}\n\\mathbb{E}g^2(U,V,h)&\\leq &4h\\left\\|C_v\\right\\|\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|^2\\left|t\\right|+4h\\left\\|C_u\\right\\|\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|\\left|s\\right|\\\\\n&\\leq & 4h\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|\\left(\\left\\|C_v\\right\\|+\\left\\|C_u\\right\\|\\right).\n\\end{eqnarray*}\nThus condition (G.ii) holds, with $C_0=4\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|\\left(\\left\\|C_v\\right\\|+\\left\\|C_u\\right\\|\\right).$\\\\\n\n\\textbf{Case 4.} $u\\wedge\\phi(\\phi^{-1}(u)-sh)=u\\ \\text{et}\\ v\\wedge \\phi(\\phi^{-1}(v)-th)=\\phi(\\phi^{-1}(v)-th)$.\\\\ \nThis case is analogous to Case 3, where the roles of $u$ and $v$ are interchanged. Hence, condition (G.ii) is fulfilled, with the same constant  $C_0=4\\left\\|\\phi'\\right\\|\\left\\|k\\right\\|\\left(\\left\\|C_v\\right\\|+\\left\\|C_u\\right\\|\\right).$\\\\\n\n\\textbf{Checking for (F.i)}. We have to check the uniform entropy condition for the class of functions\n$$\n\\mathcal{G}=\\left\\{\\begin{array}{c} \nK\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1,n}(s))}{h}, \\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2,n}(t))}{h}\\right)- \\mathbb{I}\\{s\\leq u,t\\leq v\\},\\\\\n u,v\\in[0,1], 0<h<1 \\,\\text{and }\\, \\zeta_{1,n}\\zeta_{2,n}:[0,1]\\mapsto [0,1] \\,\\text{nondecreasing.}\n\\end{array} \\right\\}\n$$\n To this end, we consider the following classes of functions, where $\\varphi$ is an increasing function :\\\\\n\\noindent\n$ \\mathbb{F}=\\left\\{(\\varphi(x)+m)\/\\lambda,\\lambda > 0,\\;m\\in\\mathbb{R}\\right\\}$\\\\\n$\\displaystyle \\mathbb{K}_0=\\left\\{K((\\varphi(x)+m)\/\\lambda),\\lambda>0, \\;m\\in\\mathbb{R}\\right\\}$\\\\\n$\\displaystyle \\mathbb{K}=\\left\\{K((\\phi(x)+m)\/\\lambda,(\\phi(y)+m)\/\\lambda), \\lambda>0,\\;m\\in\\mathbb{R}\\right\\}$\\\\\n$\\displaystyle \\mathbb{H}=\\left\\{K((\\phi(x)+m)\/\\lambda,(\\phi(y)+m)\/\\lambda)-\\mathbb{I}\\left\\{x\\leq u,y\\leq v\\right\\}\\ ;\\lambda> 0, \\;m\\in\\mathbb{R}, (u,v)\\in [0,1]^2\\right\\}$.\\\\\n\nIt is clear that by applying  lemmas 2.6.15 and 2.6.18 in van der Vaart and Wellner (see \\cite{r9}, p. 146-147), the sets $\\mathbb{F},\\;\\mathbb{K}_0,\\;\\mathbb{K},\\;\\mathbb{H}$ are all VC-subgraph classes. Thus, by choosing  the constant function ${\\rm G}(x,y)\\mapsto {\\rm G}(x,y)=\\left\\|k\\right\\|^2+1 $ as an envelope function for the class $\\mathbb{H}$ ( indeed ${\\rm G}(x,y)\\geq \\sup_{g\\in\\mathbb{H}}\\left|g(x,y)\\right|,\\ \\forall (x,y))$, we can infer from Theorem 2.6.7 in \\cite{r9} that  $\\mathbb{H}$ satisfies the uniform entropy condition. Since $\\mathbb{H}$ and $\\mathcal{G}$ have the same structure, we can conclude that $\\mathcal{G}$ satisfies this property too, i.e.\n$$\\exists \\; C_0>0, \\nu_0>0\\ :\\ N\\left(\\epsilon,\\mathcal{G}\\right)\\leq C_0\\epsilon^{-\\nu_0},\\quad 0<\\epsilon<1.$$\n\n\\noindent \\textbf{Checking for (F.ii).}\\\\\n Define the class of functions \n$$\n\\mathcal{G}_0=\\left\\{\\begin{array}{c} \nK\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_{1}(s))}{h}, \\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_{2}(t))}{h}\\right)- \\mathbb{I}\\{s\\leq u,t\\leq v\\},\\\\\n u,v\\in[0,1]\\cap\\mathbb{Q}, 0<h<1 \\,\\text{and }\\, \\zeta_{1}\\zeta_{2}:[0,1]\\mapsto [0,1] \\,\\text{nondecreasing.}\n\\end{array} \\right\\}\n$$\nIt's clear that  $\\mathcal{G}_0$ is countable and  $\\mathcal{G}_0\\subset\\mathcal{G}$. Let\n$$g(s,t)=K\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_1(s))}{h},\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_2(t))}{h}\\right)- \\mathbb{I}\\{s\\leq u,s\\leq v\\}\\in\\mathcal{G}, (s,t)\\in [0,1]^2$$\nand for $m\\geq 1$, \n$$g_m(s,t)=K\\left(\\frac{\\phi^{-1}(u_m)-\\phi^{-1}(\\zeta_1(s))}{h} ,\\frac{\\phi^{-1}(v_m)-\\phi^{-1}(\\zeta_2(y))}{h}\\right)- \\mathbb{I}\\{s\\leq u_m,t\\leq v_m\\},$$\nwhere $ u_m=\\frac{1}{m^2}[m^2u]+\\frac{1}{m^2}$ and $ v_m=\\frac{1}{m^2}[m^2v]+\\frac{1}{m^2}$.\\\\\n\\indent Let $\\alpha_m=u_m-u,\\quad \\beta_m=v_m-v$. Then, we have  $\\displaystyle 0<\\alpha_m\\leq\\frac{1}{m^2}$ and $\\displaystyle 0<\\beta_m\\leq\\frac{1}{m^2}$. Hence $u_m\\searrow u$ and $v_m\\searrow v$. By continuity $\\phi^{-1}(u_m)\\searrow \\phi^{-1}(u)$ and $\\phi^{-1}(v_m)\\searrow \\phi^{-1}(v)$. Define\n$$\\delta_{m,u}=\\left(\\frac{\\phi^{-1}(u_m)-\\phi^{-1}(\\zeta_1(x))}{h} \\right)-\\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_1(x))}{h} \\right)=\\frac{\\phi^{-1}(u_m)-\\phi^{-1}(u)}{h} $$ and\n$$\\delta_{m,v}=\\left(\\frac{\\phi^{-1}(v_m)-\\phi^{-1}(\\zeta_2(y))}{h} \\right)-\\left(\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_2(y))}{h} \\right)=\\frac{\\phi^{-1}(v_m)-\\phi^{-1}(v)}{h}.$$\nThen\n$\\delta_{m,u}\\searrow 0$ and $\\delta_{m,v}\\searrow 0$, \nwhich are equivalent to $$ \\left(\\frac{\\phi^{-1}(u_m)-\\phi^{-1}(\\zeta_1(x))}{h} \\right)\\searrow \\left(\\frac{\\phi^{-1}(u)-\\phi^{-1}(\\zeta_1(x))}{h} \\right)$$\n and $$\\left(\\frac{\\phi^{-1}(v_m)-\\phi^{-1}(\\zeta_2(y))}{h} \\right)\\searrow \\left(\\frac{\\phi^{-1}(v)-\\phi^{-1}(\\zeta_2(y))}{h} \\right).$$\nBy right-continuity of the kernel $K(\\cdot,\\cdot)$, we obtain\n$$\\forall (x,y)\\in[0,1]^2, g_m(x,y)\\longrightarrow g(x,y), m\\rightarrow \\infty$$\nand conclude that $\\mathcal{G}$ is pointwise measurable class.\\\\\n\\end{proof}\n\n\\addcontentsline{toc}{section}{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{\\label{intro}Introduction}\nQuantum information science is growing rapidly not only in theory but also at the hardware level; this leads us to believe that quantum computers will soon be practically applied on a large scale. Once quantum computers are realized on a commercial scale, quantum technology (such as devices or communications) will be widespread. In such a future, an interesting question that arises is how quantum computers can change the world. For example, in quantum game theory, Meyer \\cite{PhysRevLett.82.1052} showed that for a two-player zero-sum strategic game, a player's expectation can increase when using quantum strategies. After Meyer's work, Eisert, Wilkens, and Lewenstein (EWL) \\cite{PhysRevLett.83.3077} showed that quantum entanglement changes the Nash equilibrium \\cite{Nash48} between the strategies of two persons in Prisoners' Dilemma (PD). The strategy of whether to cooperate or defect adopted by one person affects another person's strategy through quantum entanglement between the two individual strategies. Prisoners can choose quantum strategies, which are described by unitary operators; therefore, they have more strategies at hand compared to the classical game and can use them to increase their profit. There are many studies inspired by their quantization scheme (e.g. summarized in \\cite{doi:10.1142\/S0219477502000981,Khan2018,GUO2008318}), and not a few quantum games induce nontrivial results that are not expected classically.\n\nThese ideas of quantizing the strategies can also be applied to gambling theory. A fair quantum gambling game played by two players far from each other was proposed by \\cite{PhysRevLett.82.3356}. In the classical case, it is difficult to ensure fair play without introducing a third party to oversee the game. However, by providing a quantum superposition state and then observing it, it is possible to achieve fair gambling. In addition, procedures were devised to protect the security of such remote gambling from entanglement attacks \\cite{PhysRevA.64.064302,PhysRevA.66.052311}. Zhang {\\it et al.} devised and realized an optical quantum circuit that helps achieve fair gambling and experimentally showed that the error is practically under control \\cite{Zhang_2008}. In the sense of quantizing a game, some quantized games have been proposed \\cite{Schmidt_2012,SCHMIDT2013400,FRACKIEWICZ20148,qchess,Fuchs_2020,bleiler2009quantized}. Quantum duel was proposed \\cite{Schmidt_2012}, and a Russian roulette game using a quantum gun was proposed and studied \\cite{SCHMIDT2013400,FRACKIEWICZ20148}. Quantum chess was devised as a more casual game \\cite{qchess}. Quantized strategies in poker, one of the most popular card games, were also considered in \\cite{bleiler2009quantized} using Meyer's quantized procedure \\cite{PhysRevLett.82.1052}. Recently, a game similar to poker was presented on a quantum computer by \\cite{Fuchs_2020}. In their experiment, the researchers investigated the expected value under quantum noise on a practical quantum computer. They also showed that the expected value can be improved using quantum error-mitigation techniques. These studies suggest that quite a few games based on quantum mechanics have been examined.\n\nRecent studies have shown that quantum bits yield advantages to blackjack, which is one of the most popular casino games \\cite{PhysRevA.102.012425}; the researchers showed that the expectation values of two cooperating persons can increase by imparting information of each individual's card with entangled qubits. In the present paper, strategies and cards are represented as qubits and we devised a quantum circuit for reproducing classical blackjack. Classical blackjack has been studied statistically for more than half a century \\cite{doi:10.1080\/01621459.1956.10501334,BOND1974413,Thorp1966BeatTD}, and it is decided whether the player should do {\\it hit} or {\\it stand}. This is called {\\it Basic Strategy}. Moreover, it is widely known in various rules, such as number of decks, options, and refunds when the player has blackjack \\cite{Thorp1966BeatTD,wizard}. In blackjack, as the player can choose strategies (e.g., hit or stand), we can quantize not only the game itself but also the strategies involved using a method similar to \\cite{PhysRevLett.82.1052,PhysRevLett.83.3077}. This quantum circuit reproducing blackjack can be easily realized in the near future when quantum computers are commonplace. Furthermore, we showed that the player's expectation value increases substantially when quantum strategies are employed in the presence of entanglement.\n\nIn the next section \\ref{rule}, we explain the blackjack rule that we adopted (simplified for ease of calculation). In section \\ref{cbj}, we reformulate the blackjack in order to regard it as a object of the quantum game theory. In section \\ref{qbj}, we propose a quantum circuit that reproduces blackjack, (especially a toy model,) and express the possibility of entanglement. In section \\ref{res}, we show classical expectation value and basic strategy; we also show that using quantum strategies in the presence of entanglement, the player's expectation value increases compared to the classical game. Some conclusions are shown in section \\ref{conclusion}.\n\n\\section{\\label{rule}Blackjack rules and toy model}\nBlackjack is a one-on-one card game between a dealer (Alice) and a player (Bob). One deck comprises 52 cards excluding jokers. Aces are worth 1 (hard) or 11 (soft), face cards (JQK) are worth 10, and every other card has its number value.\nAfter the player has placed his bet, the dealer distributes two cards to the player. Similarly, she distributes two cards to herself. At this time the dealer has one card face-up and the other face-down. The objective is to get a hand with a total closer to 21. It is worth noting however, that the 21 hand which is made of one ace and one face card or 10 is the strongest (called natural). \n\nFirst, the player must choose to hit or stand. If he chooses to hit, he draws a card. If he exceeds 21 (bust), he loses his bet regardless of the dealer's hand. Next, the dealer opens the face-down card, and if her total is soft 16 or less, she must hit. She repeats this procedure until her total reaches soft 17 or more. If the dealer busts and the player does not, or if the player's hand is closer to 21 than the dealer's, the player wins the bet. If the player's total equals the dealer's total, the bet will be returned (push). In blackjack, the player can do his best by calculating the expected value of the profit that he will obtain in each initial state. This is widely known as a basic strategy and has been calculated in various rules \\cite{Thorp1966BeatTD,wizard}. Note that only the player can choose whether to hit or stand. Conversely, the dealer has no choice of a strategy since she just draws the card mechanically until her total reaches soft 17 or more. At first glance, the player who has many choices seems to have a competitive edge. However, the dealer has the advantage because the player loses if both hands exceed 21. The reason many casinos adopt option rules, such as double down, split, and surrender is to increase the player's expectation.\n\n\nFor simplicity, we quantized a toy model of blackjack proposed by Ethier \\cite{Ethier2010} and called {\\it snackjack} by Epstein \\cite{epstein}. As snackjack is highly simplified compared to blackjack which uses 52 cards, it is easy to calculate the basic strategy by hand. Recently, Ethier calculated the expectation and basic strategy under various options, with card counting, and with bet variation in this classical snackjack \\cite{ethier2019snackjack}. Snackjack uses eight cards AA223333, aiming toward a maximum value of 7, with an ace having a value of either 1 or 4, and 2 and 3 having their own values. In this paper, we ignored various options and simplified them for calculation. Tab. \\ref{tab1} shows the rules we followed.\n\n\\begin{table}[h]\n    \\centering\n    \\caption{Our rulesets}\n    \\begin{tabular}{|c||c|} \\hline\n     ace means & 1 or 4 \\\\ \\hline\n     aim to & 7 \\\\ \\hline\n     natural is & ace and 3 \\\\ \\hline\n     player chooses  & hit or stand (only one time) \\\\ \\hline\n     dealer stands  & soft 6 or more \\\\ \\hline\n     blackjack pays & 1 to 1  \\\\ \\hline\n    \\end{tabular}\n    \\label{tab1}\n\\end{table}\n\n\n\\section{\\label{cbj}Classical snackjack as a quantum game}\nReferring to \\cite{doi:10.1080\/09500340008232180,doi:10.1142\/S0219749904000092}, we formulate the blackjack played by a player and a dealer as a {\\it 2 player quantum game}.\nBlackjack is regarded as the game in which the pay-off is determined by the initial hand cards of the player and the dealer.\nMore precisely, blackjack is a mapping that assigns a {\\it 2 player quantum game} $\\Gamma_{p,d}$ to an element $(p,d)$ of $T_{1}\\times T_{2}$ where $T_{1}$ is a set of a pair of initial hand cards of the player, $T_{2}$ is a set of an up card of the dealer.\nThe quantum games $\\Gamma_{p,d}=(\\mathcal{H},\\rho,\\Sigma_1,\\Sigma_2,P_1,P_2)$ are specified by Hilbert space $\\mathcal{H}$ for the states, the initial state $\\rho\\in\\mathcal{H}$, the sets $\\Sigma_1$ and $\\Sigma_2$ of the strategies of the player and the dealer, and the player's utility $P_{1}$ and the dealer's utility $P_{2}$.\nEach player has a qubit and the state of the system is represented as the tensor product of their qubit.\nThe strategies $\\hat{S}_1(\\in \\Sigma_1)$ and $\\hat{S}_2(\\in \\Sigma_2)$ selected by players are regarded as $2\\times 2$ unitary operators acting on their own qubit.\nIn this formalism, classical blackjack is the quantum game specified by\n\\begin{gather}\n\t\\mathcal{H}=\\left\\{a_{00}\\ket{0}\\otimes\\ket{0}+a_{01}\\ket{0}\\otimes\\ket{1}+a_{10}\\ket{1}\\otimes\\ket{0}+a_{11}\\ket{1}\\otimes\\ket{1}|\\,a_{00},a_{01},a_{10},a_{11}\\in\\mathbb{C}\\right\\}, \\notag \\\\\n\t\\rho = \\ket{0}\\otimes\\ket{1}, ~~~\\Sigma_1 = \\{\\hat{I},\\hat{X}\\}, ~~~\\Sigma_2 = \\{\\hat{I}\\},\n\\end{gather}\nwhere $\\ket{0}$ ($\\ket{1}$) denotes stand (hit) and $\\hat{X}$ is the first Pauli matrix. The player has the first qubit and the dealer has the second. We will call each of them {\\it strategy bit}.\nNote that the hit $\\ket{1}$ for the dealer means that he draws a card if the score of his hand is less than 6.\nSince the utilities are determined by $(p,d)\\in T_{1}\\times T_{2}$ and player's strategy $\\hat{S}_{1}\\in\\Sigma_{1}$, we can write them as $P_1(p,d,\\hat{S}_1)$ and $P_2(p,d,\\hat{S}_1)$.\nConcretely,\n\t\\begin{align}\n\tP_1(p,d,\\hat{S}_1)\n\t=E_{\\mathrm{std}}\\bra{0}\\otimes\\bra{1}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\rho+E_{\\mathrm{hit}}\\bra{1}\\otimes\\bra{1}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\rho,\n\t\\label{utiladd1}\n\t\\end{align}\nwhere $E_{\\mathrm{std}}$ and $E_{\\mathrm{hit}}$ are pay-offs corresponding to stand and hit.\nIt is a point to notice that $E_{\\mathrm{std}}$ and $E_{\\mathrm{hit}}$ depend on the initial hand cards $(p,d)$.\nThe utilities do not have $\\hat{S}_{2}$ dependence since $\\Sigma_{2}=\\{\\hat{I}\\}$ in the classical blackjack. Especially in our snackjack (Tab. \\ref{tab1}), $P_2(p,d,\\hat{S}_1)=-P_1(p,d,\\hat{S}_1)$, thus it is a zero-sum game.\n\n\\section{\\label{qbj}Entanglement and quantum circuits}\n\\subsection{\\label{entangle}The case of the presence of entangle}\nIn this section, using EWL quantization protocol \\cite{PhysRevLett.83.3077}, we investigate the quantum blackjack including an entanglement between the player and the dealer. Eisert {\\it et al.} prepared the entangled player's strategies in 2 players symmetric game, especially in PD game (Fig. \\ref{add1}). They showed that the dilemma disappears by using restricted quantum strategies and an entanglement. After their work, in PD game, some studies about generalized strategy spaces \\cite{PhysRevLett.87.069801,PhysRevLett.88.137902,Du_2003,ICHIKAWA2007531,Ichikawa_2008} or repeated games \\cite{IQBAL2002541,Frackiewicz_2012,aoki2020repeated} have been worked. In generalized strategy spaces, the players are not able to escape from the dilemma \\cite{PhysRevLett.87.069801,Ichikawa_2008}, and van Enk {\\it et al.} \\cite{PhysRevA.66.024306} pointed out that EWL protocol does not preserve the non-cooperative aspects of PD. Our blackjack game, however, does not have any dilemmas due to the asymmetry of only the player seeking a win and trying to beat the dealer who has limited strategy space $\\Sigma_2=\\{\\hat{I}\\}$. Thus we straightforwardly adopt EWL protocol. In section \\ref{conclusion}, we will comment on the degree of freedom of the dealer's strategy in this quantized blackjack game.\n \\begin{figure}[t]\n     \\centering\n     \\includegraphics[scale=0.8]{circuit4.pdf}\n     \\caption{EWL quantization scheme}The gate $\\mathcal{J}$ entangles the two qubits and $\\mathcal{J}^\\dagger$ disentangles those. The players choose an unitary operator $U$ as a quantum strategy.\n     \\label{add1}\n \\end{figure}\n \nWe consider entanglement between the player's and dealer's strategies. When the player selects the strategy $\\hat{S_{p}}$, the state of the system just before the observation is\n\\begin{equation}\n    \\mathcal{J}^\\dagger (\\hat{S}_p \\otimes \\hat{I})\\mathcal{J} \\ket{0} \\otimes \\ket{1},\n\\end{equation}\nwhere $\\mathcal{J}$ is $4\\times4$ complex matrix acting on both qubits.\n\nClassically, all strategies the player can choose are $\\hat{X}$ or $\\hat{I}$. We assume \n\\begin{equation}\n\\mathcal{J} = \\mathrm{exp}(-i\\frac{\\gamma}{2} \\hat{X} \\otimes \\hat{U})\n\\end{equation}\nso that the game reproduces the classical game if the player selects the classical strategies.\nThe first part is the player's strategy sector and the second is the dealer's, and operator $\\hat{U}$ is unitary and hermitian. $\\gamma$ is the measure for the game's entanglement \\cite{PhysRevLett.83.3077} and we will call $\\gamma$ {\\it entangle intensity} below. When $\\hat{S}_1 = \\hat{X} \\mspace{5mu} \\mathrm{or} \\mspace{5mu} \\hat{I}$, it follows\n\\begin{equation}\n\\mathcal{J}^\\dagger ( \\hat{X} \\otimes \\hat{I} ) \\mathcal{J} = \\hat{X} \\otimes \\hat{I},\n\\end{equation}\n\\begin{equation}\n\\mathcal{J}^\\dagger ( \\hat{I} \\otimes \\hat{I} ) \\mathcal{J} = \\hat{I} \\otimes \\hat{I}.\n\\end{equation}\nIn this way, without breaking the classical game, we can insert the operator making entangle strategies.\n\nA unitary and hermitian operator is written by\n\\begin{equation}\n\\hat{U} = s_0 \\hat{I} + s_1 \\hat{X} + s_2 \\hat{Y} + s_3 \\hat{Z} \n\\label{Uop1}\n\\end{equation}\nwhere\n\\begin{equation}\n(s_0^2 + s_1^2 + s_2^2 + s_3^2)\\hat{I} +2s_0(s_1\\hat{X} + s_2\\hat{Y} + s_3\\hat{Z}) = \\hat{I} \\ \\  (s_\\mu \\in \\mathbb{R}).\n\\label{Uop2}\n\\end{equation}\nIn this $\\Sigma_2=\\{\\hat{I}\\}$ game, except for relative phases, $\\hat{I}$, $\\hat{Z}$ and $\\hat{X}$, $\\hat{Y}$ have the same effect on a strategy bit respectively, hence we can set $s_0 = s_2 = 0,s_1 = \\sin\\theta , s_3 = \\cos\\theta$ without losing generality. Here $\\theta$ is {\\it game parameter}.\n\nIn the case of general player's strategy $\\hat{S}_1$, it follows\n\\begin{equation}\n\\begin{split}\n\\left( \\mathcal{J^\\dagger}(\\hat{S_1} \\otimes \\hat{I}) \\mathcal{J} \\right) \\ket{0} \\otimes \\ket{1} = \\left\\{ \\cos^2{\\frac{\\gamma}{2}} \\hat{X}\\{\\hat{X},\\hat{S_1} \\} + (\\sin^2{\\frac{\\gamma}{2}} - \\cos^2{\\frac{\\gamma}{2}}) \\hat{X}\\hat{S_1}\\hat{X} \\right\\} \\ket{0} &\\otimes \\ket{1}  \\\\\n+ i\\cos{\\frac{\\gamma}{2}}\\sin{\\frac{\\gamma}{2}}\\sin{\\theta} [ \\hat{X} , \\hat{S_1} ] \\ket{0} &\\otimes \\ket{0} \\\\ -i\\cos{\\frac{\\gamma}{2}}\\sin{\\frac{\\gamma}{2}}\\cos{\\theta} [ \\hat{X} , \\hat{S_1} ] \\ket{0} &\\otimes \\ket{1}.\n\\end{split}\n\\end{equation}\nFrom here, we limit $\\Sigma_1$ to $\\{\\hat{I},\\hat{X},\\hat{Y},\\hat{Z}\\}$. Both strategies bits after through $\\mathcal{J}^\\dagger$ are\n\\begin{align}\n\\begin{aligned}\n\\label{iop}\n\\hat{S_1} &= \\hat{I} ~:~\\ket{0}\\otimes\\ket{1} \\\\\n\\hat{S_1} &= \\hat{X} :~\\ket{1}\\otimes\\ket{1},\n\\end{aligned}\n\\end{align}\n\\begin{align}\n\\begin{aligned}\n\\label{zop}\n\\hat{S_1} &= \\hat{Y} :~i\\cos\\gamma \\ket{1} \\otimes \\ket{1} + \\sin\\gamma\\cos\\theta \\ket{0} \\otimes \\ket{1} - \\sin\\gamma\\sin\\theta \\ket{0} \\otimes \\ket{0} \\\\\n\\hat{S_1} &= \\hat{Z} :~\\cos\\gamma \\ket{0} \\otimes \\ket{1} - i\\sin\\gamma\\cos\\theta \\ket{1} \\otimes \\ket{1} + i\\sin\\gamma\\sin\\theta \\ket{1} \\otimes \\ket{0}.\n\\end{aligned}\n\\end{align}\nThe state of 01 means stand, and 11 means hit. As it is clear from (\\ref{zop}), when the player chooses $\\hat{Y}$ or $\\hat{Z}$, he can do operations 00 and 10 that do not exist in the classical game.\n\nWe summarize the quantized blackjack, employing the formalism of the 2 player quantum game $\\Gamma_{p,d}=(\\mathcal{H},\\rho,\\Sigma_1,\\Sigma_2,P_1,P_2)$ introduced in the previous subsection.\nIn order to get the quantized blackjack, we modify two settings of the classical blackjack.\nThe modifications are expanding the set of player's strategy to $\\Sigma_{1}=\\{\\hat{I},\\hat{X},\\hat{Y},\\hat{Z}\\}$ and changing the utilities to\n\t\\begin{align}\n\tP_1(p,d,\\hat{S}_1)\n\t~=~~&E_{00}\\bra{0}\\otimes\\bra{0}\\mathcal{J}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\mathcal{J}^{\\dagger}\\rho\n\t +E_{\\mathrm{std}}\\bra{0}\\otimes\\bra{1}\\mathcal{J}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\mathcal{J}^{\\dagger}\\rho \\notag \\\\\n\t+~&E_{10}\\bra{1}\\otimes\\bra{0}\\mathcal{J}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\mathcal{J}^{\\dagger}\\rho\n\t +E_{\\mathrm{hit}}\\bra{1}\\otimes\\bra{1}\\mathcal{J}\\left(\\hat{S}_1\\otimes\\hat{I}\\right)\\mathcal{J}^\\dagger \\rho\n\t \\label{qutility}\n\t\\end{align}\nwhere $E_{00}$ and $E_{10}$ are the expectation values corresponding to operations 00 and 10 respectively. From (\\ref{iop})-(\\ref{qutility}), $P_1$ corresponding to each strategy $\\{\\hat{I}, \\hat{X}, \\hat{Y}, \\hat{Z}\\}$ is \n\\begin{equation}\n\\label{exp}\n\\{E_\\mathrm{std},E_\\mathrm{hit},\\sin^2\\gamma(\\cos^2\\theta E_\\mathrm{std} +\\sin^2\\theta E_{00}) +\\cos^2\\gamma E_\\mathrm{hit}, \\cos^2\\gamma E_\\mathrm{std} +\\sin^2\\gamma(\\cos^2\\theta E_\\mathrm{hit}+\\sin^2\\theta E_{10}) \\}.\n\\end{equation}\n\n\\subsection{\\label{q-circuit}Quantum circuit reproducing blackjack}\nIn this section, we show the circuit which reproduces the games discussed above. First of all, we introduce the circuit which reproduces the classical blackjack defined in section \\ref{cbj}. Fig. \\ref{circuit1} and Fig. \\ref{circuit2} show the quantum circuit.\nLet $\\mathcal{H}_{\\rm whole}$ be Hilbert space, on which the circuit in Fig. \\ref{circuit1} works and consists of some spaces as follows:\n    \\begin{align}\\label{whole Hilbert}\n    \\mathcal{H}_{\\rm whole}=\\mathcal{H}_{\\rm deck}\\otimes\\mathcal{H}_{\\rm deck\\,copy}\\otimes\\mathcal{H}_{\\rm p\\mathchar`-hand}\\otimes\\mathcal{H}_{\\rm control}\\otimes\\mathcal{H}_{\\rm p\\mathchar`-strategy}\\otimes\\mathcal{H}_{\\rm d\\mathchar`-hand}\\otimes\\mathcal{H}_{\\rm d\\mathchar`-strategy}.\n    \\end{align}\nWe in turn explain the component spaces of $\\mathcal{H}_{\\rm whole}$ appearing in eq. \\eqref{whole Hilbert}.\nWe first define a deck state $\\ket{D}\\in\\mathcal{H}_{\\rm deck}$, a copied deck state $\\ket{D}_{\\rm copy}\\in\\mathcal{H}_{\\rm deck\\,copy}$, an initial player's hand $\\ket{p}\\in\\mathcal{H}_{\\rm p\\mathchar`-hand}$, and dealer's hand $\\ket{d}\\in\\mathcal{H}_{\\rm d\\mathchar`-hand}$ to regard $p\\in T_{1}$ and $d\\in T_{2}$ as qubits. These take this form\n\\begin{equation}\n\\ket{A^{(1)}} \\otimes \\ket{A^{(2)}} \\otimes \\ket{2^{(1)}} \\otimes \\ket{2^{(2)}} \\otimes \\ket{3^{(1)}} \\otimes \\ket{3^{(2)}} \\otimes \\ket{3^{(3)}} \\otimes \\ket{3^{(4)}},\n\\label{deck}\n\\end{equation}\nwhere,\n\\begin{gather}\n\\ket{A^{(n)}},\\ket{2^{(n)}},\\ket{3^{(m)}} = \\alpha \\ket{0} + \\beta \\ket{1} \\;\\;\\;\\;(n=1,2,~m=1,2,3,4,~ \\alpha,\\beta \\in \\mathbb{C}), \\\\\n|\\alpha|^2 + |\\beta|^2 = 1,\n\\end{gather}\nand then $\\ket{1}$ means an existing card corresponding to each card sector and $\\ket{0}$ means not. $\\mathcal{H}_{\\rm deck},\\mathcal{H}_{\\rm deck\\,copy},\\mathcal{H}_{\\rm p\\mathchar`-hand}$ and $\\mathcal{H}_{\\rm d\\mathchar`-hand}$ are spanned by these $2^8$ bases. Secondly, $\\mathcal{H}_{\\rm p\\mathchar`-strategy}\\otimes\\mathcal{H}_{\\rm d\\mathchar`-strategy}$ is the same as the space $\\mathcal{H}$ in section \\ref{cbj} and section \\ref{entangle}. Finally, $\\ket{\\Psi}$ consists of three q-bits and lies on $\\mathcal{H}_{\\rm control}$, which is a vector space of dimension $2^{3}$. Note that in the actual blackjack there is no object to which the element of $\\mathcal{H}_{\\rm deck\\,copy}$ or $\\mathcal{H}_{\\rm control}$ corresponds. We, however, need to incorporate these two spaces into our circuit due to some technical reasons. We refer to this in detail below.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=15.5cm]{circuit1.pdf}\n\\caption{Quantum circuit reproducing the classical game}\n\\label{circuit1}\n\\end{figure*}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=11cm]{circuit2.pdf}\n\\caption{Detail of {\\it Hit} operator }\n\\label{circuit2}\n\\end{figure}\n\nWe explain the outline of Fig. \\ref{circuit1} and Fig. \\ref{circuit2} as follows. The player's strategy is expressed by a unitary operator $\\hat{S_1}$ acting on its strategy bit. In Fig. \\ref{circuit1}, first the dealer and player get the cards classically, then the player chooses the strategy $\\hat{S}_1 = \\hat{X}$ or $\\hat{I}$ which means hit or stand respectively. If the player chooses hit, he has $\\hat{X}$ operate to his strategy bit. In that case {\\it Hit} operator $\\hat{\\mathcal{H}}$ acts on the deck bit and his hand bit (in Fig. \\ref{circuit2}). This operation signifies drawing a card from the deck. Now, we can expands the player's strategy space $\\Sigma_1$ to the set of $2 \\times 2$ unitary matrices thus he can overlap his strategy bit. For example, when he chooses the Hadamard gate, the cases of hit and stand overlap with a 1\/2 probability. As is clear from (\\ref{utiladd1}), this does not affect the maximum value expected by the player. After the player's operation, the dealer turns over her face-down card by $\\hat{\\mathcal{H}}$ operating, and if the total of her hand ($\\equiv |d|$) is 5 or less, she must draw a card one more time. This operation is repeated until $|d| \\ge 6$.\n\nWe next explain the detail of {\\it Hit} operator, surrounded by the dotted line in Fig. \\ref{circuit1}. Control bit $\\ket{\\Psi} (\\in \\mathcal{H}_{\\rm control})$ is responsible for the probability of appearance of cards and it is comprised of three qubits and takes the following form\n\\begin{eqnarray}\n\\ket{\\Psi} &=& \\textstyle\\sum\\limits_{ijk=0}^1 \\frac{1}{\\sqrt{8}}\\ket{ijk} \\\\ \\nonumber\n&=& \\frac{1}{\\sqrt{8}}  ( \\ket{000} + \\ket{001}  + \\ket{010} + \\ket{100} + \\ket{110} + \\ket{101} + \\ket{011} + \\ket{111} ).\n\\end{eqnarray}\nEach state corresponds to a sector of the card, e.g., 000 to the first ace, 001 to the second ace, 010 to the first deuce , and so on. Therefore, the probability that every particular card is drawn is equal when the unitary {\\it Hit} operator acts on the deck and hand bit. In Fig. \\ref{circuit2}, $\\ket{\\cdot_i} (i=1,2,\\cdots,8)$ means the sector of the card in (\\ref{deck}), e.g., $\\ket{\\cdot_1}$ corresponds to the sector of the first ace $\\ket{A^{(1)}}$. When $\\ket{D_1} = \\ket{D_1}_{copy} = \\ket{1}$, (which indicates the existence of the first ace,) $\\ket{D_1}$ and $\\ket{p_1}$, corresponding to $\\ket{000}$ which is a component of $\\ket{\\Psi}$, are swapped. Note that, the deck states can be copied without violating the no-cloning theorem \\cite{Wootters1982,DIEKS1982271} because the deck state comprises only $\\ket{0}$ and $\\ket{1}$, which are orthogonal. After measuring the control bit, it is determined which card has been drawn.\n\n\nNote that, $\\ket{\\Psi}$ converges on a state that corresponds to the sector of the card that has already been drawn with a 3\/8 probability at the time of first hit even though $\\hat{\\mathcal{H}}$ operates. For example, when a control bit converges on $\\ket{000}$ despite $\\ket{D_1} = \\ket{D_1}_{copy} = \\ket{0}$, there is no exchange. To avoid this problem, we ensure that this procedure is repeated until $\\ket{p} \\neq \\ket{p^\\prime}$. Even if a strategy bit has $\\ket{0}$ component, we can employ this procedure as we know which sector of the card bit was exchanged, by measuring not the hand bit, but the control bit. This is explained below. For example, in the case of the initial cards states \n\\begin{align}\n&\\ket{p}=\\ket{A^{(1)},2^{(1)}} \\equiv \\ket{1}\\ket{0}\\ket{1}\\ket{0}\\ket{0}\\ket{0}\\ket{0}\\ket{0}, \\notag \\\\ \n&\\ket{d}=\\ket{3^{(1)}}\\equiv \\ket{0}\\ket{0}\\ket{0}\\ket{0}\\ket{1}\\ket{0}\\ket{0}\\ket{0}, \\notag\\\\\n&\\ket{D}=\\ket{A^{(2)},2^{(2)},3^{(2)},3^{(3)},3^{(4)}} \\equiv \\ket{0}\\ket{1}\\ket{0}\\ket{1}\\ket{0}\\ket{1}\\ket{1}\\ket{1},\n\\end{align}\nand the overlapped player's strategy bit with Hadamard operator $\\hat{H}$ acting on, the first state is  \n\\begin{equation}\n    \\ket{p}\\ket{D}\\ket{\\Psi}\\ket{d}\\hat{H}\\ket{0} = \\ket{p}\\ket{D}\\ket{\\Psi}\\ket{d}\\frac{1}{\\sqrt{2}}(\\ket{0}+\\ket{1}),\n\\end{equation}\nwhere the last state is the player's strategy bit.\nHere, we omitted the dealer's strategy bit, and the copy bits. After thorough $\\hat{\\mathcal{H}}$, the state is\n\\begin{equation}\n\\frac{1}{\\sqrt{2}}\\left\\{\\ket{p}\\ket{D}\\ket{\\Psi}\\ket{d}\\ket{0} + \\hat{\\mathcal{H}}\\left(\\ket{p}\\ket{D}\\ket{\\Psi}\\right)\\ket{d}\\ket{1}\\right\\},\n\\end{equation}\nand after measuring the control bit, it will be\n\\begin{eqnarray}\n    \\begin{cases} \\ket{p}\\ket{D}\\ket{d}\\ket{0} + \\ket{p}\\ket{D}\\ket{d}\\ket{1} &: \\mspace{6mu} \\ket{\\Psi} = \\ket{000},  \\ket{010} \\mathrm{or} \\ket{110} \\\\  \\ket{p}\\ket{D}\\ket{d}\\ket{0} + \\ket{A^{(1)},A^{(2)},2^{(1)}}\\ket{2^{(2)},3^{(2)},3^{(3)},3^{(4)}}\\ket{d}\\ket{1} &: \\mspace{6mu} \\ket{\\Psi} =  \\ket{001} \\\\ \\ket{p}\\ket{D}\\ket{d}\\ket{0} + \\ket{A^{(1)},2^{(1)},2^{(2)}}\\ket{A^{(2)},3^{(2)},3^{(3)},3^{(4)}}\\ket{d}\\ket{1} &: \\mspace{6mu} \\ket{\\Psi} = \\ket{100} \\\\ \\mspace{30mu} \\vdots \\\\ \n     \\ket{p}\\ket{D}\\ket{d}\\ket{0} + \\ket{A^{(1)},2^{(1)},3^{(4)}}\\ket{A^{(2)},2^{(2)},3^{(2)},3^{(3)}}\\ket{d}\\ket{1} &: \\mspace{6mu} \\ket{\\Psi} = \\ket{111}. \\end{cases}\n\\end{eqnarray}\n\nThe overall constant has been dropped. When the control bit converges $\\ket{000}$, $\\ket{010}$ or $\\ket{110}$ in the first line, we repeat this one more time. In this way, we can judge whether we must repeat this procedure without measuring the strategy bit or hand bit. In the following discussions, though we omit specifying this, we always perform this procedure.\n\\clearpage\n\n\nDealer's operation {\\it Hit} is the same. First, the dealer hits regardless of her strategy bit (corresponding to turn over her face-down card), and if her total $|d|$ is 5 or less and her strategy bit is $\\ket{1}$, she repeats the hit operation. Again, we can determine her hand total only by measuring the control bit; therefore, we can build the algorithm where she must hit until her total $|d|$ reaches 6 without measuring her hand bit. Note that when the strategy bit has $\\ket{0}$ component, the same algorithm is used. The summary of the algorithm is shown by Fig. \\ref{flowchart}.\n\nFinally, after measuring both strategy bits, all states converge. As explained in section \\ref{rule}, the player wins when his hand does not exceed 7 and is closer to 7 than the dealer's hand. This quantum circuit reproduces classical snackjack, and therefore the player's expectation and basic strategy must be the same as the classical ones. Such a circuit can be applied to blackjack in a same way by increasing qubits, although it is not considered after this.\n\nWe can also design the quantum circuit which reproduces the quantized blackjack formulated in section \\ref{entangle}, inserting EWL protocol in  our circuit. The modified quantum circuit is shown in Fig. \\ref{fig3}. There is no difference between the circuit in Fig. \\ref{fig3} and Fig. \\ref{circuit1} except the insertion of EWL protocol.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.6]{flowchart.pdf}\n    \\caption{The Process of the game}\n    \\label{flowchart}\n\\end{figure}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=15.3cm]{circuit3.pdf}\n\\caption{Entangled strategy}\n\\label{fig3}\n\\end{figure}\n\\clearpage\n\n\n\\section{\\label{res}Results}\n\nClassically, Tab. \\ref{tab2} shows $E_{\\mathrm{std}},E_{\\mathrm{hit}}$ in all initial states and basic strategy. Here, we assume that the player bets unit per one game. Some calculations of $E_{\\mathrm{std}}$ and  $E_{\\mathrm{hit}}$ in the case of No. 14 game are shown in Fig. \\ref{stdandhit}.\n\n\\begin{table}[h]\n\\centering\n\\scalebox{0.8}[0.8]{\n\\begin{tabular}{ccccccccc}\nNo & initial state (As,2s,3s) & dealer up& $E_{\\mathrm{std}}$ & $E_{\\mathrm{hit}}$ & number of cases & CBS \\\\ \\hline \\hline\n1 & (2,0,0) & 2 & 1\/5 & 1\/5 & 2 & I,X \\\\\n2 & & 3 & -2\/5 & 1\/5 & 4 & X \\\\\n3 & (0,2,0) & 1 & -3\/5 & -3\/5 & 2 & I,X \\\\\n4 & & 3 & -1 & -2\/5 & 4 & X \\\\\n5 &(0,0,2) & 1 & -8\/15 & -4\/5 & 12 & I \\\\\n6 & & 2 & -1\/30 & -1\/3 & 12 & I \\\\\n7 & & 3 & -2\/5 & -7\/15 & 12 & I \\\\\n8 & (1,1,0) & 1 & -4\/5 & -4\/5 & 4 & I,X \\\\\n9 & & 2 & 3\/5 & 3\/5 & 4 & I,X \\\\\n10 & & 3 & -1\/20 & -1\/20 & 16 & I,X \\\\\n11 & (1,0,1) & 1 & 2\/5 & -3\/10 & 8 & I \\\\\n12 & & 2 & 1 & 1\/2 & 16 & I \\\\\n13 & & 3 & 4\/5 & 1\/30 & 24 & I \\\\\n14 & (0,1,1) & 1 & -4\/5 & -17\/20 & 16 & I \\\\\n15 & & 2 & -2\/5 & -2\/5 & 8 & I,X \\\\\n16 & & 3 & -4\/5 & -13\/30 & 24 & X \\\\ \\hline\ntotal &&&&&168& \\\\\n\\end{tabular}\n}\n\\caption{Classical basic strategy (CBS)}No. 1 to 16 are the types of initial states. Each initial state is determined by classical probability.\n\\label{tab2}\n\\end{table}\n\\begin{figure}[h]\n  \\begin{minipage}[b]{0.45\\linewidth}\n    \\centering\n    \\mbox{\\raisebox{7.5mm}{\\includegraphics[scale=0.6]{stand.pdf}}}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.45\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.6]{hit.pdf}\n  \\end{minipage}\n  \\caption{}The calculations of $E_{\\mathrm{std}}$ and $E_{\\mathrm{hit}}$ in No. 14. The hand states are denoted as (As,2s,3s;total).\n  \\label{stdandhit}\n\\end{figure}\n\nIn the classical game, when the player chooses the basic strategy, the expectation value of this game is -1.7\\%; thus, the dealer has the advantage in this game. Although the player can also choose $\\hat{Y}$ or $\\hat{Z}$ in the absence of entanglement, these effects on the strategy bit are the same as $\\hat{X}$ and $\\hat{I}$, respectively. Therefore, it does not affect the overall expectation value and basic strategy. Note here, that our classical results are different from \\cite{ethier2019snackjack}, because of differences in the rulesets (Tab. \\ref{tab1}) of snackjack.\n\nIn the presence of entanglement, the player has a meaningful quantum strategy, $\\hat{Y}$ or $\\hat{Z}$. By the previous section (\\ref{zop})-(\\ref{exp}), when parameter $\\gamma = \\theta = \\frac{\\pi}{2}$, expectation value $E_Y$ which player chooses strategy $\\hat{Y}$, equal to $E_{00}$. And also $E_Z = E_{10}$. Tab. \\ref{tab3} shows quantum basic strategy (QBS) which the player can choose in entangled strategy. Fig. \\ref{00and10} shows the calculations of $E_{00}$ and $E_{10}$ in No. 6 game.\n\\clearpage\n\\begin{table}[h]\n\\centering\n\\scalebox{0.7}[0.7]{\n\\begin{tabular}{ccccccccc}\nNo & initial state (As,2s,3s) & dealer up& $E_{\\mathrm{std}}$ & $E_{\\mathrm{hit}}$ & $E_{Y} = E_{00}$ & $E_{Z} = E_{10}$ & number of cases &QBS \\\\ \\hline \\hline\n1 & (2,0,0) & 2 & 1\/5 & 1\/5 & 1\/5 & 2\/5 & 2 & Z \\\\\n2 & & 3 & -2\/5 & 1\/5 & -3\/5 & 1\/10 & 4 & X \\\\\n3 & (0,2,0) & 1 & -3\/5 & -3\/5 & -1 & -3\/5 & 2 & I,X,Z \\\\\n4 & & 3 & -1 & -2\/5 & -1 & -2\/5 & 4 & X,Z \\\\\n5 &(0,0,2) & 1 & -8\/15 & -4\/5 & -1\/5 & -4\/5 & 12 & Y \\\\\n6 & & 2 & -1\/30 & -1\/3 & 3\/5 & -1\/5 & 12 & Y \\\\\n7 & & 3 & -2\/5 & -7\/15 & 0 & -2\/5 & 12 & Y \\\\\n8 & (1,1,0) & 1 & -4\/5 & -4\/5 & -4\/5 & -4\/5 & 4 & I,X,Y,Z \\\\\n9 & & 2 & 3\/5 & 3\/5 & 4\/5 & 4\/5 & 4 & Y,Z \\\\\n10 & & 3 & -1\/20 & -1\/20 & 0 & 0 & 16 & Y,Z \\\\\n11 & (1,0,1) & 1 & 2\/5 & -3\/10 & 2\/5 & -3\/10 & 8 & I,Y \\\\\n12 & & 2 & 1 & 1\/2 & 1 & 4\/5 & 16 & I,Y \\\\\n13 & & 3 & 4\/5 & 1\/30 & 4\/5 & 1\/10 & 24 & I,Y \\\\\n14 & (0,1,1) & 1 & -4\/5 & -17\/20 & -4\/5 & -17\/20 & 16 & I,Y \\\\\n15 & & 2 & -2\/5 & -2\/5 & -2\/5 & -3\/10 & 8 & Z \\\\\n16 & & 3 & -4\/5 & -13\/30 & -4\/5 & -2\/5 & 24 & Z \\\\ \\hline\ntotal &&&&&&&168& \\\\\n\\end{tabular}\n}\n\\caption{Quantum basic strategy ($\\gamma = \\theta = \\pi\/2$)}\n\\label{tab3}\n\\end{table}\n\\begin{figure}[h]\n  \\begin{minipage}[b]{0.45\\linewidth}\n    \\centering\n    \\mbox{\\raisebox{7.5mm}{\\includegraphics[scale=0.6]{00.pdf}}}\n  \\end{minipage}\n  \\begin{minipage}[b]{0.45\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.6]{10.pdf}\n  \\end{minipage}\n  \\caption{}\n  \\label{00and10}\n\\end{figure}\n\nWhen the player chooses QBS in Tab. \\ref{tab3}, the overall expected value is +10.2\\%; therefore, the player has the advantage in this quantum game. In the $\\gamma = \\theta = \\frac{\\pi}{2}$ game, the player has other choices that let the dealer definitely stand (prohibiting the second draw) therefore expectation value increases compared to the classical game. We can set various values in not only entangle intensity $\\gamma$ but also game parameter $\\theta$. For example in $\\gamma = \\frac{\\pi}{4},\\; \\theta= \\frac{\\pi}{2}$, the player's utilities are $\\{E_\\mathrm{std}, E_\\mathrm{hit}, \\frac{1}{2}(E_\\mathrm{hit}+E_{00}), \\frac{1}{2}(E_\\mathrm{std}+E_{10})\\}$ by (\\ref{exp}), and the best choice for the player becomes QBS in each initial hands. The whole expectation for the player in the game is the sum of the largest utility times the appearance ratio of the initial state from No. 1 to No. 16. Fig. \\ref{fig4} shows numerical results regarding the player's expectation in various parameters.\nAs is clear from (\\ref{iop})-(\\ref{exp}), in $\\theta = 0$ game, one can see any $\\gamma$ do not change the expectation value from the classical one. Otherwise, when $\\gamma = \\frac{\\pi}{4},\\; \\theta = \\frac{\\pi}{2}$, half entangled game, the expectation is +1.8\\%. From these results, we conclude that the player has the advantage in the appropriate parameter game.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=9cm]{qbjgraph1.1.pdf}\n\\caption{Player's expectation in various parameters \\\\ \\centering ($0 \\le \\gamma , \\theta \\le \\frac{\\pi}{2}$)}\n\\label{fig4}\n\\end{figure}\n\\clearpage\n\n\\section{\\label{conclusion}Conclusion and Discussion}\nIn section \\ref{intro}, we introduced quantum effects in game theory \\cite{PhysRevLett.83.3077,PhysRevLett.82.1052,doi:10.1142\/S0219477502000981,Khan2018,GUO2008318} and the application to gambling \\cite{PhysRevLett.82.3356,PhysRevA.64.064302,PhysRevA.66.052311,Zhang_2008,Schmidt_2012,SCHMIDT2013400,FRACKIEWICZ20148,qchess,Fuchs_2020,bleiler2009quantized,PhysRevA.102.012425}. We then proposed a toy model of blackjack, called {\\it snackjack} \\cite{Ethier2010,ethier2019snackjack,epstein} as a quantum game in section \\ref{rule}, \\ref{cbj}. In section \\ref{qbj}, we found that we can possibly entangle strategies without breaking the classical game and proposed a quantum circuit which reproduced classical blackjack. In section \\ref{res}, we showed that the player's expectation increases and the player has an advantage compared to the classical blackjack with QBS.\n\nThere remain some interesting questions: (1) The dealer's strategy space $\\Sigma_2=\\{\\hat{I}\\}$ can be expanded into more general strategy space, e.g. $\\Sigma_2 = \\{\\hat{I},\\hat{Z}\\}$. In other words, the dealer inherently has infinite options changing the phase of the initial state $\\ket{1}$. This phase affects the player's strategy bit and perhaps cause a dilemma between them depending on the first hand state. In this case, QBS may not be the best choice for the player. (2) It is a nontrivial question that how this quantized blackjack on noisy intermediate scale quantum computers affects the player's expectation, as in their work \\cite{Fuchs_2020}. (3) The operation of drawing cards in the circuit we proposed can be applied to other card games. It is also an open question of what kind of quantized casino game will make a difference from the classical one. In the proposed quantum blackjack, we have demonstrated that entanglement between strategies could be used to bankrupt a casino. In the forthcoming quantum era, casinos will have to set house rules that did not exist in the era of classical physics to avoid bankruptcy.\n\n\\section*{Acknowledgments}\nThis study was inspired by a discussion with K. Ikeda and S. Aoki about quantum economics. We acknowledge K. Ikeda for supporting us by proofreading and offering counsel. We also thank T. Onogi for proofreading the manuscript and providing us insightful comments. The authors would like to thank Enago (www.enago.jp) for the English language review.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\nWe consider problems of scattering of time-harmonic electromagnetic waves by a locally\nperturbed infinite plane (which is called a locally rough surface).\nSuch problems occurs in many applications such as radar, remote sensing, geophysics, medical\nimaging and nondestructive testing (see, e.g. \\cite{SS01,V99,WC01}).\n\nIn this paper, we are restricted to the two-dimensional transverse electric (TE) case for a perfectly reflecting,\nlocally rough surface by assuming that the local perturbation is invariant in the $x_3$ direction.\nDenote by $\\G:=\\{(x_1,x_2)\\;:\\;x_2=h_\\G(x_1),x_1\\in\\R\\}$ the locally rough surface with a smooth surface profile\nfunction $h_\\G\\in C^2(\\R)$ having a compact support in $\\R$, and by $D_+$ the unbounded domain above $\\G$.\nLet $\\Sp^1_\\pm:=\\{x=(x_1,x_2):|x|=1,x_1\\gtrless0\\}$ be the upper and lower parts of the unit circle\n$\\Sp^1:=\\{x=(x_1,x_2):|x|=1\\}$. Suppose a time-harmonic ($e^{-i\\omega t}$ time dependence) plane wave\n\\ben\nu^i=u^i(x;d,k):=\\exp({ikd\\cdot x})\n\\enn\nis incident on the locally rough surface from the upper unbounded domain $D_+$,\nwhere $d=(\\sin\\theta,-\\cos\\theta)^T\\in\\Sp^1_-$ is the incident direction with $\\theta\\in(-\\pi\/2,\\pi\/2)$\nthe angle of incidence, $k=\\omega\/c>0$ is the wave number, $\\omega$ and $c$ are the wave frequency and\nspeed in $D_+$, respectively.\nThen the total field $u(x)$ is given as the sum of the incident wave $u^i(x)$,\nthe reflected wave $u^r(x)$ and the unknown scattered wave $u^s(x)$. Here,\n$u^r$ is the reflected wave with respect to the infinite plane $x_2=0$ given by\n\\ben\nu^r=u^r(x;d,k):=-\\exp({ikd'\\cdot x})\n\\enn\nwhere $d'=(\\sin\\theta,\\cos\\theta)\\in\\Sp^1_+$ is the reflected direction.\nFurthermore, the scattered field $u^s$ is required to satisfy the Helmholtz equation in $D_+$,\nthe boundary condition on $\\Gamma$ and the so-called Sommerfeld radiation condition at infinity, respectively:\n\\begin{align}\\label{eq1_nr}\n\\Delta u^s+k^2 u^s=0&\\quad \\textrm{in}\\;\\;D_+\\\\\n\\label{eq2_nr}u^s=f&\\quad \\textrm{on}\\;\\;\\G \\\\\n\\label{eq3_nr}\\lim_{r\\to\\infty}r^\\half\\left(\\frac{\\pa u^s}{\\pa r}-ik u^s\\right)=0&\\quad r=|x|\n\\end{align}\nwhere $f=-(u^i+u^r)$ has a compact support on $\\Gamma$. Here, $u^{near}:=u^r+u^s$ is called the near field.\nIn addition, (\\ref{eq3_nr}) implies that $u^s$ has the following asymptotic behavior\n(see \\cite{Willers1987,ZhangZhang2013}):\n\\be\\label{eq4}\nu^s(x;d,k)=\\frac{e^{ik|x|}}{\\sqrt{|x|}}\\left(u^\\infty(\\hat{x};d,k)+O\\Big(\\frac{1}{|x|}\\Big)\\right),\\qquad |x|\\to\\infty,\n\\en\nuniformly for all observation directions $\\hat{x}=x\/|x|\\in\\Sp^1_+$, where $u^\\infty$ is\ncalled the far-field pattern of the scattered field $u^s$.\nThe geometry of the scattering problem is presented in Figure \\ref{fig1}.\n\n\\begin{figure}\\label{fig1}\n  \\centering\n    \\includegraphics[width=4in]{example\/rough.eps}\n    \\vspace{-0.4in}\n \\caption{The scattering problem from a locally rough surface}\n\\end{figure}\n\nThe existence and uniqueness of solutions to the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr})\nhas been studied by using the integral equation method in \\cite{Willers1987} and\na variational method in \\cite{BaoLin11}. Recently in \\cite{ZhangZhang2013},\na novel integral equation formulation was proposed for this scattering problem,\nwhich leads to a fast numerical solution of the scattering problem including the case with a large wave number.\nIt should be noted that, in the past years, the mathematical and computational aspects of the\nscattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}) and other boundary conditions\nhave been studied extensively by using the integral equation method and variational approaches\nfor the case when the surface $\\G$ is a non-local (or global) perturbation of the infinite plane $x_2=0$\n(which is called the rough surface scattering in the engineering community)\n(see, e.g. \\cite{CZ98,CRZ99,CM05,CHP06,CE10,SS01,V99,WC01,ZC03}).\n\nOn the other hand, many reconstruction algorithms have been developed for the inverse problem of reconstructing\nlocally rough surfaces from the scattered near-field or far-field data, corresponding to incident\npoint sources or plane waves (see, e.g. \\cite{AKY,BaoLin11,BaoLin13,CGHIR,DPT06,KressTran2000,LSZ16}).\nFor example, a Newton method was proposed in \\cite{KressTran2000} to reconstruct a locally rough surface\nfrom the far-field pattern under the condition that the local perturbation is both star-like\nand {\\em above} the infinite plane.\nAn optimization method was introduced in \\cite{BaoLin13} to recover a mild, locally rough surface\nfrom the scattered field measured on a straight line within one wavelength above the locally\nrough surface, under the assumption that the local perturbation is {\\em above} the infinite plane.\nIn \\cite{BaoLin11}, a continuation approach over the wave frequency was developed for\nreconstructing a general, locally rough surface from the scattered field measured on an upper\nhalf-circle enclosing the local perturbation, based on the choice of the descent vector field.\nA regularized Newton method was proposed in \\cite{ZhangZhang2013} to reconstruct a general, locally rough surface\nfrom multi-frequency far-field data, where the novel integral equation introduced in \\cite{ZhangZhang2013}\nis used to solve the forward scattering problem in each iteration.\nThe reconstruction results obtained in \\cite{BaoLin11,ZhangZhang2013} are stable and accurate even for\nmulti-scale surface profiles in view of using multiple frequency data and considering multiple scattering.\nWe point out that many reconstruction algorithms have also been developed for reconstructing non-locally rough surfaces\nfrom the scattered near-field data (see, e.g. \\cite{BaoLi13,BaoLi14,BP2010,BP2009,CL05,DeSanto1,DeSanto2,LS15}).\n\nIn practical applications, it is much harder to obtain data with accurate phase information\ncompared with just measuring the intensity (or the modulus) of the data, and therefore it is often desirable to\nreconstruct the scattering surface profile from the phaseless near-field or far-field data.\nHowever, not many results are available for such problems both mathematically and numerically.\nKress and Rundell first studied such inverse problems in \\cite{KR97} and proved that\nfor the sound-soft bounded obstacle case with one incidence plane wave, the modulus of the far-field pattern\nis invariant under translations of the obstacle, and therefore it is impossible to reconstruct the location of the\nobstacle from the phaseless far-field data for one incident plane wave.\nIt was further proved in \\cite{KR97} that this ambiguity cannot be remedied by using the phaseless\nfar-field pattern for finitely many incident plane waves with different wave numbers or different incident directions.\nRegularized Newton and Landweber iteration methods have also been discussed in \\cite{KR97} for recovering the\nshape of the obstacle from the phaseless far-field data.\nIn \\cite{Ivanyshyn07,IvanyshynKress2010}, a nonlinear integral equation method was proposed\nto reconstruct the shape of the obstacle from the phaseless far-field data.\nFurther, in \\cite{IvanyshynKress2010} after the shape of the obstacle is reconstructed from the phaseless far-field data,\nan algorithm is proposed for the localization of the obstacle by utilizing the translation invariance property\ntogether with several full far-field measurements at the backscattering direction.\nIn \\cite{IvanyshynKress2011}, a nonlinear integral equation method was developed to reconstruct the real-valued surface\nimpedance function from the phaseless far-field data provided that the bounded obstacle is known in advance.\nRecently, a continuation algorithm was proposed in \\cite{BaoLiLv13} to reconstruct the shape of a perfectly reflecting\nperiodic surface from the phaseless near-field data, in \\cite{BLT11} to deal with the phaseless measurements for\nan inverse source problem, and in \\cite{BaoZhang16} to recover the shape of multi-scale sound-soft large rough surfaces\nfrom phaseless measurements of the scattered field generated by tapered waves with multiple frequencies.\nRecently, for inverse acoustic scattering with bounded obstacles it was proved in \\cite{ZZ16}\nthat the translation invariance property of the phaseless far-field pattern can be broken by using\nsuperpositions of two plane waves as the incident fields in conjunction with all wave numbers in a finite interval.\nFurther, a recursive Newton-type iteration algorithm in frequencies was developed in \\cite{ZZ16} to numerically\nreconstruct both the location and the shape of the obstacle simultaneously from multi-frequency phaseless far-field data.\n\nThe purpose of this paper is to develop an efficient imaging algorithm to reconstruct the locally rough surface $\\G$\nfrom phaseless data associated with incident plane waves. Two types of phaseless data will be considered:\nthe phaseless far-field data and the phaseless near-field data (see Figure \\ref{fig1}).\nSimilarly as in the bounded obstacle case, the phaseless far-field pattern, $|u^\\infty(\\hat{x};d,k)|$,\nis also invariant under translations of the local perturbation along the $x_1$ direction for one incident plane wave,\nthat is, $|u^\\infty_\\ell(\\hat{x};d,k)|=|u^\\infty(\\hat{x};d,k)|$, $\\hat{x}\\in\\Sp^1_+$ for all $\\ell\\in\\R,$\nwhere $u^\\infty_\\ell(\\hat{x};d,k)$ is the far-field pattern of the scattering solution with\nrespect to the shifted surface $\\G^{\\ell}:=\\{(x_1+\\ell,h_\\G(x_1)):x_1\\in\\R\\}$ of $\\G$ along the $x_1$ direction\n(see Theorem \\ref{th2} below).\nThus, it is impossible to recover the location of the locally rough surface $\\G$ from phaseless far-field data\ncorresponding to one incident plane wave. To overcome this difficulty, motivated by \\cite{ZZ16},\nwe will use the following superposition of two plane waves rather than one plane wave as the incident field:\n\\be\\label{IW}\nu^i=u^i(x;d_1,d_2,k):=\\exp({ikd_1\\cdot x})+\\exp({ikd_2\\cdot x})\n\\en\nwhere, for $l=1,2$, $\\theta_l\\in(-\\pi\/2,\\pi\/2)$ is the incidence angle and\n$d_l=(\\sin\\theta_l,-\\cos\\theta_l)^T\\in\\Sp^1_-$ is the incident direction.\nThen the reflected wave with respect to the infinite plane $x_2=0$ will be given by\n\\ben\nu^r=u^r(x;d_1,d_2,k):=-\\exp({ikd_1'\\cdot x})-\\exp({ikd_2'\\cdot x})\n\\enn\nwhere, for $l=1,2$, $d'_l=(\\sin\\theta_l,\\cos\\theta_l)\\in\\Sp^1_+$ is the reflected direction,\nand the scattered field $u^s$ will have the asymptotic behavior\n\\be\\label{asymp-2}\nu^s(x;d_1,d_2,k)=\\frac{e^{ik|x|}}{\\sqrt{|x|}}\\left(u^\\infty(\\hat{x};d_1,d_2,k)\n+O\\Big(\\frac{1}{|x|}\\Big)\\right),\\qquad |x|\\to\\infty,\n\\en\nuniformly for all observation directions $\\hat{x}=x\/|x|\\in\\Sp^1_+$.\n\nWe will prove that, if the incident field is taken as $u^i=u^i(x;d_1,d_2,k)$ with\n$d_1\\not=d_2$ and all wave numbers $k$ in a finite interval,\nthen the translation invariance property of the phaseless far-field pattern does not hold\nfor non-trivial locally rough surfaces (that is, $h_\\G\\not\\equiv0$) (see Theorem \\ref{th3} below).\nThus, both the location and the shape of the local perturbation of the surface $\\G$ can be reconstructed from\nthe phaseless far-field data, corresponding to such incident fields with multiple wave numbers\n(see the numerical experiments in Section \\ref{se4}).\nFurthermore, a recursive Newton iteration algorithm in frequencies is developed to reconstruct both\nthe location and the shape of the surface $\\G$ simultaneously from multi-frequency phaseless far-field data.\nA similar Newton iteration algorithm is also developed for reconstructing the location and shape of\nthe surface $\\G$ from multi-frequency phaseless near-field data.\nIn our reconstruction algorithms the fast integral equation solver developed in \\cite{ZhangZhang2013}\nis used to solve the forward scattering problem in each iteration.\n\nThis paper is organized as follows. Section \\ref{se1} gives a brief introduction to the integral equation\nformulation of the forward problem proposed in \\cite{ZhangZhang2013}, and\nthe inverse scattering problem is studied in Section \\ref{se2} with phaseless far-field and near-field data.\nIn Section \\ref{se3}, a recursive Newton-type iteration algorithm in frequencies is proposed to solve the\ninverse problems. Numerical examples are carried out in Section \\ref{se4} to illustrate the effectiveness of\nour inversion algorithm. Concluding remarks are presented in Section \\ref{se5}.\n\n\n\\section{The integral equation formulation for the scattering problem}\\label{se1}\n\\setcounter{equation}{0}\n\n\nIn this section we give a brief introduction to the integral equation formulation\nproposed in \\cite{ZhangZhang2013} for the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr})\nwhich leads to a fast numerical solution of the problem and will be used in\nour inversion algorithm. To this end, we need the following notations.\n\nLet $B_R\\coloneqq\\{x=(x_1,x_2)\\;|\\;|x|<R\\}$ be a circle with $R>0$ large enough so that\nthe local perturbation $\\{(x_1,h_{\\G}(x_1))\\;|\\;x_1\\in\\textrm{supp}(h_{\\G})\\}\\subset B_R$.\nThen $\\G_R\\coloneqq\\G\\cap B_R$ represents the part of $\\G$ containing the local perturbation of the infinite plane.\nDenote by $x_A:=(-R,0),x_B:=(R,0)$ the endpoints of $\\G_R$.\nWrite $\\mathbb{R}^2_\\pm\\coloneqq\\{(x_1,x_2)\\in\\R^2\\;|\\;x_2\\gtrless0\\}$,\n$D^\\pm_R\\coloneqq B_R\\cap D_\\pm$ and $\\pa B^\\pm_R\\coloneqq\\pa B_R\\cap D_\\pm$,\nwhere $D_-:=\\{(x_1,x_2)\\;|\\;x_2<h_\\G(x_1),x_1\\in\\R\\}.$ See Figure \\ref{fig2} for the problem geometry.\n\\begin{figure}\\label{fig2}\n  \\centering\n    \\includegraphics[width=4in]{example\/rough2.eps}\n    \\vspace{-0.4in}\n \\caption{Problem geometry}\n\\end{figure}\n\nFor $\\varphi\\in C(\\pa D_R^-)$,\ndefine $S_k,S_k^{re},K_k,K_k^{re}$ to be the boundary integral operators of the following form:\n\\ben\n(S_k\\varphi)(x)&:=&\\int_{\\pa D^-_R}\\Phi_k(x,y)\\varphi(y)ds(y),\\quad x\\in\\pa D^-_R\\\\\n(S_k^{re}\\varphi)(x)&:=&\n     \\left\\{\\begin{array}{l}\n     \\ds\\int_{\\pa D^-_R}\\Phi_k(x,y)\\varphi(y)ds(y),\\quad x\\in\\Gamma_R\\\\\n     \\ds\\int_{\\pa D^-_R}\\Phi_k(x^{re},y)\\varphi(y)ds(y),\\quad x\\in\\pa B^-_R\\cup\\{x_A,x_B\\}\n     \\end{array}\\right.\\\\\n(K_k\\varphi)(x)&:=&\\int_{\\pa D^-_R}\\frac{\\pa\\Phi_k(x,y)}{\\pa\\nu(y)}\\varphi(y)ds(y),\n     \\quad x\\in\\pa D^-_R\\\\\n(K_k^{re}\\varphi)(x)&:=&\n\\left\\{\\begin{array}{l}\\ds\\int_{\\pa D^-_R}\\frac{\\pa\\Phi_k(x,y)}{\\pa\\nu(y)}\\varphi(y)ds(y),\n    \\quad x\\in\\Gamma_R\\\\\n    \\ds\\int_{\\pa D^-_R}\\frac{\\pa\\Phi_k(x^{re},y)}{\\pa\\nu(y)}\\varphi(y)ds(y),\n    \\quad x\\in\\pa B^-_R\\cup\\{x_A,x_B\\}\n    \\end{array}\\right.\n\\enn\nwhere $x^{re}=(x_1,-x_2)$ is the reflection of $x=(x_1,x_2)$ about the $x_1$-axis,\n$\\Phi_k(x,y)$ is the fundamental solution of the Helmholtz equation $\\Delta w+k^2w=0$\nwith the wavenumber $k$, and $\\nu$ is the unit outward normal on $\\pa D^-_R$.\n\nIt was proved in \\cite{ZhangZhang2013} that the scattering solution $u^s$ of the scattering\nproblem (\\ref{eq1_nr})-(\\ref{eq3_nr}) is given as follows\n\\be\\label{eq10_nr}\nu^s(x)=\\int_{\\pa D^-_R}\\left[\\frac{\\pa\\Phi_k(x,y)}{\\pa\\nu(y)}-i\\eta\\Phi_k(x,y)\\right]\\varphi(y)ds(y),\\quad x\\in\\pa D^-_R,\n\\en\nwith $\\eta\\neq0$ a real coupling parameter, if and only if $\\varphi\\in C(\\pa D_R^-)$ is the solution of\nthe integral equation $P\\varphi(x)=g(x)$. Here,\n\\be\\label{eq11_nr}\nP\\varphi\\coloneqq\\left\\{\\begin{array}{ll}\n\\ds\\varphi+\\left(K_k\\varphi-i\\eta S_k\\varphi\\right)\n   +\\left(K^{re}_k\\varphi-i\\eta S_k^{re}\\varphi\\right), &x\\in\\G_R\\\\\n\\ds\\half\\varphi+\\left(K_k\\varphi-i\\eta S_k\\varphi\\right)\n   +\\left(K^{re}_k\\varphi-i\\eta S^{re}_k\\varphi\\right), &x\\in\\pa B_R^-\\cup\\{x_A,x_B\\}\n\\end{array}\\right.\n\\en\nand\n\\be\\label{eq12_nr}\ng(x):=\\left\\{\\begin{array}{ll}\n-2\\big(u^i(x)+u^r(x)\\big), &x\\in\\G_R\\\\\n0, &x\\in\\pa B_R^-\\cup\\{x_A,x_B\\}\n\\end{array}\\right.\n\\en\n\nIn \\cite[theorem 2.4]{ZhangZhang2013}, it was shown that the integral equation $P\\varphi=g$ is uniquely solvable\nin $C(\\pa D_R^-)$, which is solved numerically by using the Nystr\\\"{o}m method with a graded mesh\nintroduced in \\cite[Section 3.5]{ColtonKress1998} (see \\cite{ZhangZhang2013} for details).\n\n\\begin{remark}\\label{re3}{\\rm\nBy (\\ref{eq10_nr}) and the asymptotic behavior of the fundamental solution $\\Phi_k$, it follows that the\nfar-field pattern of the scattered field $u^s$ is given by\n\\be\\label{eq14_nr}\nu^\\infty(\\hat{x};d_1,d_2,k)=\\frac{e^{-i\\pi\/4}}{\\sqrt{8\\pi k}}\\int_{\\pa D_R^-}\n[k\\nu(y)\\cdot\\hat{x}+\\eta]e^{-ik\\hat{x}\\cdot y}\\varphi(y)ds(y),\\qquad \\hat{x}\\in\\Sp^1_+,\n\\en\nwhich is an analytic function on the unit circle $\\Sp^1_+$, where $\\varphi$ is the solution of the\nintegral equation $P\\varphi=g$.\n}\n\\end{remark}\n\n\n\n\\section{The inverse problems}\\label{se2}\n\\setcounter{equation}{0}\n\n\nIn this paper we consider two types of scattered field measurement data without phase information:\nthe phaseless far-field data and the phaseless near-field data (see Figure \\ref{fig1}).\nThe far-field pattern $u^\\infty$ is already defined in (\\ref{eq4}), while the near-field $u^{near}$\nis defined as $u^{near}(x;d,k)=u^r(x;d,k)+u^s(x;d,k)$, $x\\in\\G_{H,L}$ for positive constants $H$ and $L$,\nwhere $\\G_{H,L}:=\\{(x_1,H),x_1\\in[-L,L]\\}$ is the measurement straight line segment above the surface $\\G$.\n\nWe first study properties of the phaseless far-field pattern and the phaseless near-field under\ntranslations of the locally rough surface. To this end, for $\\ell\\in\\R$ define\n$\\G^{\\ell}:=\\{(x_1+\\ell,h_\\G(x_1)):x_1\\in\\R\\}$ to be the shifted surface of $\\G$ along the $x_1$ direction.\nThen we have the following theorem.\n\n\\begin{theorem}\\label{th2}\nGiven the wavenumber $k>0$, let the incident wave be given by $u^i=u^i(x;d,k)$ with the incident direction\n$d=(\\sin\\theta,-\\cos\\theta)$ and the incident angle $\\theta\\in(-\\pi\/2,\\pi\/2)$ and let $u^s(x;d,k),u^s_\\ell(x;d,k)$\nbe the scattering solutions of the scattering problem $(\\ref{eq1_nr})-(\\ref{eq3_nr})$, corresponding to the locally\nrough surface $\\G$ and the shifted one $\\G^\\ell$, respectively.\nAssume that $u^\\infty(\\hat{x};d,k),u^\\infty_\\ell(\\hat{x};d,k)$ ($u^{near}(x;d,k),u^{near}_\\ell(x;d,k)$) are the\nfar-field pattern (the near-field) of the scattering solutions $u^s,u^s_\\ell$, respectively.\nThen we have $u^s_\\ell(x^{\\ell};d,k)=e^{ik\\ell\\sin\\theta}u^s(x;d,k)$,\n$u^{near}_\\ell(x^{\\ell};d,k)=e^{ik\\ell\\sin\\theta}u^{near}(x;d,k)$, $x\\in D_+$\nand $u^\\infty_\\ell(\\hat{x};d,k)=e^{ik\\ell(\\sin\\theta-\\hat{x}_1)}u^\\infty(\\hat{x};d,k)$, $\\hat{x}\\in\\Sp^1_+$\nwhere $x^{\\ell}:=x+(\\ell,0)^T$ for $x\\in\\R^2$ and $\\hat{x}_1$ is the first component of $\\hat{x}$.\n\\end{theorem}\n\n\\begin{proof}\nAssume that $D^\\ell_+$ is the unbounded domain above $\\G^\\ell$.\nLet $v(x):=e^{ik\\ell\\sin\\theta}u^s(x^{-\\ell};d,k)$ for $x\\in D^\\ell_+$.\nThen it is easily seen that $v$ is well defined in $D^\\ell_+$.\nThus, by the properties of the scattered field $u^s$ it follows that $v$ satisfies the Helmholtz equation (\\ref{eq1_nr})\nin $D^\\ell_+$ and the Sommerfeld radiation condition (\\ref{eq3_nr}).\nFurther, it can be seen that\n\\ben\nu^i(x,d,k)=e^{ik\\ell\\sin\\theta}u^i(x^{-\\ell},d,k),\\quad\nu^r(x,d,k)=e^{ik\\ell\\sin\\theta}u^r(x^{-\\ell},d,k)\n\\enn\nThen from the boundary condition (\\ref{eq2_nr}) on $u^s$, we have that $u^i(x,d,k)+u^r(x,d,k)+v(x)=0$ for $x\\in\\G^\\ell$.\nNow, the uniqueness result of the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}) implies that\n$v(x)=u^s_\\ell(x;d,k)$ for $x\\in D^\\ell_+$.\nTherefore, we obtain that $u^s_\\ell(x^{\\ell};d,k)=e^{ik\\ell\\sin\\theta}u^s(x;d,k)$,\n$u^{near}_\\ell(x^{\\ell};d,k)=e^{ik\\ell\\sin\\theta}u^{near}(x;d,k)$ for $x\\in D_+$.\n\nFinally, from the asymptotic behavior (\\ref{eq4}) of the scattered field and the fact that for $a\\in\\R^2$,\n\\ben\n|x-a|-|x|=-a\\cdot\\hat{x}+O\\left(\\frac{1}{|x|}\\right),\\quad |x|\\rightarrow\\infty\n\\enn\nuniformly for all directions $\\hat{x}=x\/|x|\\in\\Sp^1$, it is derived that\n$u^\\infty_\\ell(\\hat{x};d,k)=e^{ik\\ell(\\sin\\theta-\\hat{x}_1)}u^\\infty(\\hat{x};d,k)$ for $\\hat{x}\\in\\Sp^1_+$.\nThe proof is thus completed.\n\\end{proof}\n\nTheorem \\ref{th2} indicates that the modulus of the far field pattern (or the phaseless far-field pattern)\nfor one incident plane wave is invariant under translations along the $x_1$ direction of the boundary,\nthat is, $|u^\\infty_\\ell(\\hat{x};d,k)|=|u^\\infty(\\hat{x};d,k)|$, $\\hat{x}\\in\\Sp^1_+$ for all $\\ell\\in\\R.$\nThis means that the location of the local perturbation on the boundary $\\G$ can not be determined from\nthe phaseless far-field pattern for one incident plane wave, as shown in Example \\ref{fig4-1}.\nIn the next theorem we prove that, if the locally rough surface is non-trivial (that is, $x_2\\not=0$)\nthen the translation invariance property of the phaseless far-field pattern only holds for a countably infinite\nnumber of real numbers $\\ell$ in the case when the incident field is taken as a superposition of two plane waves\nwith different directions, that is, $u^i=u^i(x;d_1,d_2,k)$ with $d_1\\not=d_2$.\n\n\\begin{theorem}\\label{th3}\nLet the incident wave be given by $u^i=u^i(x;d_1,d_2,k)$ with $d_j=(\\sin\\theta_j,-\\cos\\theta_j),$\n$\\theta_j\\in(-\\pi\/2,\\pi\/2),\\;j=1,2$ and $\\theta_1\\neq\\theta_2$.\nAssume that $u^s(x;d_1,d_2,k),u^s_l(x;d_1,d_2,k)$ are the scattering solutions of the problem\n$(\\ref{eq1_nr})-(\\ref{eq3_nr})$, corresponding to the locally rough surface $\\G$ and the shifted one $\\G^\\ell$,\nrespectively, with $u^\\infty(\\hat{x};d_1,d_2,k),u^\\infty_\\ell(\\hat{x};d_1,d_2,k)$ the corresponding far-field\npatterns.\nAssume further that the locally rough surface $\\G$ is non-trivial (that is, $x_2\\not=0$ or $h_\\G\\not\\equiv0$).\nThen we have\n\\be\\label{TI}\n|u^\\infty(\\hat{x};d_1,d_2,k)|=|u^\\infty_\\ell(\\hat{x};d_1,d_2,k)|,\\;\\;\\hat{x}\\in\\Sp^1_+\n\\en\nfor all $\\ell=\\ell_n:=2\\pi n\/[k(\\sin\\theta_1-\\sin\\theta_2)]$ with any $n\\in\\Z$.\nFurther, except for $\\ell_n$, there may exist at most one real constant $\\tau$ with $0<\\tau<2\\pi$ such that\n(\\ref{TI}) holds for $\\ell=\\ell_{n\\tau}:=(2\\pi n+\\tau)\/[k(\\sin\\theta_1-\\sin\\theta_2)]$ with any $n\\in\\Z$.\n\\end{theorem}\n\n\\begin{proof}\nLet $v_j(\\hat{x}):=u^\\infty(\\hat{x};d_j,k)$, where $u^\\infty(\\hat{x};d_j,k)$ is the far-field pattern of the\nsolution $u^s(x;d_j,k)$ to the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}), corresponding to the incident field\n$u^i=u^i(x;d_j,k)$, $j=1,2.$ From the linearity of the scattering problem and the definition of $u^i({x};d_1,d_2,k)$,\nwe have that $u^\\infty(\\hat{x};d_1,d_2,k)=v_1(\\hat{x})+v_2(\\hat{x})$.\nFurther, by Theorem \\ref{th2} it follows that\n$$\nu^\\infty_\\ell(\\hat{x};d_1,d_2,k)=e^{ik\\ell(\\sin\\theta_1-\\hat{x}_1)}v_1(\\hat{x})\n+e^{ik\\ell(\\sin\\theta_2-\\hat{x}_1)}v_2(\\hat{x}).\n$$\nThen (\\ref{TI}) becomes\n\\ben\n|v_1(\\hat{x})+v_2(\\hat{x})|=|e^{ik\\ell(\\sin\\theta_1-\\hat{x}_1)}v_1(\\hat{x})\n+e^{ik\\ell(\\sin\\theta_2-\\hat{x}_1)}v_2(\\hat{x})|,\\quad\\hat{x}\\in\\Sp^1_+\n\\enn\nfor $\\ell\\in\\R$. By a direct calculation, the above equation is reduced to\n\\be\\label{c5eq6}\n\\Rt\\left(v_1(\\hat{x})\\ol{v_2(\\hat{x})}\\right)=\n\\Rt\\left(e^{ik\\ell(\\sin\\theta_1-\\sin\\theta_2)}v_1(\\hat{x})\\ol{v_2(\\hat{x})}\\right),\\quad\\hat{x}\\in\\Sp^1_+\n\\en\nfor $\\ell\\in\\R$.\n\nWe can claim that $v_1(\\hat{x})\\ol{v_2(\\hat{x})}\\not\\equiv 0,\\;\\;\\hat{x}\\in\\Sp^1_+.$\nIn fact, if this is not true, then we have\n\\be\\label{c5eq7}\nv_1(\\hat{x})\\ol{v_2(\\hat{x})}=0,\\quad\\hat{x}\\in\\Sp^1_+\n\\en\nWe now distinguish between the following two cases.\n\n{\\bf Case 1.} $v_1\\equiv0$ on $\\Sp^1_+$.\nSince $v_1(\\hat{x})$ is the far-field pattern of $u^s(x;d_1,k)$, then, by \\cite[Lemma 3.2]{Ma05}\nwe have that $u^s(x;d_1,k)\\equiv0$ in $D_+$. Further, from the boundary condition (\\ref{eq2_nr})\nfor $u^s(x;d_1,k)$ it follows that $u^i(x;d_1,k)+u^r(x;d_1,k)\\equiv0$ on $\\G$. This implies that\n$\\exp(2ikh_\\G(x_1)\\cos\\theta_1)\\equiv 1$ for all $x_1\\in\\R$. Thus, $kh_\\G(x_1)\\cos\\theta_1\\equiv n\\pi$\nfor all $x_1\\in\\R$, where $n=0,\\pm1,\\pm2,\\cdots$, so $h_\\G\\equiv 0$. This is a contradiction.\n\n{\\bf Case 2.} There exists an $\\hat{x}_0\\in\\Sp^1_+$ such that $v_1(\\hat{x}_0)\\neq0$.\nThen we have $v_1\\neq0$ in a neighborhood of $x_0$, which, together with (\\ref{c5eq7}), implies that\n$v_2=0$ in a neighborhood of $x_0$. Since $v_2$ is an analytic function on $\\Sp^1_+$ (see Remark \\ref{re3}),\nwe have $v_2\\equiv0$ on $\\Sp^1_+$. Arguing similarly as in Case 1 gives that $h_\\G\\equiv 0$,\ncontradicting to the assumption of the theorem.\n\nNow, by (\\ref{c5eq6}) it is easy to see that (\\ref{c5eq6}) or equivalently (\\ref{TI}) holds if $\\ell$\nsatisfies the condition\n\\be\\label{TI-C1}\nk\\ell(\\sin\\theta_1-\\sin\\theta_2)=2\\pi n,\\qquad n\\in\\Z,\n\\en\nor if $\\ell=2\\pi n\/[k(\\sin\\theta_1-\\sin\\theta_2)]$ with any $n\\in\\Z$.\nFurther, assume that $v_1(\\hat{x}_0)\\ov{v_2(\\hat{x}_0)}\\not=0$ for some $\\hat{x}_0\\in\\Sp^1_+$ and write $v_1(\\hat{x}_0)\\ov{v_2(\\hat{x}_0)}=|v_1(\\hat{x}_0)\\ov{v_2(\\hat{x}_0)}|\\exp(i\\theta(\\hat{x}_0))$\nwith $0<\\theta(\\hat{x}_0)<\\pi$. Then (\\ref{c5eq6}) is reduced to the equation\n\\be\\label{TI-C1a}\n\\cos[k\\ell(\\sin\\theta_1-\\sin\\theta_2)+\\theta(\\hat{x}_0)]-\\cos[\\theta(\\hat{x}_0)]=0.\n\\en\nBy (\\ref{TI-C1a}) we know that, except for the above $\\ell$ satisfying the condition (\\ref{TI-C1}),\n(\\ref{c5eq6}) or equivalently (\\ref{TI}) also holds for $\\ell$ satisfying the condition\n\\be\\label{TI-C1+}\nk\\ell(\\sin\\theta_1-\\sin\\theta_2)=2\\pi n-2\\theta(\\hat{x}_0),\\qquad n\\in\\Z,\n\\en\nor equivalently for\n\\be\\label{TI-C1b}\n\\ell=\\frac{2\\pi n-2\\theta(\\hat{x}_0)}{k(\\sin\\theta_1-\\sin\\theta_2)},\\;\\; n\\in\\Z.\n\\en\nThus, except for $\\ell=\\ell_n$, there may exist at most one real number $\\tau$ with $0<\\tau<2\\pi$ such that\n(\\ref{c5eq6}) or equivalently (\\ref{TI}) holds for $\\ell=(2\\pi n+\\tau)\/[k(\\sin\\theta_1-\\sin\\theta_2)]$\nwith any $n\\in\\Z$. In fact, by (\\ref{TI-C1b}) we have $\\tau=2\\pi-2\\theta(\\hat{x}_0)$.\nThe proof is thus complete.\n\\end{proof}\n\n\nTheorem \\ref{th3} indicates that the translation invariance property of the phaseless far-field pattern can be broken\nfor non-trivial locally rough surfaces if the incident field is taken as $u^i=u^i(x;d_1,d_2,k)$ with\n$d_1\\not=d_2$ and all wave numbers $k\\in[k_1,k_N]$ for some wave numbers $k_N>k_1>0.$\nThen it is expected that both the location and the shape of the local perturbation part of the boundary $\\G$ can be\nreconstructed simultaneously from the phaseless far-field data, corresponding to such incident fields with multiple\nwave numbers, as demonstrated in the numerical experiments.\n\nOn the other hand, from Theorem \\ref{th2} and Figure \\ref{fig4-7} it is expected that the translation invariance\nproperty of the phaseless near-field data measured on the line segment $\\G_{H,L}$ does not hold even for one incident\nplane wave though we can not prove this rigorously.\nThus, both the location and the shape of the local perturbation part of the boundary $\\G$ can also be\nreconstructed from the phaseless near-field data, corresponding to one incident plane wave (see the numerical examples).\n\nBased on the above discussions, we consider the following two inverse problems.\n\n{\\bf Inverse problem (IP1):} Given the incident fields $u^i=u^i(x;d_1,d_2,k)$ with multiple wave numbers\n$k=k_1,\\cdots,k_N$, where $d_1,d_2\\in\\Sp^1_-$ and $d_1\\neq d_2$, to reconstruct the locally rough surface $\\G$\nfrom the corresponding intensity-only far-field data $|u^\\infty(\\hat{x};d_1,d_2,k)|^2$, $\\hat{x}\\in\\Sp^1_+$,\n$k=k_1,\\cdots,k_N$.\n\n{\\bf Inverse problem (IP2):} Given the wave number $k>0$ and the incident field $u^i=u^i(x;d,k)$\nwith $d\\in\\Sp^1_-$, to reconstruct the locally rough surface $\\G$ from the corresponding intensity-only\nnear-field data $|u^{near}(x;d,k)|^2$, $x\\in\\G_{H,L}$.\n\nIn the next section we develop a recursive Newton-type iteration algorithm in frequencies for solving\nthe inverse problems (IP1) and (IP2).\nTo this end, given the incident wave $u^i=u^i(x;d_1,d_2,k)$ with $d_1,d_2\\in\\Sp^1_-$ and $d_1\\neq d_2$,\ndefine the far-field operator $\\mathcal{F}_{(d_1,d_2,k)}$ mapping the function $h_\\G$ which describes the locally\nrough surface $\\G$ to the intensity of the corresponding far-field pattern,\n$|u^\\infty(\\hat{x};d_1,d_2,k)|^2$ in $L^2(\\Sp^1_+)$ of the scattered wave $u^s(x;d_1,d_2,k)$ of the\nscattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}), that is,\n\\be\\label{FFO}\n\\big(\\mathcal{F}_{(d_1,d_2,k)}[h_\\G]\\big)(\\hat{x})=|u^\\infty(\\hat{x};d_1,d_2,k)|^2,\\quad\n\\hat{x}\\in\\Sp^1_+\n\\en\nSimilarly, given the incident wave $u^i=u^i(x;d,k)$ with $d\\in\\Sp^1_-$, define the near-field operator\n$\\mathcal{N}_{(d,k)}$ mapping the function $h_\\G$ to the intensity of the corresponding near-field,\n$|u^{near}(x;d,k)|^2$ in $L^2(\\G_{H,L})$, of the scattering problem (\\ref{eq1_nr})-(\\ref{eq3_nr}), that is,\n\\be\\label{eq5}\n\\big(\\mathcal{N}_{(d,k)}[h_\\G]\\big)(x)=|u^{near}(x;d,k)|^2,\\quad x\\in\\G_{H,L}\n\\en\n\nOur Newton-type iterative algorithm consists in solving the nonlinear and ill-posed equation (\\ref{FFO})\nor (\\ref{eq5}) for the unknown $h_\\G$. To this end, we need to investigate the Frechet differentiability of\n$\\mathcal{F}_{(d_1,d_2,k)}$ and $\\mathcal{N}_{(d,k)}$ at $h_\\G.$\nNow, let $\\triangle h\\in C^2_{0,R}(\\R)\\coloneqq\\{h\\in C^2(\\R):\\textrm{supp}(h)\\subset(-R,R)\\}$ be a small\nperturbation and let $\\G_{\\triangle h}\\coloneqq \\{(x_1,h_\\G(x)+\\triangle h(x)):x_1\\in\\R\\}$ denote the\ncorresponding boundary for $h_\\G(x)+\\triangle h(x)$.\nThen $\\mathcal{F}_{(d_1,d_2,k)}$ is said to be Frechet differentiable at $h_\\G$ if there exists a linear\nbounded operator $\\mathcal{F}'_{(d_1,d_2,k)}:C^2_{0,R}(\\R)\\rightarrow L^2(\\Sp^1_+)$ such that\n\\ben\n\\left\\|\\mathcal{F}_{(d_1,d_2,k)}[h_\\G+\\triangle h]-\\mathcal{F}_{(d_1,d_2,k)}[h_\\G]\n-\\mathcal{F}'_{(d_1,d_2,k)}[h_\\G;\\triangle h]\\right\\|_{L^2(\\Sp^1_+)}=o\\left(||\\triangle h||_{C^2(\\R)}\\right)\n\\enn\nThe Frechet differentiable at $h_\\G$ of $\\mathcal{N}_{(d,k)}$ is defined similarly.\nWe have the following theorem.\n\n\\begin{theorem}\\label{th1_nr}\n(i) Given the incident wave $u^i=u^i(x;d_1,d_2,k)$ with $d_1,d_2\\in\\Sp^1_-$ and $d_1\\neq d_2$,\nlet $u(x)=u^i(x)+u^r(x)+u^s(x)$, where $u^s$ solves the scattering problem $(\\ref{eq1_nr})-(\\ref{eq3_nr})$\nwith the boundary data $f=-(u^i+u^r)$. If $h_\\G\\in C^2(\\R)$, then $\\mathcal{F}_{(d_1,d_2,k)}$ is Frechet\ndifferentiable at $h_\\G$ with the derivative given by\n$\\mathcal{F}'_{(d_1,d_2,k)}[h_\\G;\\triangle h]=2\\Rt\\left[\\ov{u^\\infty}(u')^\\infty\\right]$ for\n$\\triangle h\\in C^2_{0,R}(\\R)$.\nHere, $(u')^\\infty$ is the far-field pattern of $u'$ which solves the scattering problem\n$(\\ref{eq1_nr})-(\\ref{eq3_nr})$ with the boundary data $f=-(\\triangle h\\cdot\\nu_2){\\pa u}\/{\\pa\\nu}$,\nwhere $\\nu_2$ is the second component of the unit normal $\\nu$ on $\\G$ directed into the infinite domain $D_+$.\n\n(ii) Given the incident wave $u^i=u^i(x;d,k)$ with $d\\in\\Sp^1_-$, let $u$ and $u^{near}$ be\nthe total and near-field, respectively, corresponding to the scattering problem $(\\ref{eq1_nr})-(\\ref{eq3_nr})$.\nIf $h_\\G\\in C^2(\\R)$, then $\\mathcal{N}_{(d,k)}$ is Frechet differentiable at $h_\\G$ with the derivative\ngiven by $\\mathcal{N}'_{(d,k)}[h_\\G;\\triangle h]=2\\Rt\\left[\\ov{u^{near}}u'\\right]$\nfor $\\triangle h\\in C^2_{0,R}(\\R)$, where $u'$ is the solution of the problem ($(\\ref{eq1_nr})-(\\ref{eq3_nr})$\nwith the boundary data $f=-(\\triangle h\\cdot\\nu_2){\\pa u}\/{\\pa\\nu}$.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is similar to that of Theorems 2.1 and 2.2 in \\cite{BaoLiLv13} for\ninverse diffraction grating problems with appropriate modifications.\n\\end{proof}\n\n\n\n\\section{Reconstruction algorithm}\\label{se3}\n\\setcounter{equation}{0}\n\n\nIn this section, we describe the Newton-type iteration algorithm for the inverse problem (IP1).\nFor the inverse problem (IP2), the approach is similar, so we omit it.\n\nLet $d_{1l},d_{2l}\\in\\Sp^1_-,l=1,2,\\ldots,n_d,$ be the incident directions and let $k>0$\nbe the fixed wave number. Assume that $h^{app}$ is an approximation to the function $h_\\G$.\nWe replace (\\ref{FFO}) by the linearized equations:\n\\be\\label{LFFO}\n\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x})\n+\\big(\\mathcal{F}'_{(d_{1l},d_{2l},k)}[h^{app};\\triangle h]\\big)(\\hat{x})\n\\approx|u^\\infty(\\hat{x};d_{1l},d_{2l},k)|^2,\\quad l=1,2,\\ldots,n_d,\n\\en\nwhere $\\triangle h$ is the update function to be determined. Our Newton iterative algorithm consists in\niterating the equations (\\ref{LFFO}) by using the Levenberg-Marquardt algorithm (see, e.g. \\cite{Hohage1999}).\n\nIn the numerical examples, the noisy phaseless far-field pattern\n$|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|,j=1,2,\\ldots,n_f,l=1,2,\\ldots,n_d,$\nare considered as the measurement data which satisfies\n\\ben\n\\big\\||u^\\infty_{\\delta}(\\,\\cdot\\,;d_{1l},d_{2l},k)|^2-|u^\\infty(\\,\\cdot\\,;d_{1l},d_{2l},k)|^2\\big\\|_{L^2(\\Sp^1_+)}\n\\leq\\delta\\big\\||u^\\infty(\\,\\cdot\\,;d_{1l},d_{2l},k)|^2\\big\\|_{L^2(\\Sp^1_+)},\\quad l=1,2,\\ldots,n_d,\n\\enn\nwhere $\\delta>0$ is called the noise ratio and the observation directions $\\hat{x_j},j=1,2,\\ldots,n_f,$\nare the equidistant points on $\\Sp^1_+$.\nIn practical computation, $h^{app}$ has to be taken from a finite-dimensional subspace $R_M$.\nHere, $R_M=\\textrm{span}\\{\\phi_{1,M},\\phi_{2,M},\\cdots,\\phi_{M,M}\\}$ is a subspace of $C^2_{0,R}(\\R)$,\nwhere $\\phi_{j,M},j=1,2,\\ldots,M,$ are spline basis functions with support in $(-R,R)$ (see Remark \\ref{re1_nr}).\nThen, by the strategy in \\cite{Hohage1999}, we seek an updated function\n$\\triangle h=\\sum^M_{i=1}\\triangle a_i\\phi_{i,M}$ in $R_M$ so as to solve the minimization problem:\n\\be\\no\n&&\\min_{\\triangle a_i}\\left\\{\\sum_{l=1}^{n_d}\\sum_{j=1}^{n_f}\n\\Big|\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x}_j)\n+\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}'[h^{app};\\triangle h]\\big)(\\hat{x}_j)\\right.\\\\ \\label{eq15_nr}\n&&\\qquad\\qquad\\left. -|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^2\\Big|^2+\\beta\\sum_{i=1}^M|\\triangle a_i|^2\\right\\}\n\\en\nwhere $\\beta>0$ is chosen such that\n\\be\\label{eq16_nr}\n\\left[\\sum_{l=1}^{n_d}\\sum_{j=1}^{n_f}\\Big|\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x}_j)\n+\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}'[h^{app};\\triangle h]\\big)(\\hat{x}_j)\n-|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^2\\Big|^2\\right]^{\\half}\\no\\\\\n=\\rho\\left[\\sum_{l=1}^{n_d}\\sum_{j=1}^{n_f}\\Big|\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x}_j)\n-|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^2\\Big|^2\\right]^{\\half}\n\\en\nfor a given constant $\\rho\\in(0,1)$. Then the approximation function $h^{app}$ is updated by $h^{app}+\\triangle h$.\nFurther, define the error function\n\\ben\nErr_k:=\\frac{1}{n_d}\\sum_{l=1}^{n_d}\n\\begin{array}{c}\n{\\left[\\sum\\limits_{j=1}^{n_f}\\Big|\\big(\\mathcal{F}_{(d_{1l},d_{2l},k)}[h^{app}]\\big)(\\hat{x}_j)\n-|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^2\\Big|^2\\right]^{1\/2}} \\\\\n\\hline\n{\\left[\\sum\\limits_{j=1}^{n_f}|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k)|^4\\right]^{1\/2}}\n\\end{array}\n\\enn\nThen the iteration is stopped if $Err_k< \\tau\\delta$, where $\\tau>1$ is a fixed constant.\n\n\\begin{remark}\\label{re1_nr} {\\rm\nFor a positive integer $M\\in\\N^+$ let $h=2R\/(M+5)$ and $t_i=(i+2)h-R$.\nThen the spline basis functions of $R_M$ are defined by\n$\\phi_{i,M}(t)=\\phi((t-t_i)\/h),i=1,2,\\ldots,M,$ where\n\\ben\n\\phi(t):=\\sum^{k+1}_{j=0}\\frac{(-1)^j}{k!}\\left(\\begin{array}{c}k+1\\\\\n                                      j\\end{array}\\right)\n       \\left(t+\\frac{k+1}{2}-j\\right)^k_+\n\\enn\nwith $z^k_+=z^k$ for $z\\geq0$ and $=0$ for $z<0$.\nIn this paper, we choose $k=4$, that is, $\\phi$ is the quartic spline function.\nNote that $\\phi_{i,M}\\in C^3(\\R)$ with support in $(-R,R)$. See \\cite{DeBoor} for details.\n}\n\\end{remark}\n\n\\begin{remark}\\label{re2_nr} {\\rm\nThe integral equation method proposed in \\cite{ZhangZhang2013} (cf. Section \\ref{se1}) is used to solve\nthe direct scattering problem in each iteration.\n}\n\\end{remark}\n\nOur recursive Newton iteration algorithm in frequencies is given in Algorithm \\ref{alg1} for the inverse problem (IP1).\n\n\\begin{algorithm}\\label{alg1}\nGiven the phaseless far-field data $|u^\\infty_{\\delta}(\\hat{x_j};d_{1l},d_{2l},k_m)|$,\n$j=1,2,\\ldots,n_f$, $l=1,2,\\ldots,n_d$, $m=1,2,\\ldots,N$ with $d_{1l},d_{2l}\\in\\Sp^1_-$, $d_{1l}\\neq d_{2l}$\nand $k_1<k_2<\\cdots<k_N$.\n\n1) Let $h^{app}$ be the initial guess of $h_\\G$. Set $i=0$ and go to Step 2).\n\n2) Set $i=i+1$. If $i>N$, then stop the iteration; otherwise, set $k=k_i$ and go to Step 3).\n\n3) If $Err_k<\\tau\\delta$, go to Step 2); otherwise, go to Step 4).\n\n4) Solve (\\ref{eq15_nr}) with the strategy (\\ref{eq16_nr}) to get an updated function $\\triangle h$.\nLet $h^{app}$ be updated by $h^{app}+\\triangle h$ and go to Step 3).\n\\end{algorithm}\n\n\\begin{remark}\\label{re3_nr} {\\rm\nSince $\\mathcal{F}'_{(d_1,d_2,k)}[0;\\triangle h]=0$ for any $d_1,d_2\\in\\Sp^1_-$, $k>0$ and\n$\\triangle h\\in C^2_{0,R}(\\R^2)$, then the initial guess $h^{app}$ should not be zero for\nthe inverse problem (IP1). However, in all the numerical examples for the inverse problem (IP2),\nthe initial guess of $h_\\G$ is chosen to be $h^{app}=0$.\n}\n\\end{remark}\n\n\n\\section{Numerical experiments}\\label{se4}\n\n\nIn this section, several numerical experiments are carried out to illustrate the effectiveness of\nthe inversion algorithm. The following assumptions are made in all numerical experiments.\n\n1) For each example we use multi-frequency data with the wave numbers $k=1,3,\\ldots,2N-1,$ where $N$\nis the total number of frequencies.\n\n2) To generate the synthetic data, the integral equation method is used to the direct scattering problem.\nFor the inverse problem (IP1), the intensity of the far-field pattern is measured along the upper half-aperture\n(that is, the measurement angle is between $0$ and $\\pi$) with $200$ equidistant measurement points.\nFor the inverse problem (IP2), the intensity of the near-field is measured on the straight line segment\n$\\G_{1,1}:=\\{(x_1,1)\\;:\\;x_1\\in[-1,1]\\}$ also with $200$ equidistant measurement points.\nThe corresponding noisy data $|u^\\infty_{\\delta}|$ and $|u^{near}_{\\delta}|$ are simulated as\n$|u^\\infty_{\\delta}|^2=|u^\\infty|^2(1+\\delta\\zeta)$ and $|u^{near}_{\\delta}|^2=|u^{near}|^2(1+\\delta\\zeta)$,\nrespectively, where $\\zeta$ is a normally distributed random number in $[-1,1].$\nIn all the numerical examples, the noise level is taken as $\\delta=5\\%$.\n\n3) The parameters are taken as $\\rho=0.8$ and $\\tau=1.5$.\n\n4) In all the numerical examples, the local perturbation of the infinite plane is assumed to be restricted to\nthe range $-1<x_1<1$, that is, $\\textrm{supp}(h_\\G)\\in(-1,1)$.\n\n\\textbf{Example 1 (Shape reconstruction).} We first demonstrate numerically that the location of the local\nperturbation on the infinite plane can not be determined from the intensity far-field data if only one plane\nwave is used as the incident field, that is, $u^i(x)=u^i(x;d,k)$. The locally rough surface is given by\n\\ben\nh_\\G(x_1)=\\phi(({x_1+0.2})\/{0.3}),\n\\enn\nwhere $\\phi$ is defined as in Remark \\ref{re1_nr}, and the incident direction is taken as\n$d=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$. The number of the spline basis functions is chosen to be $M=10$,\nand the total number of frequencies used is $N=10$. Further, we choose two different initial guesses\nfor the iteration inversion algorithm: 1) $h^{app}=0.1\\sum_{i=2}^4\\phi_{i,M}$, and\n2) $h^{app}=0.1\\sum_{i=5}^7\\phi_{i,M}$, where the corresponding reconstructed curves\nare denoted as \"Reconstructed curve 1\" and \"Reconstructed curve 2\".\nFigure \\ref{fig4-1} presents the initial curves and the reconstructed curves at the wavenumbers $k=1,7,19$.\nFrom the reconstructions presented in Figure \\ref{fig4-1} it is seen that\nthe shape of the local perturbation can be well reconstructed numerically, but its reconstructed location\nis translated along the $x_1$ direction, depending on the choice of initial guess. This means that\nthe location of the local perturbation can not be recovered from the intensity far-field data\nif one plane wave is taken as the incident field.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP0\/case3\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP0\/case3\/rough3_k_1_error5.eps}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP0\/case3\/rough3_k_7_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP0\/case3\/rough3_k_19_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Shape reconstruction of a locally rough surface from $5\\%$ noisy intensity far-field data\nwith one incident plane wave $u^i=u^i(x;d,k)$, where $d=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=1,7,19.$\n}\\label{fig4-1}\n\\end{figure}\n\n\\textbf{Example 2 (Shape and location reconstruction).} We consider the same locally rough surface as\nin Example 1. Here, we demonstrate that by using a superposition of two plane waves with different\ndirections as the incident field, that is, $u^i=u^i(x;d_1,d_2,k)$ with $d_1\\not=d_2$,\nboth the location and the shape of the local perturbation on the rough surface can be reconstructed from\nthe corresponding intensity far-field data.\nIn the numerical experiment, the two incident directions are taken as $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and\n$d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$, the number of the spline basis functions is chosen to be $M=10$, and\nthe total number of frequencies used is $N=13$. Further, we use the same initial guesses and\nthe same terms for the corresponding reconstructed curves of $h_\\G$ as in Example 1.\nThe initial and reconstructed curves are presented in Figure \\ref{fig4-2}, at the wavenumbers $k=1,7,25,$\nrespectively. From Figures \\ref{fig4-1} and \\ref{fig4-2} it can be seen that both the location and the shape\nof the local perturbation are accurately reconstructed with two different initial guesses.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case3\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case3\/rough3_k_1_error5.eps}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case3\/rough3_k_7_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case3\/rough3_k_25_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Location and shape reconstruction of a locally rough surface from $5\\%$ noisy intensity far-field data\nwith a superposition of two plane waves as the incident field $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=1,7,25$.\n}\\label{fig4-2}\n\\end{figure}\n\n\\textbf{Example 3 (Piecewise linear curve).} We now consider the inverse problem (IP1) with a piecewise linear local perturbation\n(the solid line in Figure \\ref{c5fig4-9}). The number of the spline basis functions is chosen as $M=40$, the total number of\nfrequencies used is taken to be $N=18$, and the initial guess for the rough surface profile $h_\\G$ is chosen as\n$h^{app}=0.05\\sum_{i=5}^{15}\\phi_{i,M}$. Figure \\ref{c5fig4-9} presents the initial and reconstructed curves\nat the wavenumbers $k=7,21,35$, respectively, obtained by using the intensity far-field data with $5\\%$ noise,\ncorresponding to the incident wave $u^i=u^i(x;d_1,d_2,k)$ with\n$d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nThe reconstruction results show that the piecewise linear surface profile is very accurately reconstructed even\nat the corners of the boundary.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case9\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case9\/rough9_k_7_error5.eps}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case9\/rough9_k_21_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case9\/rough9_k_35_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Location and shape reconstruction of a piecewise linear local perturbation from $5\\%$ noisy intensity\nfar-field data with a superposition of two plane waves as the incident field $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=7,21,35$.\n}\\label{c5fig4-9}\n\\end{figure}\n\n\\textbf{Example 4 (multi-scale curve).} Consider the inverse problem (IP1) again with a multi-scale surface profile given by\n\\ben\nh_\\G(x_1)=\\left\\{\\begin{array}{ll}\n\\ds \\exp\\left[16\/(25x_1^2-16)\\right]\\left[0.5+0.1\\sin(16\\pi x_1)\\right], &|x_1|<4\/5\\\\\n\\ds 0, &|x_1|\\geq4\/5.\n\\end{array}\\right.\n\\enn\nThis function has two scales: the macro-scale represented by the function $0.5\\exp[16\/(25x_1^2-16)]$ and the micro-scale\nrepresented by the function $0.1\\exp[16\/(25x_1^2-16)]\\sin(16\\pi x_1)$.\nFor the inverse problem, the number of the spline basis functions is chosen to be $M=40$, the total number of\nfrequencies used is $N=30$, and the initial guess for $h_\\G$ is $0.05\\sum_{i=10}^{30}\\phi_{i,M}$.\nFigure \\ref{fig4-6} presents the initial curve and the reconstructed curves at $k=19,33,39,49,59$,\nobtained by using the intensity far-field data with $5\\%$ noise, with respect to the incident wave\n$u^i=u^i(x;d_1,d_2,k)$, where $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nIt is observed from Figure \\ref{fig4-6} that the macro-scale of $h_\\G$ is recovered at low frequencies (e.g. $k=19$),\nbut the micro-scale of $h_\\G$ is not recovered completely even in the case when data with higher frequencies are used.\n\nIn order to get a better reconstruction, more measurement data are used.\nFigure \\ref{fig4-3} presents the initial and reconstructed curves at $k=19,33,39,49,59,$\nobtained by using more intensity far-field data generated with the incident waves $u^i(x)=u^i(x;d_{1l},d_{2l},k),$ $l=1,2$,\nwhere $d_{11}=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6)),d_{21}=(0,-1),d_{12}=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$ and $d_{22}=(0,-1)$.\nCompared with Figure \\ref{fig4-6}, the micro-scale of the surface profile $h_\\G$ is accurately recovered.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_19_error5}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_33_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_39_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_49_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/two_wave\/rough6_k_59_error5}}\n\\caption{Reconstruction of a two-scale surface profile from $5\\%$ noisy intensity far-field data\nwith a superposition of two plane waves as the incident field $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=19,33,39,49,59$.\n}\\label{fig4-6}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_19_error5}}\n \n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_33_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_39_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_49_error5}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP1\/case6\/four_wave\/rough6_k_59_error5}}\n\\caption{Reconstruction of a two-scale surface profile from $5\\%$ noisy intensity far-field data\ngenerated with two superpositions of two plane waves as the incident fields $u^i=u^i(x;d_{1l},d_{2l},k),l=1,2$,\nwhere $d_{11}=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6)),d_{21}=(0,-1),d_{12}=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$ and $d_{22}=(0,-1)$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=19,33,39,49,59$.\n}\\label{fig4-3}\n\\end{figure}\n\n\\textbf{Example 5 (multi-scale curve).} We consider the inverse problem (IP1) again with the multi-scale surface profile given by\n\\ben\nh_\\G(x_1)=\\left\\{\\begin{array}{ll}\n\\ds \\exp\\left[16\/(25x_1^2-16)\\right]\\left[0.5+0.1\\sin(16\\pi x_1)\\right]\\sin(\\pi x_1), &|x_1|<4\/5\\\\\n\\ds 0, &|x_1|\\geq4\/5\n\\end{array}\\right.\n\\enn\nThis profile consists of a macro-scale represented by $0.5\\exp\\left[16\/(25x_1^2-16)\\right]\\sin(\\pi x_1)$\nand a micro-scale represented by $0.1\\exp\\left[16\/(25x_1^2-16)\\right]\\sin(16\\pi x_1)\\sin(\\pi x_1)$.\nFor the inverse problem, the number of the spline basis functions is chosen to be $M=40$, the total number of frequencies\nused is $N=35$, and the initial guess for $h_\\G$ is $0.05\\sum_{i=25}^{35}\\phi_{i,M}$.\nFigure \\ref{c5fig4-10} presents the initial and reconstructed curves at $k=19,39,49,59,69$, respectively,\nobtained by using $5\\%$ noisy intensity far-field data generated with the incident wave $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nFrom Figure \\ref{c5fig4-10} it is observed that the macro-scale features are captured at $k=19$\nand the reconstruction improves as the measurement data with higher frequencies are used.\nFinally, the whole locally rough surface (including the micro-scale features of the boundary)\nis accurately recovered at $k=69$.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_19_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n    \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_39_error5.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_49_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_59_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=2.8in]{example\/IP1\/case7\/rough7_k_69_error5.eps}}\n  \\vspace{0in}\n\\caption{Reconstruction of a two-scale surface profile from $5\\%$ noisy intensity far-field data\ngenerated with a superposition of two plane waves as the incident field $u^i=u^i(x;d_1,d_2,k)$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=19,39,49,59,69$.\n}\\label{c5fig4-10}\n\\end{figure}\n\n\\textbf{Example 6 (piecewise linear curve).} We now consider the inverse problem (IP2) with the locally rough surface\ngiven as in Example 3. For the inverse problem, the number of the spline basis functions is chosen to be $M=40$,\nthe total number of frequencies is assumed to be $N=18$, and the initial guess for the reconstructed curve is taken as\nthe infinite plane $x_2=0$. In Figure \\ref{fig4-7}, we present the initial curve and the reconstructed curves at $k=7,15,31,$\nrespectively, which are obtained from the $5\\%$ intensity near-field data generated with the incident wave $u^i=u^i(x;d,k)$,\nwhere $d=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$. It can be seen that the reconstruction at the illuminated part of the boundary is\nbetter than that at the shadowed part. Thus, in order to improve the reconstruction, more measurement data are needed.\nFigure \\ref{fig4-4} presents the initial curve and the reconstructed curves at $k=7,15,35,$ respectively, obtained by using\nthe $5\\%$ intensity near-field data corresponding to two incident waves $u^i=u^i(x;d_l,k),l=1,2$,\nwhere $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nIt is found that the piecewise linear surface profile is accurately reconstructed even at the corners of the surface by\nusing two incident plane waves with different directions.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/one_wave\/rough9_k_7_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/one_wave\/rough9_k_15_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/one_wave\/rough9_k_31_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Reconstruction of a piecewise linear surface profile from $5\\%$ noisy intensity near-field data\ngenerated with one incident plane wave $u^i=u^i(x;d,k)$, where $d=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wavenumbers $k=7,15,31,$ respectively.\n}\\label{fig4-7}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{-0.5in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/two_wave\/rough9_k_7_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/two_wave\/rough9_k_15_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case9\/two_wave\/rough9_k_35_error5.eps}}\n  \\vspace{-0.5in}\n\\caption{Reconstruction of a piecewise linear surface profile from $5\\%$ noisy intensity near-field data\ngenerated with two incident plane waves $u^i=u^i(x;d_l,k),l=1,2$, where $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$\nand $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at the wave numbers $k=7,15,35$.\n}\\label{fig4-4}\n\\end{figure}\n\n\\textbf{Example 7 (multi-scale curve).} We consider the inverse problem (IP2) again with the multi-scale surface profile\ngiven as in Example 5. For the inverse problem, the number of the spline basis functions is chosen to be $M=40$,\nthe total number of frequencies is set to be $N=30$, and the initial guess for the reconstructed curve is taken as\nthe infinite plane $x_2=0$. In Figure \\ref{fig4-8},\nwe present the initial curve and the reconstructed curves at $k=13,29,39,49,59$,\nobtained from the $5\\%$ noisy intensity near-field data generated with one incident plane wave $u^i=u^i(x;d,k)$,\nwhere $d=(0,-1)$. It is observed that the micro-scale of the boundary surface is not recovered accurately.\nFigure \\ref{fig4-5} presents the initial curve and the reconstructed curves at $k=13,29,39,49,59,$ respectively,\nobtained from the $5\\%$ noisy intensity near-field data corresponding to two incident plane waves $u^i=u^i(x;d_l,k),l=1,2$,\nwith two different directions $d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nCompared with Figure \\ref{fig4-8} it can be seen that the multi-scale surface profile can be accurately recovered\nby using two incident plane waves with different directions.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_13_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_29_error5.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_39_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_49_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/one_wave\/rough7_k_59_error5.eps}}\n  \\vspace{0in}\n\\caption{Reconstruction of a multi-scale surface profile from $5\\%$ noisy intensity near-field data\ngenerated by one incident plane wave $u^i=u^i(x;d,k)$ with a normal incidence ($d=(0,-1)$).\nHere, the initial and reconstructed curves are presented at $k=13,29,39,49,59$.\n}\\label{fig4-8}\n\\end{figure}\n\n\\begin{figure}[htbp]\n  \\centering\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/initial_curve.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_13_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_29_error5.eps}}\n  \\hspace{-0.35in}\n  \\vspace{0in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_39_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_49_error5.eps}}\n  \\vspace{0in}\n  \\hspace{-0.35in}\n  \\subfigure{\\includegraphics[width=3in]{example\/IP2\/case7\/two_wave\/rough7_k_59_error5.eps}}\n  \\vspace{0in}\n\\caption{Reconstruction of a multi-scale surface profile from $5\\%$ noisy intensity near-field data\ngenerated by two incident plane waves $u^i=u^i(x;d_l,k),l=1,2$ with different directions\n$d_1=(\\sin(-\\pi\/6),-\\cos(-\\pi\/6))$ and $d_2=(\\sin(\\pi\/6),-\\cos(\\pi\/6))$.\nHere, the initial and reconstructed curves are presented at $k=13,29,39,49,59.$\n}\\label{fig4-5}\n\\end{figure}\n\n\n\n\\section{Conclusion}\\label{se5}\n\n\nIn this paper, it is proved that the translation invariance along the $x_1$-direction of the phaseless far-field pattern\ncan be broken down if superpositions of two plane waves with different directions are used as the incident fields.\nAn efficient multi-frequency Newton iterative algorithm is then proposed for reconstructing the locally rough surface profile\nfrom the intensity of the far-field patterns (called the intensity-only far-field data) generated by such incident fields.\nThe algorithm can also be applied to recover the locally rough surface profile from the intensity of the scattering waves\nmeasured at a straight line segment with a constant height above the surface (called the intensity-only near-field data),\ngenerated by one incident plane wave.\nAt each iteration, an efficient integral equation solver is used to solve the direct scattering problem which\nwas proposed previously in \\cite{ZhangZhang2013}, so multiple scattering is considered.\nSeveral numerical examples presented here indicate that our inversion algorithm with multi-frequency phaseless far-field and\nnear-field data gives a stable and accurate reconstruction of the local perturbation of the infinite plane,\neven in the multi-scale case.\nThe reconstruction results are similar to those obtained by the inversion algorithms in \\cite{BaoLin11,BaoLin13,ZhangZhang2013}\nwith using the full far-field and near-field data (including the phase information).\nThe main reason for this, we think, is that our inversion algorithm considers both multiple scattering (through the use of\nthe fast integral equation solver for the direct problem) and multiple frequency data.\nIt was indicated in \\cite{pre06,josa08} that multiple scattering plays a key role in subwavelength imaging from far-field data.\nOur reconstruction results here together with those given in \\cite{BaoLin11,BaoLin13,ZhangZhang2013}\nillustrate that multiple scattering in conjunction with using multiple frequency data plays an essential role in\nsubwavelength imaging from both far-field and near-field data.\n\n\n\\section*{Acknowledgements}\n\nThis work was partly supported by the NNSF of China grants 61379093, 91430102 and 11501558.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}