diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzoqj" "b/data_all_eng_slimpj/shuffled/split2/finalzoqj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzoqj" @@ -0,0 +1,5 @@ +{"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 $q0}\\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 $10$ 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 $k0$ 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{}\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{}) 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 $00$, 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 $00$, 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 $00$, 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 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 \\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 $00$ 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 $00\\right\\}$ 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 $0 0,\\;m\\in\\mathbb{R}\\right\\}$\\\\\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}, 00$ 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|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_20$, 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 $k_1N$, 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