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+{"text":"\\section{Introduction}\\label{sec:introduction}\nIn this paper, we consider a one-dimensional chain of coupled harmonic oscillators with masses $m_x$ and Hamiltonian \n$$\n\tH \\; = \\; \\sum_{x\\in \\mathbb{Z}} \\left( \\frac{p_x^2}{2 m_x} + g\\frac{(q_{x+1} - q_{x})^2}{2} \\right) \n$$\nBy changing units, one can assume that the stiffness coefficient $g$ is equal to $1$.\nThe dynamics is governed by Hamilton's equations: \n$$\nm_x \\dot q_x =p_x, \\qquad \\dot p_x = (\\Delta q)_x,\n$$ \nwhere we have used the notation $\\Delta = \\nabla_- \\nabla_+ = \\nabla_+ \\nabla_-$ for the discrete Laplacian, \nwith $(\\nabla_+ f)_x = f_{x+1} - f_{x}$ and $(\\nabla_- f)_x = f_x - f_{x-1}$. \n{For the sake of simplicity, we consider for now the system on the infinite lattice $\\mathbb{Z}$. \nThis is by no means necessary, and starting from the next section, we will restrict ourselves to a finite box and take the thermodynamic limit properly.}\n\nThis system was first analyzed in finite volume when all masses $m_x$ are equal. \nPutting the chain in a non-equilibrium stationary state (NESS) between two heat reservoirs at different temperatures, \nit was found in \\cite{rieder} that the energy current does not decay with the size of the system, indicating that energy propagates ballistically.\nThe situation changes if the masses are taken to be i.i.d.\\@ random variables. \nThis case was first investigated in \\cite{rubin_greer,casher_lebowitz} and subsequently studied in \\cite{verheggen,dhar,ajanki_huveneers}. \nAs it turns out, the disordered harmonic chain is an Anderson insulator in disguise \\cite{anderson}. \nHowever, as a consequence of the conservation of momentum, \nthe ground state of the operator $M^{-1}\\Delta$, featuring in Newton's equation $\\ddot q = M^{-1}\\Delta q$ with $M$ the diagonal matrix of the masses,\nis a ``symmetry protected mode'' \\cite{halperin}, implying a divergent localization length in the lower edge of the spectrum. \nThis leads to a rich and unexpected phenomenology. \nIn particular, if the chain is again in a NESS, the scaling of the energy current with the system size \nhappens to depend on boundary conditions and spectral factors of the reservoirs \\cite{dhar}.\nThis finding reveals also the complete lack of local thermal equilibrium, \nthat results eventually from integrability (see Section \\ref{subsec: invariant quantities}).\n\nThe harmonic chain has three ``obvious'' conserved quantities: \nthe total energy $H$, the total momentum $P = \\sum_x p_x$ and the total stretch or elongation $R = \\sum_x r_x$ with $r_x = (\\nabla_+ q)_x$.   \nThis gives rise to the following microscopic conservation laws: \n$$\n\t\\dot r_x = \\frac{p_{x+1}}{m_{x+1}} - \\frac{p_x}{m_x}, \n\t\\qquad\n\t\\dot p_x = r_x - r_{x-1}, \n\t\\qquad \n\t\\dot e_x = \\frac{r_x p_{x+1}}{m_{x+1}} - \\frac{r_{x-1}p_x}{m_x}\n$$\nwith $e_x = \\frac{1}{2} \\left( \\frac{p_x^2 }{ m_x } + r_x^2 \\right)$. \nAfter a hyperbolic rescaling of space and time, we ask in this paper whether the empirical densities of these conserved quantities converge to the densities $\\mathbf r, \\mathbf p$ and $\\mathbf e$ governed by the macroscopic laws\n$$\n\t\\partial_t \\mathbf r (y,t) = \\frac{1}{\\overline{m}} \\partial_y \\mathbf p (y,t), \n\t\\quad \n\t\\partial_t \\mathbf p (y,t) = \\partial_y \\mathbf r (y,t), \n\t\\quad \n\t\\partial_t \\mathbf e (y,t) = \\frac{1}{\\overline{m}}  \\partial_y \\big(\\mathbf r (y,t) \\mathbf p (y,t)\\big) ,\n$$\ncorresponding to Euler equations in Lagrangian coordinates, with $\\overline{m}$ the average mass. \n\nInstances of rigorous derivation of Euler equations in the smooth regime rest on the ergodicity of the microscopic dynamics.\nIn \\cite{eo,ovy,komorowski_olla}, the Hamiltonian dynamics is perturbed by some stochastic noise \nacting in such a way that conserved quantities are not destroyed but that the ergodicity of the dynamics can be established rigorously.  \n\n{One of the main motivations of this work is to show that ergodicity \nis not in general a necessary assumption for Euler equations to hold. Indeed,} \nthe dynamics considered here is purely Hamiltonian, non-ergodic, \nand possesses actually a full set of invariant quantities (see Section \\ref{subsec: invariant quantities}). \nIn the clean case, i.e.\\@ when the masses are all equal, we show in Section \\ref{subsec: clean} that Euler equations hold if and only if the temperature profile is constant.\nInstead, in Section \\ref{subsec: disordered}, we argue that Euler equations hold even out of thermal equilibrium if there is disorder on the masses. \nWe briefly discuss the fate of other conserved quantities in Section \\ref{subsec: other conserved}. \nTheorem \\ref{the: main result} in Section \\ref{sec: model and results} constitutes the main result of our paper: \nWe show the convergence to Euler equations for the disordered harmonic chain, almost surely with respect to the masses and on average with respect to an initial local Gibbs state. \nThe rest of the paper is devoted to the proof of this theorem.  \n\n\n\n\\subsection{Clean harmonic chain}\\label{subsec: clean}\nLet us assume that all masses $m_x$ are equal, say $m_x = 1$ for simplicity. \nIn this case, the equations of motion read\n\\begin{equation}\\label{eq:1}\n\t\\dot q_x = p_x , \\qquad \\dot p_x = \\Delta q_x .\n\\end{equation}\nLet us first consider the thermal equilibrium case: \nAssume that the initial configuration of the chain is random and distributed according to a Gaussian law $\\mu_0$ with covariance matrix\n\\begin{equation}\\label{eq:2}\n\t\\lgs{(\\nabla_+ q)_x ; (\\nabla_+ q)_y} =  \\lgs{ p_x ; p_y} = \\beta^{-1} \\delta_{x,y}, \\qquad \n  \t\\lgs{q_x; p_y} = 0,\n\\end{equation}\nfor some inverse temperature $\\beta$. \nIt is easy to prove that at time $t>0$, the distribution $\\mu_t$ in the phase space is still given by a Gaussian law with the same covariance matrix.  \nTo see this, just use Fourier transforms to diagonalize the dynamics:\n\\begin{equation}\\label{eq:3}\n  \t\\hat q(k,t) = \\sum_{x\\in \\ZZ} e^{i 2\\pi k x} q_x(t), \\qquad  \\hat p(k,t) = \\sum_{x\\in \\ZZ} e^{i 2\\pi k x} p_x(t),\n\\end{equation}\nand define the wave function \n\\begin{equation}\\label{eq:4}\n  \t\\hat \\phi (k,t) = \\omega(k)  \\hat q(k,t) + i  \\hat p(k,t) \n\\end{equation}\nwhere $\\omega(k) = |2\\sin(\\pi k)|$ is the dispersion relation. Then, the explicit solution of \\eqref{eq:1} is given by\n\\begin{equation}\n  \\label{eq:5}\n   \\hat \\phi (k,t) = e^{-i\\omega(k) t} \\,  \\hat \\phi (k,0).\n\\end{equation}\nThe correlations \\eqref{eq:2} imply that \n\\begin{equation}\\label{eq:5-0}\n\t\\lgs{ \\hat \\phi (k,0)^* ;  \\hat \\phi (k',0)}  = 2 \\beta^{-1} \\delta(k-k'), \\qquad  \\lgs{ \\hat \\phi (k,0) ; \\hat \\phi (k',0)}=0.\n\\end{equation}\nConsequently\n\\begin{equation}\n\\label{eq:6}\n\\begin{split}\n& \\lgs{ \\hat \\phi (k,t)^* ;  \\hat \\phi (k',t)} =   e^{i(\\omega(k)- \\omega(k')) t}\\,  \\lgs{ \\hat \\phi (k,0)^* ;  \\hat \\phi (k',0)}\n= 2\\beta^{-1} \\delta(k-k'),\\\\\n&\\lgs{ \\hat \\phi (k,t) ;  \\hat \\phi (k',t)} =   e^{-i(\\omega(k)+\\omega(k')) t} \\, \\lgs{ \\hat \\phi (k,0) ;  \\hat \\phi (k',0)}\n=0. \n\\end{split} \n\\end{equation}\nFrom (\\ref{eq:5-0}) and (\\ref{eq:6}), we deduce that the covariances  \\eqref{eq:2} are the same at any time $t$:\n\\begin{equation}\n \\label{eq:2222}\n  \\lgs{(\\nabla_+ q)_x (t) ; (\\nabla_+ q)_y (t) } =  \\lgs{ p_x (t)  ; p_y (t) } = \\beta^{-1} \\delta_{x,y}, \\qquad \n  \\lgs{q_x (t) ; p_y (t) } = 0,\n\\end{equation}\nwhich implies that the Gaussian distribution $\\mu_t$ differs from $\\mu_0$ only by the averages $\\bar r_x (t) = \\lgs{r_x (t)} = \\lgs{(\\nabla_+ q)_x(t)}$ and $\\bar p_x (t) = \\lgs{p_x(t)}$ that, \nby linearity of the dynamics, evolve following the same equation \\eqref{eq:1}. \n\nAssume now that the initial averages of the momentum $p_x$ and stretch $r_x = (\\nabla_+ q)_x$ are slowly varying on a macroscopic scale. \n{More precisely, let $N$ be an integer representing a macroscopic number of sites in the chain, \nand let $\\mathsf p, \\mathsf r : \\mathbb{R} \\to \\mathbb{R}$ be smooth and fast decaying initial macroscopic profiles. \nWe let}\n\\begin{equation}\n  \\label{eq:7}\n  {\\bar r_{[Ny]}(0) = \\mathsf r (y),\\qquad\n  \\bar p_{[Ny]}(0) = \\mathsf p (y)} . \n\\end{equation}\nLet $\\widehat {\\mathsf p} (\\xi)$ and $\\widehat {\\mathsf r} (\\xi)$ be the Fourier transforms (in $\\RR$) of $\\mathsf p(y)$ and $\\mathsf r(y)$. \nThen, as $N\\to\\infty$, \n$\\frac 1N \\widehat{\\bar p}(\\tfrac \\xi N) \\longrightarrow \\widehat {\\mathsf p}(\\xi)$ and \n$\\frac 1N \\widehat{\\bar r}(\\tfrac \\xi N) \\longrightarrow \\widehat {\\mathsf r} (\\xi)$. After a straightforward analysis \nwe have that\n\\begin{equation}\n  \\label{eq:8}\n  \\begin{split}\n    \\frac 1N \\widehat{\\bar p}\\Big(\\tfrac \\xi N, Nt\\Big) \\longrightarrow \\widehat {\\mathbf p} \\, (\\xi,t), \\qquad \n    \\frac 1N \\widehat{\\bar r} \\Big(\\tfrac \\xi N, Nt \\Big)\n    \\longrightarrow \\widehat {\\mathbf r}\\, (\\xi,t )\n  \\end{split}\n\\end{equation}\nwhere \n\\begin{equation}\n  \\label{eq:9}\n \\partial_t  \\widehat {\\mathbf r} \\, (\\xi,t) = -i {2\\pi} \\xi  \\,  \\widehat {\\mathbf p} \\, (\\xi,t), \\qquad \n \\partial_t  \\widehat {\\mathbf p} \\, (\\xi,t) = -i {2\\pi} \\xi \\,  \\widehat {\\mathbf r} \\, (\\xi,t).\n\\end{equation}\nConsequently {$\\bar r_{[Ny]}(Nt)$ and $\\bar p_{[Ny]}(Nt)$} converge {(as distributions)} to the solution of the linear wave equation\n\\begin{equation}\n  \\label{eq:10}\n  \\partial_t  \\mathbf r \\,(y,t) = \\partial_y \\mathbf p \\, (y,t), \\qquad \n \\partial_t \\mathbf p \\,(y,t) = \\partial_y   \\mathbf r \\, (y,t).\n\\end{equation}\nLet us now consider the energy per particle $e_x = \\frac 12\\left(p_x^2 + r_x^2\\right)$. \nIts average under the distribution $\\mu_t$ is \n$\\lgs{ e_x (t)} = \\beta^{-1} + \\frac 12\\left(\\bar p_x^2(t) + \\bar r_x^2(t)\\right)$ since by (\\ref{eq:2222}), the variance of $p_x$ and $r_x$, i.e. the temperature, remains constant in time. In the limit $N\\to \\infty$ we have\n$$\\lgs{ e_{[Ny]}(Nt)} \\longrightarrow {\\bf e} \\, (y,t) = \\beta^{-1} + \\frac 12\\left( {\\bf p}^2 (y,t) + {\\bf r}^2(y,t)\\right),$$\ni.e. it solves the equation\n\\begin{equation}\n  \\label{eq:11}\n   \\partial_t {\\bf e}\\, (y,t) = \\partial_y \\left( {\\bf p}\\, (y,t)   \\, {\\bf r}\\, (y,t)\\right).\n\\end{equation}\nWe recognize that (\\ref{eq:10} - \\ref{eq:11}) are the Euler equations. \nThe above is the simplest example of propagation of local equilibrium and hydrodynamic \nlimit in hyperbolic scaling: in a harmonic chain in thermal equilibrium at temperature $\\beta^{-1}$, \nand the mechanical modes not in equilibrium, we  will have a persistence of \nthe thermal equilibrium {at any time $t$}, \nwhile the mechanical modes evolve independently from the thermal mode  following the linear wave equation. \n\nNotice that the argument above does not require the distribution $\\mu_0$ to be the thermal equilibrium measure defined by \\eqref{eq:2}, \nand that it holds for any measure $\\mu_0$ with translation invariant covariance given by\n\\begin{equation}\n  \\label{eq:2-c}\n  \\lgs{\\nabla q_x ; \\nabla q_y} =  \\lgs{ p_x ; p_y} = C(x-y), \\qquad \n  \\lgs{q_x; p_y} = 0\n\\end{equation}\nfor a positive definite function $C(x)$. \nThe only difference is then that in (\\ref{eq:5-0}- \\ref{eq:6}) the term $\\beta^{-1}$ has to be replaced by the Fourier transform ${\\widehat C} (k)$. \nActually, the measure $\\mu_t$ is not even a local equilibrium state \\cite{BO_book}, underlining that the validity of Euler equations in this example does not require the propagation of local equilibrium.\n\nThe above argument rests on the translation invariance of the distribution of the thermal modes and fails if it is space inhomogeneous, \nfor example if the starting distribution is given by a local Gibbs state with a slowly varying temperature $\\beta_{N,x}^{-1} = \\beta^{-1}(x\/N)$, \ni.e.\\@ a Gaussian measure with covariances\n\\begin{equation}\n  \\label{eq:12}\n  \\lgs{\\nabla q_x ; \\nabla q_y} =  \\lgs{ p_x ; p_y} = \\beta_{N,x}^{-1}\\delta_{x,y}, \\qquad \n  \\lgs{q_x; p_y} = 0.\n\\end{equation}\nIn this case, even though the wave equation \\eqref{eq:10} still holds, \ngenerally the energy \nequation \\eqref{eq:11} is not valid. \nIn fact the energies of each mode $k$ evolves autonomously, \nas we can see studying the limit evolution of the Wigner distribution defined by\n\\begin{equation}\\label{eq:wigner1}\n\\begin{split}\n    \t\\widehat W_N(\\xi, k,t) &:= \\frac 2N \n    \t\\Blgs{\\widehat\\phi^*\\left(k - \\tfrac{\\xi}{2N}, Nt\\right)\n       \t\\widehat\\phi\\left(k + \\tfrac{\\xi}{2N}, Nt\\right)}\\\\\n   \t W_N(y, k,t) &:= \\int e^{-i 2\\pi \\xi y}   \\widehat W_N (\\xi, k,t) \\; d\\xi,\n\\end{split}\n\\end{equation}\n(the above definitions should be understood as distributions on $\\mathbb{R}\\times \\Pi$ with $\\Pi = \\mathbb{R}\\backslash \\mathbb{Z}$).\n\nIn the limit as $N\\to\\infty$ the Wigner distribution converge to a positive distribution\nwith an absolutely continuous part, the local distribution of the thermal modes, and \na singular part concentrated on $k=0$, the mechanical modes:\n\\begin{equation}\n  \\label{eq:18}\n  \\lim_{N\\to\\infty} \\widehat W_N(\\xi, k, t) = \\widehat W_{th}(\\xi, k,t) + \\widehat W_m(\\xi, t)\\;  \\delta_0(dk) \n\\end{equation}\nThe mechanical part $\\widehat W_m(\\xi, t)$ is the Fourier transform of \n$\\frac 12\\left( {\\bf p}^2 (y,t) + {\\bf r}^2(y,t)\\right)$.\n\nA straightforward calculation gives for the thermal part (see \\cite{dobru} or \\cite{bos} for a rigorous argument):\n\\begin{equation}\n  \\label{eq:14}\n  \\widehat W_{th}(\\xi, k,t) = e^{-i\\omega'(k)\\xi t} \\; \\widehat W_{th}(\\xi, k,0).\n\\end{equation}\nThis implies that the inverse Fourier transform  $W_{th}(y, k,t)$ satisfies the transport equation\n\\begin{equation}\n  \\label{eq:13}\n  \\partial_t W_{th}(y, k,t) + \\frac{\\omega'(k)}{2\\pi} \\partial_y W_{th}(y, k,t) = 0.  \n\\end{equation}\nIt also follow that \n\\begin{equation}\n  \\label{eq:15}\n  \\int  W_{th}(y, k,t) \\; dk \\ = \\ \\tilde {\\bf e}(y,t)\n\\end{equation}\nwhere $\\tilde {\\bf e}(y,t)$ is the limit profile of thermal energy (or temperature) defined as\n\\begin{equation}\n  \\label{eq:16}\n  \\frac 12 \\left(\\lgs{ r_{[Ny]}(Nt); r_{[Ny]}(Nt)} + \\lgs{ p_{[Ny]}(Nt); p_{[Ny]}(Nt)}\\right) \\rightharpoonup \\tilde {\\bf e}(y,t).\n\\end{equation}\nConsequently the thermal energy  $\\tilde {\\bf e}(y,t)$ evolves non autonomously following the equation\n\\begin{equation}\n  \\label{eq:17}\n  \\partial_t \\tilde {\\bf e}(y,t) + \\partial_y J(y,t) = 0, \\qquad J(y,t) = \\int \\omega'(k)  W_{th}(y, k,t) \\; dk.\n\\end{equation}\nWe say that the system is in {\\it local equilibrium} if $W_{th}(y,k) = \\beta^{-1}(y)$ constant in $k$. \nThis correspond to the fact that Gibbs measure gives uniform distribution on the modes.\nStarting in thermal equilibrium means $W_{th}(y,k,0) = \\beta^{-1}$ and trivially $W_{th}(y,k,t) = \\beta^{-1}$ \nfor any $t>0$.\nBut starting with local equilibrium, i.e. $W(y,k,0) = \\beta^{-1}(y)$ constant in $k$, \nwe have a non autonomous evolution of $\\tilde e(y,t)$. \n\n\n\n\n\n\n\n\n\n\\subsection{Disordered harmonic chain}\\label{subsec: disordered}\nThe situation so far can be summarized as follows. \nBy linearity, the variables $r_x$ and $p_x$ admit a macroscopic limit described by \\eqref{eq:10} independently of the initial temperature profile. \nThe macroscopic equation \\eqref{eq:11} predicts that the evolution of the energy is purely mechanical and that the temperature does not evolve with time. \nAs it turns out, the evolution of the mechanical energy is correctly described by Euler equation (see the term $\\mathcal A_N (t)$ in our decomposition \\eqref{eq: average and fluctuation} below), \nbut thermal fluctuations do in general evolve with time as well, except if the temperature profile is initially flat. \n\nThis picture gets strongly modified if the masses are taken to be random. \nOn the one hand, deriving the macroscopic evolution of the fields $r_x$ and $p_x$ becomes less obvious because some homogenization over the masses is required. \nThis difficulty can be solved by the elegant method of the \n``corrected empirical measure'', see \\cite{goncalves_jara,jara_landim,bernardin} \n(though we will actually solve it another way). \nOn the other hand, and this is the main point in considering random masses, \nthe evolution of the energy $e_x$ is now much better approximated by Euler equation.\nIndeed, at a microscopic level, all thermal fluctuations are frozen thanks \nto Anderson localization and the evolution of the energy becomes purely mechanical. \n\nTo understand this a little bit better, it is good to realize how the disorder modifies the nature \nof the eigenmodes $(\\psi^k)_{1 \\le k \\le N}$ of the operator $M^{-1}\\Delta$ for a finite chain of size $N$. \nAs a consequence of Anderson localization \\cite{anderson}, all modes at positive energy are spatially localized. \nHowever the localization length $\\zeta_k$ diverges as one approaches the ground state: \n$$\n\t\\zeta_k^{-1} \\sim \\omega^2_k \\sim \\Big(\\frac{k}{N}\\Big)^2,\n$$\nso that only the modes with $k\\gtrsim \\sqrt N$ are actually localized, while the modes $k \\lesssim \\sqrt N$ remain comparable to the modes of the clean chain \\cite{verheggen,ajanki_huveneers}.\nBy imposing a smooth initial profile $\\mathsf r, \\mathsf p$, the initial local Gibbs state attributes a weight of order 1 to a few first modes above the ground state, and a weight of order 1 to all other modes together. \nThe first ones are responsible for the transport of mechanical energy; \nall modes with $k \\gg \\sqrt N$ are localized and do not transport any thermal energy;  \nall modes with $1  \\ll k \\le o (N)$ have a vanishing weight in the thermodynamic limit and can be neglected in the analysis. \n\nFinally, we would like to mention that, while the disorder considered here \nand the stochastic velocity exchange noise considered in \\cite{eo,komorowski_olla} \nact in an obviously very different way, \ne.g.\\@ the disorder preserves integrability while the stochastic noise makes the dynamic ergodic, \nthey do produce the same effects in some respect. \n{Indeed the noise has only a very slow (negligeable) effect on the macroscopic modes.}\n{ This bares some similarity with the fact that the disorder has very little influence on the low modes of the disordered chain,\nwhile the dynamical noise provides an active hopping mechanism among the high modes. \nConsequently the dynamical noise produces a superdiffusive sub-ballistic spreading of the thermal energy \\cite{jara_kom_olla,komorowski_olla}, \nthat is not visible in the hyperbolic scaling.\nThus, the dynamical noise plays here as well a role analogous to the disorder only in the hyperbolic scaling by freezing the temperature profile.\n\nIt is important to notice that in the disordered case the temperature profile remains frozen at any time scale, \nincluding the diffusive time scale and further, see remark \\ref{frozenforever}.\nIn particular this implies a vanishing thermal diffusivity for the disordered unpinned harmonic chain. This is not in contradiction with the divergence of the thermal conductivity observed in the NESS of the same unpinned system when connected to Langevin thermostats at different temperatures with free boundary conditions, see \\cite{casher_lebowitz,ajanki_huveneers}, and our result sheds actually some light on the various behaviors for the conductivity found in \\cite{dhar} depending on the boundary conditions. In fact the thermal conductivity divergence in the NESS is due to the fluctuations of the low mechanical modes, and in our analysis there is a clear separation of the behavior of the mechanical modes (responsable for the ballistic motion) and the high thermal modes that give the temperature profile.}\n\n\n\n\n\n\n\\subsection{Other conserved quantities}\\label{subsec: other conserved}\nBefore moving on, let us briefly comment on the issue of the other conserved quantities of the system. \nThese can also be written as a sum of local terms and lead thus to additional conservation laws. \nFor example, \n$$\n\tI \\; = \\; \\sum_{x} d_x \\; = \\; \\frac{1}{2}\\sum_x\\left( \\frac{(r_x - r_{x-1})^2}{m_x} + \\left( \\frac{p_{x+1}}{m_{x+1}} - \\frac{p_x}{m_x} \\right)^2 \\right)\n$$\nis conserved (see Sections \\ref{subsec: a priori} and \\ref{subsec: invariant quantities}) and leads to the microscopic conservation law\n$$\n\t\\dot d_x = \\left( \\frac{p_{x+1}}{m_{x+1}} - \\frac{p_{x}}{m_{x}} \\right) \\frac{r_{x+1} - r_x}{m_{x+1}} -  \\left( \\frac{p_{x}}{m_{x}} - \\frac{p_{x-1}}{m_{x-1}} \\right) \\frac{r_{x} - r_{x-1}}{m_x} .\n$$\n\nIt is thus natural to ask whether this relation generates also some macroscopic law. \nIn the cases where we can derive the macroscopic evolution equation \\eqref{eq:11} for the energy, it is easy to argue that the corresponding macroscopic density $\\mathbf d (y,t)$ does not evolve with time in the hyperbolic scaling.\nIndeed, we can decompose $d_x$ as the sum of a mechanical and a thermal contribution, as we do in \\eqref{eq: average and fluctuation} below for the energy.\nIn this case, contrary to what happens for the energy, the mechanical contribution vanishes in the thermodynamic limit since $d_x$ depends on $r$ and $p$ only through their gradients, \nwhile the contribution from the thermal modes does not evolve with time, for the same reasons as it does not for the energy.  \n\nAll the other conserved quantities in this model that can be written as a sum of local terms are obtained by taking further gradients in the variables $r$ and $p$ (see Section \\ref{subsec: invariant quantities}), \nand have thus no evolution either in the hyperbolic scaling.   \n\n\n\n\n\n\n\\section{Model and results}\\label{sec: model and results}\n\nWe define the model studied in this paper and we state our main result. \nFor technical reasons, it is easier to work on a finite system of size $N$ and then let $N\\to \\infty$.\n\n\\subsection{Hamiltonian model}\nThe Hamiltonian $H$ on $\\mathbb{R}^{2N}$ is defined by \n$$\n\tH (q,p) = \\frac{1}{2} \\sum_{x=1}^N \\left( \\frac{p_x^2}{m_x} + ((\\nabla_+q)_{x})^2 \\right).\n$$\n{For concreteness, we assume} free boundary conditions, i.e.\\@ $q_0 = q_1$ and $q_{N+1} = q_N$; \n{other boundary conditions such as fixed or periodic could be considered just as well.} \nThe masses $(m_x)_{1 \\le x \\le N}$ are i.i.d.\\@ random variables. \nIn order to avoid any technical difficulty in exploiting known results from the Anderson localization literature, we assume that the law of $m_x$ admits a smooth density compactly supported in $[m_-,m_+]$ with $m_->0$. \n\nThe equations of motion read $M \\dot q = p$ and $\\dot p = \\Delta q$ where $M$ is the square diagonal matrix of size $N$ with entries defined by $M_{x,y} = \\delta (x-y) m_x$ ($\\delta(z)$ is defined by $1$ for $z=0$ and $0$ otherwise).\nIt is more convenient to express the equations of motion in terms of the displacement variables \n\\begin{equation*}\n\tr_x  =  (\\nabla_+ q)_x \\qquad (1 \\le x \\le N-1) .  \n\\end{equation*}\nThe equations of motion become\n\\begin{equation}\\label{eq: equations of motion}\n\t\\dot r_x = \\big( \\nabla_+ M^{-1}p \\big)_x \\quad (1 \\le x \\le N-1), \n\t\\qquad \\dot p_x = (\\nabla_- r)_x \\quad (1 \\le x \\le N)\n\\end{equation}\nwhere we use fixed boundary conditions for $r$ in the second equation: $r_0 = r_N = 0$.\n\n\\subsection{Gibbs and locally Gibbs states}\nWe consider three locally conserved quantities in the bulk: \n$$ H = \\sum_{x=1}^N e_x =  \\sum_{x=1}^N \\left( \\frac{p^2_x}{2m_x} + \\frac{r_x^2}{2} \\right), \\qquad P = \\sum_{x=1}^N p_x, \\qquad R = \\sum_{x=1}^{N-1} r_x. $$\nThe energy $H$ and the momentum $P$ are actually truly conserved, but the conservation of $R$ is broken at the boundary: $\\dot R = m_N^{-1}p_N - m_1^{-1}p_1$. \n\nThe Gibbs states are characterized by three parameters: \n$\\beta>0$ and $\\mathsf p, \\mathsf r \\in \\mathbb{R}$. Its probability density writes\n$$ \n\\rho_{\\text{G}} (r,p) = \\frac{1}{Z_{\\text{G}}} \\exp \\Big\\{- \\frac{\\beta}{2} \\sum_{x=1}^{N} m_x\\Big(\\frac{p_x}{m_x} - \\frac{\\mathsf p}{\\overline m}\\Big)^2 - \\frac{\\beta}{2} \\sum_{x=1}^{N-1} (r_x - \\mathsf r)^2  \\Big\\} .\n$$\nwhere $\\overline m$ denotes the mean mass and $Z_{\\text{G}} :=Z_{\\text{G}} (\\beta, \\mathsf p, \\mathsf r)$ is a normalizing constant.\nLocal Gibbs states are obtained by replacing the constant parameters $\\beta, \\mathsf p, \\mathsf r$ by functions \n$$\\beta,\\mathsf p, \\mathsf r : [0,1] \\to \\mathbb{R},$$  \nwith $\\beta(x)>0$ for all $x \\in [0,1]$, and by considering the measure with density \n\\begin{equation}\\label{eq: local Gibbs}\n\t\\rho_{\\text{loc}} (r,p) = \\frac{1}{Z_{\\text{loc}}} \\exp \\Big\\{- \\frac{1}{2} \\sum_{x=1}^{N} \\beta(x\/N) m_x\\Big(\\frac{p_x}{m_x} - \\frac{\\mathsf p(x\/N)}{\\overline{m}}\\Big)^2 - \\frac{1}{2} \\sum_{x=1}^{N-1} \\beta(x\/N) (r_x - \\mathsf r(x\/N))^2  \\Big\\}\n\\end{equation}\nwhere $Z_{\\text{loc}}:=Z_{\\text{loc}} (\\beta, \\mathsf p, \\mathsf r)$ is a normalizing constant.\nWe impose the following regularity conditions on $\\beta, \\mathsf p, \\mathsf r$: \n\\begin{equation}\\label{eq: regularity beta r p}\n\t\\beta \\in \\mathcal C^0([0,1]), \n\t\\quad \\mathsf r \\in \\mathcal C^1([0,1]) \\text{ with }\\mathsf r(0) = \\mathsf r(1) = 0, \n\t\\quad \\mathsf p \\in \\mathcal C^1([0,1]).\n\\end{equation}\nWe take such a local Gibbs state as initial state.\nBelow, we denote the expectation with respect to it by $\\lgs{\\cdot}$:\n$$\n\t\\lgs{F} = \\int F(r,p) \\rho_{\\text{loc}} (r, p) \\, \\mathrm{d} r \\mathrm{d} p.\n$$ \nInstead, expectation (resp.\\@ probability) with respect to the masses is denoted by $\\mathsf E$ (resp.\\@ $\\mathsf P$). \n\n\n\\subsection{Evolution of the locally conserved quantities} \nLet us fix some maximal time $T>0$. \nLet us define the fields $\\mathcal R$, $\\mathcal P$ and $\\mathcal E$ acting on functions $f \\in \\mathcal C^0([0,1])$ as \n\\begin{equation}\\label{eq:rpe}\n\\begin{split}\n&\\mathcal R(f,t) = \\int_0^1 \\mathbf r (y,t)\\,  f(y) \\, \\mathrm{d} y, \\quad  \\mathcal P(f,t) = \\int_0^1 \\mathbf p (y,t) \\, f(y)\\,  \\mathrm{d} y,\\\\\n&\\mathcal E(f,t) = \\int_0^1 \\mathbf e (y,t) \\, f(y) \\, \\mathrm{d} y. \n\\end{split}\n\\end{equation}\nfor all $t \\in [0,T]$. \nThe kernels $\\mathbf r$, $\\mathbf p$ and $\\mathbf e$ are defined as follows. \nFirst, at $t=0$, we impose\n$$\n\t\\mathbf r(y,0) = \\mathsf r(y), \\qquad \n\t\\mathbf p(y,0) = \\mathsf p(y), \\qquad\n\t\\mathbf e(y,0) =  \\frac1{\\beta(y)} +  \\frac{\\mathsf p^2 (y)}{2\\overline m} + \\frac{\\mathsf r^2 (y)}{2}. \n$$\nNext, the evolution at all further time is governed by the following system of conservation laws: \n\\begin{align}\n\t&\\partial_t \\mathbf r(y,t) = \\frac1{\\overline m} \\partial_y \\mathbf p(y,t), \\qquad \\mathbf r(0,t) = \\mathbf r(1,t) = 0, \n\t\\label{eq:equa diff r}\\\\\n\t&\\partial_t \\mathbf p(y,t) = \\partial_y \\mathbf r (y,t),\n\t\\label{eq:equa diff v}\\\\\n\t&\\partial_t \\mathbf e(y,t) = \\frac1{\\overline m} \\partial_y (\\mathbf r(y,t) \\mathbf p(y,t)).\n\t\\label{eq:equa diff e}\n\\end{align}\n\nThanks to the regularity conditions on $\\mathsf r, \\mathsf p$ in \\eqref{eq: regularity beta r p}, the solutions of these equations are classical.\nSince $(\\mathbf r, \\mathbf p)$ are solution of wave equations with suitable boundary conditions, they can be obtained explicitly by expanding them in Fourier series.  \nThen, by a time integration, $\\mathbf e$ may be expressed as a function of $(\\mathbf r, \\mathbf p)$, see (\\ref{eq: evolution energy limit}). \nLater we will use that a classical solution for the system governing $(\\mathbf r, \\mathbf p)$ coincides with the (unique) weak solution of this system. \nBecause of the boundary conditions, test functions will have to be chosen appropriately (see (\\ref{eq: R characterization}-\\ref{eq: P characterization})). \n\n\\begin{Theorem}\\label{the: main result}\nLet $t\\in [0,T]$ and $f \\in \\mathcal C^0([0,1])$. \nLet us assume that the system is initially prepared in a locally Gibbs state such that $\\beta$, $\\mathsf r$ and $\\mathsf p$ satisfy \\eqref{eq: regularity beta r p}. \nThen, as $N\\to \\infty$, almost surely (w.r.t.\\@ $\\mathsf P$), \n\\begin{align}\n\t&\\mathcal R_N (f,t) \\; = \\;  \\frac{1}{N} \\sum_{x=1}^N f(x\/N) \\; \\lgs{r_x (Nt)} \\qquad \\to \\qquad \\mathcal R(f,t), \\label{R limit}\\\\\n\t&\\mathcal P_N (f,t) \\; = \\; \\frac{1}{N} \\sum_{x=1}^N f(x\/N) \\; \\lgs{p_x (Nt)} \\qquad \\to \\qquad \\mathcal P(f,t), \\label{P limit}\\\\\n\t&\\mathcal E_N (f,t) \\; = \\; \\frac{1}{N} \\sum_{x=1}^N f(x\/N) \\; \\lgs{e_x (Nt)} \\qquad \\to \\qquad \\mathcal E(f,t) \\label{E limit}.\n\\end{align}\n\\end{Theorem}\n\\begin{Remark}\nAs pointed out in the introduction, the situation is much simpler at thermal equilibrium, i.e.\\@ for $\\beta$ constant, and these limits hold even for the non-disordered chain.\nSee Section \\ref{subsec: thermal equilibrium} for a derivation along the lines used to derive Theorem \\ref{the: main result}. \n\\end{Remark}\n\n\n\n\n\n\\section{Evolution of $\\mathcal R_N$ and $\\mathcal P_N$}\n\nIn this section, we show the limits (\\ref{R limit}-\\ref{P limit}). \nMoreover, in order to later deal with the field $\\mathcal E_N$, we show more: \n\nThe functions $\\lgs{r_{[Ny]} (Nt)}$ and $m_{[Ny]}^{-1}\\lgs{ p_{[Ny]} (Nt)}$ are uniformly (in $N$) H\\\"older regular in $y\\in [0,1]$, with exponent at least $1\/2$. \nHence they converge pointwise to $\\mathbf r(y,t)$ and $\\mathbf p (y,t)$ respectively. \n\n\\subsection{A priori estimates}\\label{subsec: a priori}\nGiven $d\\in \\mathbb{N}$, we denote the standard scalar product on $\\mathbb{R}^d$ by $\\langle \\cdot , \\cdot \\rangle_d$ (we will drop the subscript $d$ when no confusion seems possible).\nLet us consider the two following conserved quantities: \n\\begin{align}\n\tH(r,p) &= \\frac12 \\big(  \\langle p ,M^{-1 } p \\rangle_N + \\langle r ,r \\rangle_{N-1} \\big), \\\\\n\tI(r,p) &= \\frac12 \\big( \\langle \\nabla_- r, M^{-1} \\nabla_- r \\rangle_N + \\langle \\nabla_+ M^{-1} p , \\nabla_+ M^{-1}p \\rangle_{N-1} \\big).\\label{eq: I conserved}\n\\end{align}\nThe conservation of $I$ follows from the fact that, if $(r,p)$ solve \\eqref{eq: equations of motion}, then $(\\nabla_+ M^{-1}p, \\nabla_- r)$ solve the same equation, the corresponding Hamiltonian being $I$ \n(since we have that $H (\\nabla_+ M^{-1} p, \\nabla_- r) =I (r,p)$).\nNotice also that a full set of conserved quantities can be generated by further taking gradients, see Section \\ref{subsec: invariant quantities}. \n\nThanks to these two conservation laws, and to the smoothness assumptions on $\\mathsf r$ and $\\mathsf p$, we deduce\n\\begin{Lemma}\\label{lem-bounds}\nThere exists {a deterministic} $\\mathrm{C}$ such that, for any $t \\ge 0$ and any $N \\in \\mathbb{N}$, \n\\begin{align} \n\t&\\sum_{x=1}^{N-1} \\lgs{ r_x(Nt) }^{2} \\le \\mathrm{C} N, \\qquad \n\t\\sum_{x=1}^N \\lgs{ p_x(Nt) }^2 \\le \\mathrm{C} N, \n\t\\label{eq: L2 estimate}\\\\\n\t&\\sum_{x=1}^N \\lgs{ (\\nabla_- r)_x (Nt)}^2 \\le \\frac\\mathrm{C}{N}, \\qquad \n\t\\sum_{x=1}^{N-1} \\lgs{ (\\nabla_+ M^{-1} p)_x (Nt) }^2 \\le \\frac\\mathrm{C}{N}.\n\t\\label{eq: H1 estimate}\n\\end{align}\n\\end{Lemma}\n\\begin{proof}\nBy linearity of the equations of motion \\eqref{eq: equations of motion}, $(\\lgs{ r }, \\lgs{ p } )$ solve the same equations as $(r,p)$. \nTherefore, the conservation of $H(r,p)$ and $I(r,p)$ implies the conservation of $H(\\lgs{ r }, \\lgs{ p })$ and $I(\\lgs{ r }, \\lgs{ p })$. \nSince the quantities to be estimated in \\eqref{eq: L2 estimate} are bounded by $H(\\lgs{ r }, \\lgs{ p })$ \nand the quantities to be estimated in \\eqref{eq: H1 estimate} are bounded by $I(\\lgs{ r }, \\lgs{ p })$, \nwe conclude that is it enough to establish them respectively for $H(\\lgs{r}, \\lgs{p})$ and  $I(\\lgs{r}, \\lgs{p})$ at $t=0$. \nThis follows from a direct computation, thanks to the product structure of the local Gibbs state \\eqref{eq: local Gibbs} and to the hypotheses on $\\mathsf r$ and $\\mathsf p$ in \\eqref{eq: regularity beta r p} \n(in particular, this is the place where the boundary condition on $\\mathsf r$ plays a role). \n\\end{proof}\n\\noindent\n\n{\n\\begin{Remark}\nNotice that the bounds in Lemma \\ref{lem-bounds} are actually valid for any time scale $N^\\alpha t$, for any $\\alpha>0$.\n\\end{Remark}\n}\n\nAs a corollary, we deduce the existence of a constant $\\mathrm{C}\\in \\mathbb{R}$ such that, for any $x,y \\in \\mathbb{Z} \\cap [1,N]$,\n\\begin{equation}\n\\label{eq: Holder continuity}\n\\begin{split}\n&\\big| \\lgs{r_{x'}(Nt) } - \\lgs{r_x(Nt) } \\big| \\le  \\mathrm{C} \\left|\\tfrac{x'-x}N \\right|^{1\/2},\\\\\n&\\big| m_{x'}^{-1} \\lgs{ p_{x'}(Nt) } -  m_x^{-1} \\lgs{ p_x(Nt) } \\big|  \\le  \\mathrm{C} \\Big|\\tfrac{x'-x}N \\Big|^{1\/2},\n\\end{split}\n\\end{equation}\nand therefore also such that\n\\begin{equation}\\label{eq: bounded r and p}\n\t|\\lgs{ r_x(Nt) }| \\le \\mathrm{C}, \\quad |\\lgs{ p_x(Nt) }| \\le \\mathrm{C} .\n\\end{equation}\nIndeed, to get e.g.\\@ \\eqref{eq: Holder continuity} for $r$, we deduce from \\eqref{eq: H1 estimate} that \n\\begin{equation*}\n\\begin{split}\n\\big|\\lgs{ r_{x'}(Nt) } - \\lgs{ r_x(Nt) } \\big|&=\\left|\\sum_{z=x+1}^{x'} \\lgs{ (\\nabla_- r)_z(Nt)} \\right| \\le \\left( \\sum_{z=1}^N \\lgs{ (\\nabla_- r)_z (Nt) }^2 \\right)^{1\/2} |x-x'|^{1\/2} \\\\\n&\\le \\mathrm{C} \\Big|\\tfrac{x'-x}N \\Big|^{1\/2}.\n\\end{split}\n\\end{equation*}\n{Next \\eqref{eq: bounded r and p} follows from \\eqref{eq: Holder continuity} if, given $N,t$, there exists at least some $x_0$ such that the inequalities hold.\nThis in turn follows from \\eqref{eq: L2 estimate}.}\n\n\n\\subsection{Averaging lemma for the field $\\mathcal P_N$}\nThe method of the corrected empirical measure is an elegant method to deal with the randomness on the masses in deriving the hydrodynamic limit for $\\mathcal R_N$ and $\\mathcal P_N$ \\cite{goncalves_jara,jara_landim,bernardin}.\nHowever, in our case, it seems more convenient to use the following lemma: \n\n\\begin{Lemma}\\label{lem: replacement}\nLet $f \\in \\mathcal C^0([0,1])$ and $t\\ge 0$. \nAlmost surely (w.r.t.\\@ the masses), for $N\\to\\infty$, \n\\begin{align}\n\t &\\frac1N \\sum_{x=1}^N f (x\/N) \\frac{\\lgs{ p_x (Nt) }}{m_{x}} (m_x - \\overline m) \\quad \\to \\quad 0,\n\t \\label{eq: replacement}\\\\\n\t &\\frac1N \\sum_{x=1}^N f (x\/N) \\left(\\frac{\\lgs{ p_x (Nt) }}{m_{x}}\\right)^2 (m_x - \\overline m) \\quad \\to \\quad 0.\n\t \\label{eq: replacement bis}\n\\end{align}\n\\end{Lemma}\n\n\\begin{proof}\nLet us start with \\eqref{eq: replacement}.\nLet $A_N$ be the quantity in the left hand side of \\eqref{eq: replacement},\nlet $\\widetilde m_x = m_x - \\overline m$, and let \n\\begin{equation}\\label{eq: some def of varphi}\n\t\\varphi(x) =  f(x\/N) \\frac{\\lgs{ p_x (Nt) }}{m_{x}}\n\\end{equation}\n{where, for simplicity, we do not write explicitly the dependence of $\\varphi$ on $N$ and $t$}.\nLet $0 < \\tau < 1$. \n{Let}\n$$\n\t{\\Gamma_N^\\tau = \\{\u00a01 + k \\lfloor N^\\tau \\rfloor : k \\in \\mathbb{Z} \\} \\cap [1,N] } \n$$ \n{and, given $x\\in \\Gamma_N^\\tau$, let }\n$$\n\t{\u00a0x' = \\min \\{ y \\in \\Gamma_N^\\tau \\cup \\{\u00a0N+1\\}\u00a0: y > x\u00a0\\} .} \n$$\nWe decompose $A_N$ as\n\\begin{align*}\n\tA_N\n\t=& {\\frac{1}{N^{1 - \\tau}} \\sum_{x \\in \\Gamma_N^\\tau} \\frac{1}{N^\\tau} \\sum_{x \\le z < x'} \\varphi (z) \\tilde m_z }\\\\\n\t=&{ \\frac{1}{N^{1 - \\tau}} \\sum_{x \\in \\Gamma_N^\\tau} \\frac{\\varphi (x)}{N^\\tau} \\sum_{x \\le z < x'} \\tilde m_z  \n\t + \\frac{1}{N^{1 - \\tau}} \\sum_{x \\in \\Gamma_N^\\tau} \\frac{1}{N^\\tau} \\sum_{x \\le z < x'} (\\varphi (z) - \\varphi (x)) \\tilde m_z }\\\\\n\t=: & A_N^{(1)} + A_N^{(2)}.\n\\end{align*}\nTo deal with $A_N^{(1)}$, we observe that $\\varphi$ is bounded, see \\eqref{eq: bounded r and p}, so that by Jensen's inequality, \n$$\n\t\\mathsf E((A_N^{(1)})^4) \n\t\\le {\\frac{\\mathrm{C}}{N^{1-\\tau}} \\sum_{x \\in \\Gamma_N^\\tau} \\mathsf E \\bigg(\\bigg( \\frac{1}{N^\\tau} \\sum_{x\\le z <x'} \\widetilde m _{z}  \\bigg)^4\\bigg) }\n\t\\le \\frac{\\mathrm{C}}{N^{2 \\tau}}\n$$\n{(since the masses $m_x$ are i.i.d.).} \nTaking $\\tau > 1\/2$, this shows that $A_N^{(1)} \\to 0$ almost surely by Borel-Cantelli's lemma.\n{To deal with $A_N^{(2)}$, we start from the definition \\eqref{eq: some def of varphi} of $\\varphi$ and we bound}\n\\begin{equation}\\label{eq: bound varphi in proof lemma 2}\n\t{\n\t|\\varphi (z) - \\varphi(x)| \n\t\\le \n\t\\mathrm{C} \\left| m_z^{-1}\\lgs{ p_z (Nt) } -  m_x^{-1}\\lgs{ p_x (Nt) } \\right|\n\t+ \\mathrm{C} |f(z\/N) - f(x\/N)|.\n\t}\n\\end{equation}\n{The fact that $A_N^{(2)}\\to 0$ (deterministically) follows then from the bound \\eqref{eq: Holder continuity} and from the uniform continuity of $f$. }\n\n{The proof of \\eqref{eq: replacement bis} is entirely analogous, with now \n$\\varphi (x) = f(x\/N) (\\lgs{ p_x (Nt) } \/ m_{x} )^2$ instead of \\eqref{eq: some def of varphi}. \nBy \\eqref{eq: bounded r and p}, this function is bounded and \\eqref{eq: bound varphi in proof lemma 2} is still satisfied since\n\\begin{align*}\n\t&\\left| (m_z^{-1}\\lgs{ p_z (Nt) })^2 -  (m_x^{-1}\\lgs{ p_x (Nt) })^2 \\right|\\\\\n\t&\\le \n\t\\left| m_z^{-1}\\lgs{ p_z (Nt) } +  m_x^{-1}\\lgs{ p_x (Nt) } \\right| \\times \\left| m_z^{-1}\\lgs{ p_z (Nt) } -  m_x^{-1}\\lgs{ p_x (Nt) } \\right|\\\\\n\t&\\le \n\t\\mathrm{C} \\left| m_z^{-1}\\lgs{ p_z (Nt) } -  m_x^{-1}\\lgs{ p_x (Nt) } \\right|.\n\\end{align*}\n}\n\\end{proof}\n\n\n\n\n\\subsection{Proof of  the convergence to the linear wave equation (\\ref{R limit}-\\ref{P limit})}\\label{sec:proof-}\nFor any smooth functions $f, g:[0,1] \\in \\mathbb{R}$ such that $f(0)= f(1) = 0$, \nthe limiting fields $\\mathcal R$ and $\\mathcal P$ defined in (\\ref{eq:rpe}) \ncan be equivalently characterized as follows: \n\\begin{align}\n\t\\mathcal R (f,t) &= \\mathcal R (f,0) - \\frac{1}{\\overline m} \\int_0^t \\mathcal P (f',s) \\mathrm{d} s, \n\t\\label{eq: R characterization}\\\\\n\t\\mathcal P (g,t) &= \\mathcal P (g,0) - \\int_0^t \\mathcal R (g',s) \\mathrm{d} s,\n\t\\label{eq: P characterization}\n\\end{align}\nand \n\\begin{equation}\\label{eq: limiting fields t=0}\n\t\\mathcal R (f,0) = \\int_0^1 f(x) \\mathsf r(x) \\mathrm{d} x, \\qquad \\mathcal P(g,0) = \\int_0^1 g(x) \\mathsf p (x) \\mathrm{d} x.\n\\end{equation}\nLet us use this characterization to show that $\\mathcal R_N (f,t) \\to \\mathcal R(f,t)$ \nand  $\\mathcal P_N (g,t) \\to \\mathcal P(g,t)$.\n\nThe convergence at $t=0$ follows from the strong law of large numbers: $\\mathcal R_N (f,0)$ and $\\mathcal P_N (g,0)$ converge almost surely to $\\mathcal R (f,0)$ and $ \\mathcal P(g,0)$ given by \\eqref{eq: limiting fields t=0}. \n\nLet us next consider $t\\ge 0$, and let us first deal with $\\mathcal R_N$. \nIntegrating the equations of motion yields \n\\begin{align*}\n\t\\mathcal R_N(f,t) \n\t&= \\mathcal R_N (f ,0) + \\int_0^{Nt}   \\frac1{N}\\sum_{x=1}^N f(x\/N)\\, \\blgs{ \\big( \\nabla_+ M^{-1} p \\big)_x (s)} \\,\\mathrm{d} s \\\\\n\t&= \\mathcal R_N (f, 0) - \\int_0^{Nt}   \\frac1{N}\\sum_{x=1}^N \\nabla_- f(x\/N) m_x^{-1}  \\lgs{ p_x (s) } \\,\\mathrm{d} s\n\\end{align*}\nwhere we used the boundary condition $f(0) = f (1) = 0$ to perform the integration by part. \nSince $\\nabla_- f(x\/N) = N^{-1} f'(x\/N) + \\mathcal O (N^{-2})$, we obtain\n$$\n\t\\mathcal R_N (f,t) =  \\mathcal R_N (f, 0) - \\int_0^{t}   \\frac1{N}\\sum_{x=1}^N f'(x\/N) m_x^{-1}  \\lgs{ p_x (Ns) } \\mathrm{d} s + \\mathcal O \\Big(\\frac1N \\Big).\n$$\n{Using \\eqref{eq: replacement} in Lemma \\ref{lem: replacement}, as well as the dominated convergence theorem to deal with the time integral, \nwe may replace} $m_x^{-1}$ by $(\\overline m)^{-1}$ up to an error that vanishes almost surely in the limit $N\\to \\infty$.\nThus \n\\begin{equation}\\label{eq: RN relation}\n\t\\mathcal R_N (f,t) =  \\mathcal R_N (f, 0) -  \\frac{1}{\\overline{m}}\\int_0^{t} \\mathcal P_N (f',s) \\mathrm{d} s + \\varepsilon_N,\n\\end{equation}\nwhere $\\varepsilon_N \\to 0$ almost surely as $N \\to \\infty$. \nLet us next deal with $\\mathcal P_N$. This case is simpler since no homogenization over the masses is needed. Proceeding similarly, we find\n\\begin{align}\n\t\\mathcal P_N(g,t)\n\t&= \\mathcal P_N (g,0) + \\int_0^{Nt} \\frac1N \\sum_{x=1}^N g(x\/N)\\, \\blgs{\\big(\\nabla_- r\\big)_x(s)}\\, \\mathrm{d} s \n\t\\nonumber\\\\\n\t&= \\mathcal P_N (g,0) - \\int_0^{Nt} \\frac1N \\sum_{x=1}^N \\nabla_+ g(x\/N) \\lgs{ r_x (s)}\\, \\mathrm{d} s \n\t\\nonumber\\\\\n\t& = \\mathcal P_N (g,0) - \\int_0^t \\mathcal R_N (g',s) \\mathrm{d} s + \\tilde\\varepsilon_N\n\t\\label{eq: PN relation}\n\\end{align}\nwhere we used the boundary condition $r_0(s) = r_N(s) = 0$ for all time $s\\ge 0$ to perform the integration by part, and where $\\tilde\\varepsilon_N \\to 0$ deterministically as $N\\to \\infty$. \n\nThe families $\\big( \\mathcal R_N (f,\\cdot)\\big)_N$ and $\\big( \\mathcal P_N (g,\\cdot)\\big)_N$ are equicontinuous since a uniform bound on the time derivative of $\\mathcal R_N(f,\\cdot)$ and $ \\mathcal P_N (g,\\cdot)$ holds. \nHence, the relations \\eqref{eq: RN relation} and \\eqref{eq: PN relation} implies that any limiting point must satisfy (\\ref{eq: R characterization}-\\ref{eq: P characterization}).\n\n\n\n\\subsection{Pointwise convergence}\\label{sec:poit}\nThanks to the H\\\"older regularity\nof both $\\lgs{ r_{x}(Nt) }$ and $m_x^{-1}\\lgs{ p_{x}(Nt) }$ expressed by \\eqref{eq: Holder continuity}, \nwe deduce a stronger result: \n\\begin{Proposition}\\label{prop: pointwise convergence}\nLet $y\\in ]0,1[$ and let $t\\in ]0,1[$. \nAs $N\\to \\infty$, almost surely (w.r.t.\\@ $\\mathsf P$),  \n$$\n\t\\lgs{ r_{[Ny]}(Nt) } \\quad\\to\\quad \\mathbf r(y,t), \n\t\\qquad \n\t\\frac{\\lgs{ p_{[Ny]} (Nt) }}{m_{[Ny]}} \\quad\\to\\quad \\frac{\\mathbf p(y,t)}{\\overline{m}}.\n$$\n\\end{Proposition}\n\\begin{proof}\nLet us first deal with $\\lgs{ r_{[Ny]}(Nt) }$.\nLet $(\\rho_\\epsilon)_{\\epsilon > 0}$ be a regularizing family: \n$\\rho_\\epsilon \\in \\mathcal C^\\infty (\\mathbb{R})$, \n$\\mathrm{supp} (\\rho_\\epsilon) = [-\\epsilon, \\epsilon]$, $\\rho_\\epsilon \\ge 0$ and $\\int \\rho_\\epsilon (y) \\, \\mathrm{d} y = 1$.\nFor $y\\in ]\\epsilon, 1-\\epsilon[$, we decompose\n\\begin{equation*}\n\t\\begin{split}\n    \t&\\lgs{ r_{[Ny]} (Nt) } = \\int \\rho_\\epsilon \\Big( y-y' \\Big) \\lgs{ r_{[Ny]} (Nt) } \\mathrm{d} y'\\\\\n\t&=  \\int \\rho_\\epsilon \\Big( y-y' \\Big) \\lgs{ r_{[Ny']} (Nt) } \\mathrm{d} y' + \\int \\rho_\\epsilon \\Big( y-y' \\Big) \\left( \\lgs{r_{[Ny]} (Nt)} -\\lgs{ r_{[Ny']} (Nt) } \\right)\\mathrm{d} y'.\n  \t\\end{split}\n\\end{equation*}\nBy \\eqref{eq: Holder continuity}, the second term is bounded in absolute value by\n\\begin{equation}\t\\label{eq: Holder rest}\n  \\int \\rho_\\epsilon \\Big( y-y' \\Big) \\left| \\lgs{r_{[Ny]} (Nt)} -\\lgs{ r_{[Ny']} (Nt) } \\right|\\mathrm{d} y' \\ \\le\\\n  \\mathrm{C} \\sqrt\\epsilon,\n\\end{equation}\nwhile the first term is approximated uniformly in $N$ by\n\\begin{equation*}\n \\frac{1}N \\sum_{x=1}^N \\rho_\\epsilon \\Big( \\frac{x}N - y \\Big) \\lgs{ r_x (Nt) }.\n\\end{equation*}\nThus, by the result shown {in Section~\\ref{sec:proof-}}, this term converges to \n\\begin{equation}\\label{eq: regularized integral}\n\t\\int_0^1 \\rho_\\epsilon (y - y') \\mathbf r(y',t) \\mathrm{d} y'\n\\end{equation}\nas $N\\to \\infty$. \nLetting next $\\epsilon\\to 0$,  the continuity of $\\mathbf r(\\cdot , t)$ \nimplies that \\eqref{eq: regularized integral} converges to $\\mathbf r(y,t)$ \nwhile \\eqref{eq: Holder rest} converges to $0$ as $N\\to \\infty$. \nTo deal with $m_{[Ny]}^{-1}\\lgs{ p_{[Ny]} (Nt) }$, we proceed similarly, \n{using \\eqref{eq: replacement} in} Lemma \\ref{lem: replacement}, to get the analog of \\eqref{eq: regularized integral}. \n\\end{proof}\nFinally, thanks to the bound \\eqref{eq: bounded r and p} and the pointwise convergence result in Proposition \\ref{prop: pointwise convergence}, \nand thanks to {using \\eqref{eq: replacement} in} Lemma \\ref{lem: replacement} for the field $\\mathcal P_N$, \nwe derive  (\\ref{R limit}-\\ref{P limit}) by applying the dominated convergence theorem. \n\n\n\n\n\n\n\\section{Evolution of the energy $\\mathcal E_N$}\\label{sec: energy}\n\nIn this section we show the limit \\eqref{E limit}. We will assume that $f \\in \\mathcal C^1([0,1])$. \nWe can then recover the result \\eqref{E limit} for $f \\in \\mathcal C^0([0,1])$ by density, and using the a priori estimate $\\sum_{x} \\lgs{e_x (t)} \\le \\mathrm{C} N$ at all time $t \\ge 0$. \n\n\\subsection{Main decomposition of the energy}\\label{sec:textbfm-decomp-energ}\nIn order to derive the limit of $\\mathcal E_N$, we separate the contribution to the total energy from the temperature (that does not evolve with time) \nand from mechanical energy, i.e.\\@ the average kinetic and potential energy (that does evolve due to the transport of momentum and displacement). \n\nAt the macroscopic level, we deduce from (\\ref{eq:equa diff r}-\\ref{eq:equa diff e}) that \n\\begin{align}\n\t\\mathbf e(y,t) \n\t&= \\mathbf e (y,0) + \\frac{1}{\\overline m} \\int_0^t \\partial_y  (\\mathbf r(y,s) \\mathbf p(y,s)) \\mathrm{d} s \\nonumber\\\\\n\t&= \\frac1{\\beta(y)} + \\frac{\\mathbf p{^2} (y,0)}{2 \\overline m} + \\frac{\\mathbf r{^2} (y,0)}{2} + \\frac{1}{\\overline m} \\int_0^t \\partial_s \\Big( \\mathbf p^2(y,s)\/2 + \\overline m \\mathbf r{^2}(y,s)\/2 \\Big) \\mathrm{d} s \\nonumber\\\\\n\t&= \\frac{\\mathbf p^2 (y,t)}{2 \\overline m} + \\frac{\\mathbf r^2(y,t)}{2} + \\frac1{\\beta(y)}.\\label{eq: evolution energy limit}\n\\end{align}\nAt the microscopic level, we decompose\n\\begin{align}\n\t\\mathcal E_N(f,t)\n\t=& \\frac1N \\sum_{x=1}^N f(x\/N) \\left(  \\frac{\\lgs{ p_x^2 } }{2m_x} + \\frac{\\lgs{ r_x^2 }}{2}  \\right) (Nt) \n\t\\nonumber\\\\\n\t=& \\frac1N \\sum_{x=1}^N  f(x\/N) \\left(  \\frac{\\lgs{ p_x}^2}{2m_x} + \\frac{\\lgs{ r_x }^2}{2}  \\right) (Nt)\n\t+\\frac1N \\sum_{x=1}^N  f(x\/N) \\left(  \\frac{\\lgs{ \\widetilde p_x^2 }}{2m_x} + \\frac{\\lgs{ \\widetilde r_x^2 }}{2}  \\right) (Nt) \n\t\\nonumber\\\\ \n\t=:& \\mathcal A_N(t) + \\mathcal F_N(t), \n\t\\label{eq: average and fluctuation}\n\\end{align}\nwith \n$$ \n\t\\widetilde p_x = p_x - \\lgs{p_x}, \\qquad \\widetilde r_x = r_x - \\lgs{r_x},\n$$\nand where $\\mathcal A$ and $\\mathcal F$ stands respectively for ``average'' and ``fluctuations''. \nComparing \\eqref{eq: evolution energy limit} and \\eqref{eq: average and fluctuation}, we conclude that it is enough to show that, $\\mathsf P$ almost surely, as $N\\to \\infty$, \n\\begin{align} \n\t& \\mathcal A_N(t) \\quad \\to \\quad  \\int_0^1 f (y) \\left(  \\frac{\\mathbf p^2 (y,t)}{2 \\overline m} + \\frac{\\mathbf r^2(y,t)}{2} \\right) \\mathrm{d} y, \n\t\\label{eq: 1st limit energy}\\\\\n\t& \\mathcal F_N (t) - \\mathcal F_N (0) \\quad \\to \\quad  0, \\qquad \\mathcal F_N (0) \\quad \\to \\quad \\int_0^1 \\frac{f(y)}{\\beta (y)} \\mathrm{d} y. \n\t\\label{eq: 2d limit energy}\n\\end{align}\n\nThe limit \\eqref{eq: 1st limit energy} is deduced in the same ways as (\\ref{R limit}-\\ref{P limit}): \n{\nWe first use \\eqref{eq: replacement bis} in Lemma \\ref{lem: replacement} to replace $\\lgs{p_x}^2\/2m_x$ by $\\frac{\\overline m}{2}(\\lgs{p_x}\/m_x)^2$\nup to an error that vanishes as $N\\to \\infty$. }\nNext, thanks to the bound \\eqref{eq: bounded r and p} and the pointwise convergence result in Proposition \\ref{prop: pointwise convergence}, we derive \\eqref{eq: 1st limit energy}  by applying the dominated convergence theorem. \n\n{\nThe limit \\eqref{eq: 2d limit energy} express the fact that the profile of thermal energy (temperature) remains frozen in time. }\nIt will be established thanks to the localization of the high modes of the chain; \nthis is the only place where localization is used. \nMoreover, we will show in Section \\ref{subsec: thermal equilibrium} that in thermal equilibrium, \nthe equality $\\mathcal F_N (t) = \\mathcal F_N (0)$ holds without any assumption on the distribution of the masses (besides positivity).\nThis shows thus that Theorem \\ref{the: main result} holds actually also for a clean chain if $\\beta$ is constant.  \n\n{\n  \\begin{Remark}\\label{frozenforever}\n    The proof of \\eqref{eq: 2d limit energy} holds for any larger time scale $N^\\alpha t$, $\\alpha \\ge 1$, i.e. \n    \\begin{align} \n\t& \\mathcal F_N (N^{\\alpha -1} t) - \\mathcal F_N (0) \\quad \\to \\quad  0.\n\t\\label{eq: 2d limit energy-s}\n\\end{align}\n  \\end{Remark}\n}\n\n\\subsection{Convergence of $\\mathcal F_N (t)$}\nTo deal with the limit \\eqref{eq: 2d limit energy}, we will use the fact that any mode of the chain at positive energy is spatially localized in the thermodynamic limit. \nHence, we will expand the solutions to the equations of motion into the eigenmodes of the chain. \nIn Section \\ref{sec: eigenmodes expansion} below, we carry this expansion in details and we deduce the needed localization estimates. \nFor our present purposes, it suffices to know the following: \nThere exists a basis $\\{\\psi^k \\}_{0\\le k \\le N-1}$ of $\\mathbb{R}^N$, the basis of the eigenmodes of the chain, so that the solutions to the equations of motion read\n\\begin{align}\n&\\widetilde r_x (t)   = \\sum_{k=1}^{N-1} \\Big(  \\frac{\\langle  \\nabla_+ \\psi^k, \\widetilde r(0) \\rangle}{\\omega_k} \\cos \\omega_k t + \\langle \\psi^k , \\widetilde p(0) \\rangle \\sin \\omega_k t \\Big)  \\frac{(\\nabla_+ \\psi^k)_x}{\\omega_k}, \n\t\\label{eq:r solution bis}\\\\\n\t& \\widetilde p_x(t) = \\sum_{k = 0}^{N-1} \\Big( \\langle \\psi^k, \\widetilde p(0) \\rangle \\cos \\omega_k t  -  \\frac{\\langle \\nabla_+ \\psi^k ,\\widetilde r(0) \\rangle}{\\omega_k} \\sin \\omega_k t \\Big) (M \\psi^k)_x,\n\t\\label{eq:p solution bis}\n\\end{align}\nwhere $\\omega_0 = 0$  and $\\omega_k >0$ for $1 \\le k \\le N$ are the corresponding eigenfrequencies of the chain\n{(we assume that the $\\omega_k$ are sorted by increasing order)}.\nObserve that the first term starts from $k=1$ while the second one starts from $k=0$.\nMoreover, the orthogonality relation $\\langle \\psi^k , M \\psi^j \\rangle = \\delta (k - j)$ holds and $\\{ \\omega_k^{-1}\\nabla_+ \\psi^k  \\}_{1 \\le k \\le N-1}$ forms an orthonormal basis of $(\\mathbb{R}^{N-1}, \\langle \\cdot, \\cdot \\rangle_{N-1})$. \nSee Section  \\ref{subsec: eigenmodes solution} for more details. \nThis representation is useful to exploit localization: all modes with $k\\gtrsim \\sqrt N$ are spatially localized. \nSee Section \\ref{subsec: localization} for more quantitative estimates. \nHowever, low modes with $k\\lesssim \\sqrt N$ remain extended, and we will have to show that the contribution of these modes vanish since their proportion $N^{1\/2}\/N \\to 0$ in the thermodynamic limit. \nBelow, for technical reasons, we will replace $1\/2$ by $1-\\alpha$, for some $\\alpha > 0 $ that we will need to choose small enough.  \n\nLet $0 < \\alpha \\ll 1$, let \n$$ \n\tF_1 = \\mathbb{Z} \\cap [0,N^{1-\\alpha}], \\qquad F_{2} = \\mathbb{Z} \\cap ]N^{1-\\alpha}, N-1],\n$$\nand let us decompose $\\widetilde r(t) = \\widetilde r^{(1)}(t) + \\widetilde r^{(2)}(t)$ and $\\widetilde p (t) = \\widetilde p^{(1)} (t) + \\widetilde p^{(2)}(t)$ with \n$$\n\t\\widetilde r^{(i)}(t) = \\sum_{k\\in F_{i} \\backslash \\{0\\}} (\\dots), \\qquad \n\t\\widetilde p^{(i)}(t) = \\sum_{k\\in F_{i}} (\\dots),\n$$\nfor $i=1,2$ and with $(...)$ the summand featuring in \\eqref{eq:r solution bis} or \\eqref{eq:p solution bis}.\nWe insert this decomposition in $\\mathcal F_N$:\n$$ \n\t\\mathcal F_N (t) =  \\frac1N \\sum_{x=1}^N  f(x\/N) \\left(  \n\t\\frac{\\blgs{ \\big( \\widetilde p_x^{(1)} + \\widetilde p_x^{(2)} \\big)^2 }}{2m_x} \n\t+ \\frac{\\blgs{ \\big(\\widetilde r_x^{(1)} +  \\widetilde r_x^{(2)} \\big)^2 } }{2}  \\right) (Nt). \n$$\nLet us show the two following limits: \n\\begin{align}\n\t&\\mathcal F^{(1)}_N (t) = \\frac1N \\sum_{x=1}^N  f (x\/N) \\left(  \n\t\\frac{\\blgs{ \\big( \\widetilde p_x^{(1)} \\big)^2} }{2m_x} \n\t+ \\frac{\\blgs{ \\big(\\widetilde r_x^{(1)} \\big)^2 } }{2}  \\right) (Nt)  \\quad \\to \\quad  0, \n\t\\label{eq:F1}\\\\\n\t&\\mathcal F^{(2)}_N (t) = \\frac1N \\sum_{x=1}^N  f(x\/N) \\left(  \n\t\\frac{\\blgs{ \\big( \\widetilde p_x^{(2)}  \\big)^2} }{2m_x} \n\t+ \\frac{\\blgs{ \\big(\\widetilde r_x^{(2)}  \\big)^2 } }{2}  \\right) (Nt)  \\quad \\to \\quad  \\int\\frac{f(y)}{\\beta (y)} \\mathrm{d} y,\n\t\\label{eq:F2}\n\\end{align}\nwhich, by Cauchy-Schwarz inequality, implies \\eqref{eq: 2d limit energy}. \n\nLet us show \\eqref{eq:F1}.\nLet us bound $|f (x\/N)| \\leq \\mathrm{C}$, and use the explicit solution (\\ref{eq:r solution bis}-\\ref{eq:p solution bis}): \n\\begin{align*}\n\t&\\quad|\\mathcal F^{(1)}_N (t)| \\\\\n\t&\\le \\frac\\mathrm{C}{2N}\\sum_{x=1}^N \\frac{1}{m_x}\n\t\\BBlgs{\\bigg( \n\t\\sum_{k\\in F_1} \\Big(   \\langle \\psi^k ,\\widetilde p (0) \\rangle  \\cos (\\omega_k Nt) - \\tfrac{\\langle \\nabla_+ \\psi^k,\\widetilde r(0)\\rangle}{\\omega_k} \\sin (\\omega_k Nt ) \\Big) \n\tm_x \\psi^k_x \\bigg)^2} \\\\\n\t&+\\frac\\mathrm{C}{2N}\\sum_{x=1}^N\n\t\\BBlgs{\\bigg( \n\t\\sum_{k\\in F_1\\backslash \\{0\\}} \\Big( \\tfrac{\\langle \\nabla_+ \\psi^k, \\widetilde r(0)\\rangle}{\\omega_k} \\cos (\\omega_k Nt) + \\langle \\psi^k, \\widetilde p (0) \\rangle \\sin (\\omega_k Nt) \\Big) \n\t\\frac{(\\nabla_+ \\psi^k)_x}{\\omega_k}\n\t\\bigg)^2}.\n\\end{align*}\nIn both terms, one may expand the square so as to get a double sum over $k,j\\in F_1$ or $k,j\\in F_1 \\backslash \\{0\\}$, and insert the sum over $x$ inside the sum over $k,j$. \nThis yields\n$$ \n\t\\sum_{x=1}^N \\frac{m_x^2}{m_x} \\psi^j_x \\psi^k_x  = \\langle \\psi^j, M\\psi^k \\rangle = \\delta(k-j), \\qquad \n\t\\sum_{x=1}^N \\frac{(\\nabla_+ \\psi^j)_x (\\nabla_+ \\psi^k)_x}{\\omega_j \\omega_k} = \\delta (k-j).\n$$\nThus \n\\begin{align*}\n\t|\\mathcal F^{(1)}_N (t)| \\le\n\t&\\frac\\mathrm{C}{2N} \\sum_{k \\in F_1} \\Blgs{ \\Big(  \\langle \\psi^k ,\\widetilde p (0) \\rangle  \\cos (\\omega_k Nt) - \\frac{\\langle \\nabla_+ \\psi^k ,\\widetilde r(0)\\rangle}{\\omega_k} \\sin (\\omega_k Nt)  \\Big)^2 } \\\\\n\t&+\\frac\\mathrm{C}{2N} \\sum_{k \\in F_1 \\backslash \\{0\\}} \\Blgs{ \\Big(  \\frac{\\langle \\nabla_+ \\psi^k ,\\widetilde r(0)\\rangle}{\\omega_k} \\cos (\\omega_k Nt) + \\langle \\psi^k, \\widetilde p(0) \\rangle \\sin (\\omega_k Nt)  \\Big)^2}.\n\\end{align*}\nAt this point, it suffices to show that there exists a constant $\\mathrm{C}$ such that, for all $k \\in F_1$, \n\\begin{equation}\n\\label{eq:avant30}\n\\lgs{ \\langle \\psi^k,\\widetilde p(0) \\rangle^2 } \\le \\mathrm{C}, \\qquad \\frac{\\lgs{ \\langle \\nabla_+ \\psi^k, \\widetilde r(0) \\rangle^2 }}{\\omega^2_k} \\le \\mathrm{C}, \n\\end{equation}\nsince, bounding $\\sin$ and $\\cos$ by 1, and using Cauchy-Schwarz, we obtain \n$$\n\t|\\mathcal F^{(1)}_N (t)| \\le \\frac{\\mathrm{C}}{N} \\sum_{k \\in F_1} 1 = \\frac{\\mathrm{C} N^{1 - \\alpha}}{N} \\to 0.\n$$\nLet us deal with $\\lgs{ \\langle \\psi^k,\\tilde p(0) \\rangle^2 }$ (the other case is analogous): \n\\begin{equation*}\n\\begin{split}\n\\lgs{ \\langle \\psi^k , \\tilde p (0) \\rangle^2 }\n\t&= \\Blgs{\\Big( \\sum_x \\psi^k_x  \\widetilde p_x (0)  \\Big)^2}\n\t= \\Blgs{ \\sum_{x,y}   \\psi^k_x  \\psi^k_y \\widetilde p_x (0) \\widetilde p_y (0)  }\\\\\n\t&= \\sum_x (\\psi^k_x)^2  \\lgs{ (\\widetilde p_x (0))^2 }\n\\end{split}\n\\end{equation*}\nwhere we have used the fact that $\\lgs{\\cdot}$ is a product measure and that $\\lgs{\\widetilde p_x(0)} = 0$\n for all $x\\in \\{1, \\dots , N\\}$. \nWe compute $ \\lgs{ (\\widetilde p_x (0))^2 } = \\frac{m_x}{\\beta(x\/N)}$. \nSince $\\beta$ is positive and continuous on $[0,1]$, there exists $\\beta_->0$\n such that $\\beta (x\/N) \\ge \\beta_-$ for all $x\\in \\{1, \\dots , N\\}$. \nHence\n\\begin{equation}\\label{eq: beta minus equation}\n\t\\lgs{ \\langle \\psi^k, \\widetilde p (0) \\rangle^2 } \\le \\frac{1}{\\beta_-} \\sum_{x=1}^N m_x (\\psi^k_x)^2 = \\frac{1}{\\beta_-} \\langle \\psi^k , M \\psi^k \\rangle=\\frac{1}{\\beta_-}. \n\\end{equation}\nLet us now show \\eqref{eq:F2}. \nA computation using the initial measure shows that \n$\\mathcal F(0) \\to \\int \\frac{f(y)}{\\beta (y)} \\mathrm{d} y$ as $N\\to \\infty$. \nHence, thanks to \\eqref{eq:F1}, it holds that $\\mathcal F^{(2)}_N (0)\n \\to  \\int \\frac{f(y)}{\\beta (y)} \\mathrm{d} y$ as $N\\to \\infty$. \nThus it suffices to show that $\\mathcal F^{(2)}_N (t) - \\mathcal F^{(2)}_N (0) \\to 0$ as $N\\to \\infty$. \nLet us write $\\mathcal F^{(2)}_N(t)$ as a scalar product and expand it in the eigenmodes basis: \n\\begin{align*}\n\t\\mathcal F^{(2)}_N(t) \n\t=&  \\frac{1}{2N}\\blgs{ \\langle (f \\cdot \\widetilde p^{(2)})(Nt), M^{-1} \\widetilde p^{(2)} (Nt)\\rangle  + \\langle ( f \\cdot \\widetilde r^{(2)} )(Nt), \\widetilde r^{(2)}(Nt) \\rangle  }\\\\\n\t=& \\frac{1}{2N} \\sum_{k\\in F_2}\\Blgs{ \\langle ( f \\cdot \\widetilde p^{(2)})(Nt), \\psi^k \\rangle \\langle \\psi^k, \\widetilde p (Nt)\\rangle \\\\\n\t&\\phantom{ \\frac{1}{2N} \\sum_{k\\in F_2}\\langle\\langle} +\\frac{1}{\\omega_k^2}  \\langle (f \\cdot \\widetilde r^{(2)}) (Nt), \\nabla_+ \\psi^k \\rangle \\langle \\nabla_+\\psi^k ,  \\widetilde r(Nt) \\rangle} .\n\\end{align*}\nHere $g\\cdot h$ denotes a function on $\\mathbb{Z}\\cap [1,N]$ obtained by the usual multiplication in real space between a function $g$ on $[0,1]$ and $h$ on $\\mathbb{Z}\\cap [1,N]$, i.e.\\@ $(g\\cdot h)_x = g(x\/N) h_x$.\nBy Lemma \\ref{lem: localization} stated in Section \\ref{sec: eigenmodes expansion} below, one may associate a localization center $x_0(k)$ to each mode $\\psi^k$ with $k\\in F_2$: \n$x_0(k)$ is the center of the interval $J(k)$ featuring there \n(assuming that $\\alpha$ is small enough so that the hypotheses of Lemma \\ref{lem: localization} are satisfied).\nFor each $k\\in F_2$, let us decompose $f$ as \n$$\n\tf = f_0 (k) + \\widetilde f_k \\quad \\text{with} \\quad f_0 (k) = f (x_0(k)\/N) \n$$\n(thus $f_0(k)$ is a constant and $\\widetilde f_k$ vanishes at $x_0 (k)\/N$).\nWe insert this decomposition in the above expression for $\\mathcal F_N^{(2)}(t)$: \n\\begin{align}\n\t\\mathcal F^{(2)}_N(t) \n\t= &\\frac{1}{2N} \\sum_{k\\in F_2} f_0 (k) \\,\\BBlgs{ \\langle \\widetilde p(Nt) , \\psi^k \\rangle^2 + \\frac{\\langle \\widetilde r (Nt),\\nabla_+ \\psi^k \\rangle^2}{\\omega_k^2}   } \\label{eq: F2 part 1} \\\\\n\t&+ \\frac1{2N}\\sum_{k\\in F_2} \\BBlgs{ \\langle (\\widetilde f_k \\cdot \\widetilde p^{(2)})(Nt), \\psi^k \\rangle \\langle \\psi^k, \\widetilde p (Nt)\\rangle \\nonumber\\\\\n\t&\\phantom{\\frac1{2N}\\sum_{k\\in F_2} \\Blgs{.}} +\\frac{1}{\\omega_k^2}  \\langle (\\widetilde f_k \\cdot \\widetilde r^{(2)}) (Nt), \\nabla_+ \\psi^k \\rangle \\langle \\nabla_+\\psi^k ,  \\widetilde r(Nt) \\rangle}. \\label{eq: F2 part 2}\n\\end{align}\nEach expression between $\\lgs{\\dots}$ in the sum in \\eqref{eq: F2 part 1} represents the energy of the mode $\\psi^k$ and does not evolve with time, \nsee \\eqref{eq: energy of the modes} in Section \\ref{sec: eigenmodes expansion} below. \nHence, to show $\\mathcal F^{(2)}_N (t) - {\\mathcal F}^{(2)}_N (0) \\to 0$, we only need to show that the sum in \\eqref{eq: F2 part 2} converges to 0 as $N\\to \\infty$.\n\nLet us consider a single term in the sum \\eqref{eq: F2 part 2}, and let us focus on the term involving $\\widetilde p$ (the one involving $\\widetilde r$ is treated the same way). \nFirst, by Cauchy-Schwarz, \n\\begin{equation}\\label{eq: to show for localization}\n\\blgs{\\langle (\\widetilde f_k \\cdot\\widetilde p^{(2)})(Nt), \\psi^k \\rangle \\langle \\psi^k, \\widetilde p (Nt)\\rangle }\n\\le \\lgs{\\langle (\\widetilde f_k \\cdot \\widetilde p^{(2)})(Nt), \\psi^k \\rangle^{2}}^{1\/2} \\lgs{ \\langle \\psi^k, \\widetilde p (Nt)\\rangle^{2}}^{1\/2}.\n\\end{equation}\nThe second factor in \\eqref{eq: to show for localization} is bounded by a constant: \n\\begin{align}\n\t \\lgs{ \\langle \\psi^k, \\widetilde p (Nt)\\rangle^{2}}\n\t &= \\BBlgs{ \\Big( \\langle \\psi^k , \\widetilde p(0) \\rangle \\cos \\omega_k Nt - \\frac{\\langle \\nabla_+ \\psi^k , \\widetilde r(0) \\rangle}{\\omega_k} \\sin \\omega_k Nt  \\Big)^2 }  \\nonumber\\\\\n\t &\\le 2 \\,\\BBlgs{  \\langle \\psi^k , \\widetilde p(0) \\rangle^2 + \\frac{\\langle \\nabla_+ \\psi^k , \\widetilde r(0) \\rangle^2}{\\omega_k^2} }\\, \\le \\mathrm{C},\\label{eq: to show for localization next 2}\n\\end{align}\nsee \\eqref{eq:avant30}.\nFor the first factor in \\eqref{eq: to show for localization}, we use again Cauchy-Schwarz to get\n\\begin{equation}\n\\label{eq: to show for localization next}\n\\begin{split}\n\t\\lgs{\\langle (\\widetilde f_k \\cdot \\widetilde p^{(2)})(Nt), \\psi^k \\rangle^{2}}\n\t&= \\Blgs{\\Big( \\sum_x \\widetilde f_k(x\/N) \\widetilde p^{(2)}_x (Nt) \\psi^k_x \\Big)^2}\\\\\n\t&\\le \\Big( \\sum_x \\widetilde f_k^2 (x\/N) (\\psi^k_x)^2 \\Big) \\,\\Blgs{\\sum_x \\big( \\widetilde p^{(2)}_x\\big)^2 (Nt)}\\, .\n\t\\end{split}\n\\end{equation}\nFor $\\widetilde f_k$, we have the bound \n$$\n\t|\\widetilde f_k (x\/N)| = |f_k (x\/N) - f_k (x_0(k)\/N) |\\le \\mathrm{C} \\,\\frac{|x-x_0 (k) |}{N}\n$$\n(this is the only place where we use $f \\in \\mathcal C^1([0,1])$).\nHence, form Lemma \\ref{lem: localization} below, we deduce that for any $\\epsilon > 0$, the first factor in the right hand side of \\eqref{eq: to show for localization next} can be bounded by $1\/N^{2-\\epsilon}$ by taking $\\alpha > 0$ small enough.\nFrom the conservation of energy (see (\\ref{eq: energy of the modes}) and the bound (\\ref{eq:avant30})) , the second factor in \\eqref{eq: to show for localization next} is $\\mathcal O (N)$.\nHence, for $\\alpha>0$ small enough, \\eqref{eq: to show for localization next} goes to zero as $N\\to \\infty$. \nCombining this with \\eqref{eq: to show for localization next 2}, we find that  \\eqref{eq: to show for localization} goes to zero as $N \\to \\infty$, and hence that \\eqref{eq: F2 part 2} converges to 0 as $N\\to \\infty$.\n\n\n\n\n\n\\subsection{Thermal equilibrium case}\\label{subsec: thermal equilibrium}\n\nAssume here that\n there exists some $\\overline{\\beta} >0$ such that $\\beta (y) = \\overline{\\beta}$ for all $y \\in [0,1]$. \nThen, we may relax the assumptions on the masses: requiring only that they are all strictly positive, let us show that $\\mathcal F_N (t) = F_N (0)$ for all $t \\ge 0$. \nThis results from an exact computation. \n\nSince $f$ is arbitrary, it is necessary and sufficient to prove that, for any $x$, \n$$ \n\t\\frac{\\mathrm{d} }{ \\mathrm{d} t} \\left(  \\frac{\\lgs{\\tilde p_x^2(t)}}{2 m_x} + \\frac{\\lgs{\\tilde r_x^2(t)}}{2} \\right) = 0. \n$$\nWe compute \n\\begin{multline*}\n\t\\frac{\\tilde p_x^2(t)}{2 m_x} \n\t=\\frac12 \\sum_{j,k} \\Big( \\langle \\psi^j , \\tilde p(0) \\rangle \\cos \\omega_j t - \\frac{\\langle \\nabla_+ \\psi^j , \\tilde r(0)\\rangle}{\\omega_j} \\sin \\omega_j t  \\Big)\n\t {\\,\\times} \\\\\n\t \\Big( \\langle \\psi^k, \\tilde p(0) \\rangle \\cos \\omega_k t - \\frac{\\langle \\nabla_+ \\psi^k , \\tilde r(0)\\rangle}{\\omega_j} \\sin \\omega_k t  \\Big)  m_x \\psi^j_x \\psi^k_x.\n\\end{multline*}\nA similar expression holds for $\\tilde r^{2}(t)\/2$. \nIn order to obtain the expectation with respect to $\\lgs{\\cdot}$, we compute\n\\begin{eqnarray}\n\t\t\\blgs{\\langle \\psi^k , \\tilde p (0) \\rangle \\langle \\psi^j , \\tilde p (0) \\rangle} & = & \\frac{\\delta (k-j)}{\\overline{\\beta}}, \\label{cancellation beta const 1} \\\\\n\t\t\\blgs{\\langle \\psi^k , \\tilde p (0) \\rangle \\langle \\nabla_+ \\psi^j , \\tilde r (0) \\rangle} &  = & 0, \\label{cancellation beta const 2} \\\\\n\t\t\\blgs{\\langle \\nabla_+ \\psi^k , \\tilde r (0) \\rangle \\langle \\nabla_+ \\psi^j , \\tilde r (0) \\rangle} &  = & \\frac{\\omega_k^2 \\delta (k-j)}{\\overline\\beta}. \\label{cancellation beta const 3}\n\\end{eqnarray}\nThese three properties result from the fact the product structure of $\\rho_{\\mathrm{loc}}$, from the fact that the variables $\\tilde p$ and $\\tilde r$ are centered, \nand from the the fact that $\\beta$ is constant for  \\eqref{cancellation beta const 1} and \\eqref{cancellation beta const 3}.\nE.g.\\@ \\eqref{cancellation beta const 1} is obtained by \n\\begin{equation*}\n\\begin{split}\n\\blgs{\\langle \\psi^k , \\tilde p (0) \\rangle \\langle \\psi^j , \\tilde p (0) \\rangle} \n\t&= \\sum_{x,y}\\psi^k_x  \\psi^j_y \\lgs{ \\tilde p_x (0) \\tilde p_y (0)} = \\frac{1}{\\beta} \\sum_x m_x \\psi^k_x \\psi^j_x \\\\\n\t&= \\frac{\\delta (k-j)}{\\overline \\beta}.\n\\end{split}\n\\end{equation*}\nHence we have that\n$$\n\t \\frac{\\lgs{\\tilde p_x^2(t)}}{2 m_x} = \\frac{1}{2\\beta} \\sum_k( \\cos^2 \\omega_kt + \\sin^2 \\omega_k t ) m_x (\\psi^k_x)^2 = \\frac{1}{2\\beta}\n$$\nand similarly $\\lgs{\\tilde r_x^2(t)}\/2 = \\frac1{2\\beta}$. \n\n\n\n\n\n\n\\section{Eigenmodes expansion: integrability, localization}\\label{sec: eigenmodes expansion}\n\nWe describe an explicit solution to the equations of motion \\eqref{eq: equations of motion} in terms of the eigenmodes of the system. \nThis representation is useful to establish the integrability of the system and to exploit the localization at all energies above the ground states (in the thermodynamic limit).  \n\n\\subsection{Solution to the equations of motion}\\label{subsec: eigenmodes solution} \nFrom \\eqref{eq: equations of motion}, one can deduce second order equations for $r$ and $p$ separately:\n$$ \n\t \\ddot r_x = \\big(\\nabla_+ M^{-1} \\nabla_- r\\big)_x \\quad (1 \\le x \\le N-1), \\qquad \\ddot p_x = \\big( \\Delta M^{-1}p\\big)_x \\quad (1 \\le x \\le N),\n$$ \nwhere, besides the boundary conditions $r_0 = r_N = 0$, we have assumed free boundary conditions for $M^{-1}p$, i.e. $m_0^{-1}p_0 = m_1^{-1}p_1$ and $m_{N+1}^{-1}p_{N+1} = m_N^{-1}p_{N}$. \nNotice that there are two different vector spaces: a $(N-1)$-dimensional space for $r$ with fixed boundary conditions, and a $N$-dimensional space for $M^{-1}p$ with free boundary conditions.\nMoreover, we observe that $\\nabla_+ = - (\\nabla_-)^\\dagger$ with fixed boundary conditions, and that $\\Delta = \\Delta^\\dagger$ with free boundary conditions. \n\nIn order to solve the equations of motion, we need to diagonalize two matrices: \n$\\big(\\nabla_+ M^{-1} \\nabla_-\\big)^\\dagger=\\nabla_+ M^{-1} \\nabla_-$ (of size $N-1$) \nand $\\big(\\Delta M^{-1} \\big)^\\dagger=M^{-1}\\Delta$ (of size $N$). \n{Let us start with the latter:}\nThis matrix is not symmetric but the matrix $ M^{-1\/2} (-\\Delta) M^{-1\/2} $ is symmetric and non-negative.\nIt admits thus an orthonormal set of eigenvectors, $\\{\\varphi^k\\}_{0 \\le k \\le N-1}$ and corresponding eigenvalues $\\omega^2_k$\n{that we assume to be sorted by increasing order}. \nMoreover, the spectrum is $\\mathsf P$-almost surely non-degenerate \n(see e.g.\\@ Proposition II.1 in \\cite{kunz_souillard}, considering here a perturbation around the non-degenerate equal masses case). \nTherefore the vectors $\\psi^k = M^{-1\/2} \\varphi^k$ are such that \n\\begin{equation}\\label{eq: eigenmodes equation}\n\tM^{-1} (-\\Delta) \\psi^k = \\omega^2_k  \\psi^k, \\qquad \\langle \\psi^j, M \\psi^k \\rangle = \\delta(j-k).\n\\end{equation}\nBecause of free boundary conditions, $\\omega^2_0 = 0$, and {one may chose $\\psi_0$ to be given by}\n$$\n\t\\psi_0 = \\Big(\\sum_{x}m_x\\Big)^{-1\/2} (1, \\dots , 1)^\\dagger .   \n$$\n{Next}, the matrix ${-} \\nabla_+ M^{-1} \\nabla_- $ is symmetric and non-negative, \nand we denote the eigenvectors by $|\\widetilde\\psi^k \\rangle$ for $1 \\le k \\le N-1$.\nIt is readily checked that {they may be chosen to be given by}\n$$\n\t\\widetilde \\psi^k = \\frac{1}{\\omega_k} \\nabla_+ \\psi^k\n$$\nwith the corresponding eigenvalue given by $\\omega_k$ for $1 \\le k \\le N-1$. \nWe observe that, by the free boundary conditions on $\\psi^k$,  $\\widetilde \\psi^k (0) = \\widetilde \\psi^k (N) = 0$. \n \nGiven initial conditions $r(0),p(0)$, we can write an explicit solution for $r(t),p(t)$: \n$$ \n\t\\langle \\widetilde \\psi^k , \\ddot{r} \\rangle = -\\omega^2_k \\langle \\widetilde \\psi^k, r\\rangle \\quad (1 \\le k \\le N-1), \n\t\\qquad \n\t\\langle \\psi^k, \\ddot p \\rangle = -\\omega^2_k \\langle \\psi^k, p \\rangle \\quad (0 \\le k \\le N-1).\n$$\nThus\n\\begin{align*}\n\t&\\langle \\widetilde\\psi^k, r(t) \\rangle = \\langle \\widetilde \\psi^k, r(0) \\rangle \\cos \\omega_k t + \\frac{\\langle \\widetilde\\psi^k, \\nabla_+ M^{-1}p(0) \\rangle}{\\omega_k} \\sin \\omega_k t \\quad (1 \\le k \\le N- 1),\\\\\n\t&\\langle \\psi^k ,p(t) \\rangle = \\langle \\psi^k ,p(0) \\rangle \\cos \\omega_k t + \\frac{\\langle \\nabla_- r(0),\\psi^k \\rangle}{\\omega_k} \\sin \\omega_k t \\quad (0 \\le k \\le N-1)\n\\end{align*}\nwith the convention $\\frac{\\sin 0}{0} = 1$ in the second expression at $k=0$ (notice that $\\langle \\nabla_- r(0) ,\\psi^k \\rangle = - \\langle r(0) , \\nabla_+\\psi^k \\rangle = 0$ for $k=0$). This yields therefore\n\\begin{align}\n\t&r(t) = \\sum_{k=1}^{N-1} \\Big(  \\frac{\\langle  \\nabla_+ \\psi^k ,r(0) \\rangle}{\\omega_k} \\cos \\omega_k t + \\langle \\psi^k , p(0) \\rangle \\sin \\omega_k t \\Big)  \\frac{\\nabla_+ \\psi^k}{\\omega_k}, \n\t\\label{eq:r solution}\\\\\n\t&p(t) = \\sum_{k = 0}^{N-1} \\Big( \\langle \\psi^k ,p(0) \\rangle \\cos \\omega_k t  -  \\frac{\\langle \\nabla_+ \\psi^k ,r(0) \\rangle}{\\omega_k} \\sin \\omega_k t \\Big) M \\psi^k.\n\t\\label{eq:p solution}\n\\end{align}\n\n\n\\subsection{Full set of invariant quantities}\\label{subsec: invariant quantities} \nWe observe that the dynamics has $N$ invariant quantities, corresponding to the energy of each mode. \nIt is thus an integrable system. \nIndeed, let us write the full energy as \n\\begin{align*}\n\tH &= \\frac12 \\big( \\langle p, M^{-1}p \\rangle + \\langle r,r \\rangle \\big)\n\t= \\frac12 \\sum_{k=0}^{N-1} \\langle p,\\psi^k \\rangle \\langle M\\psi^k, M^{-1}p \\rangle + \\frac12 \\sum_{k=1}^{N-1} \\langle r, \\widetilde \\psi^k \\rangle \\langle \\widetilde \\psi^k, r \\rangle \\\\\n\t&= \\frac12 \\sum_{k=0}^{N-1} \\Big(  \\langle p,\\psi^k \\rangle^2 + \\frac{\\langle r ,\\nabla_+ \\psi^k \\rangle^2}{\\omega_k^2} \\Big)\n\\end{align*}\nwith the convention that the second term in the last sum is $0$ at $k=0$.\nFrom the evolution equation of the dynamics, one gets that actually \n\\begin{equation}\n\\label{eq: energy of the modes}\n\t\\frac{\\mathrm{d} }{ \\mathrm{d} t} \\left(  \\langle p ,\\psi^k \\rangle^2 + \\frac{\\langle r, \\nabla_+ \\psi^k \\rangle^2}{\\omega_k^2} \\right) = 0 \\quad\\text{for all} \\quad 0 \\le k \\le N-1. \n\\end{equation}\n\nMoreover, by taking specific linear combinations of these conserved quantities, one can obtain conserved quantities that can be written as a sum of local terms. \nThis is for instance the case of the quantity $I$ defined in \\eqref{eq: I conserved}, that reads also \n$$\n\tI(r,p) = \\frac12 \\sum_{k=1}^{N-1} \\omega_k^2 \\, \\left(  \\langle p,\\psi^k \\rangle^2 + \\frac{\\langle r,\\nabla_+ \\psi^k \\rangle^2}{\\omega_k^2} \\right).\n$$\n\n\n\n\n\n\\subsection{Localization}\\label{subsec: localization} \nLocalization can be expressed mathematically in the following strong sense, \nsee \\cite{kunz_souillard,aizenman_graff,aizenman_molchanov} for the general theory and \\cite{verheggen,ajanki_huveneers} for precise estimates on the localization length as one approaches the ground state. \nLet $0 < \\alpha < \\frac12$ and let $I(\\alpha) = ]N^{(1- \\alpha)},N-1]\\cap \\mathbb{Z}$. \nThere exist constants $\\mathrm{C}, c >0$ such that \n$$\n\t\\mathsf E \\Big( \\sum_{k \\in I(\\alpha)} |\\psi^k_x \\psi^k_y | \\Big) \\le \\mathrm{C} \\mathrm{e}^{-c|x-y|\/\\zeta(\\alpha)} \\quad \\text{with} \\quad \\zeta(\\gamma) = N^{2\\alpha}.\n$$\nWe will use this estimate to show that every mode in $k \\in I(\\alpha)$ is supported in a small subset of $[1,N] \\cap \\mathbb{Z}$ up to a small error: \n\n\\begin{Lemma}\\label{lem: localization}\nLet $\\alpha, \\gamma>0$ be such that $2 \\alpha < \\gamma < 1$.\nThere exists almost surely $N_0 \\in\\mathbb{N}$ so that for all $N \\ge N_0$, and for all $k \\in I (\\alpha)$, \nthere exists an interval $J(k)$ with $|J(k)| \\le 2N^\\gamma$ such that $|\\psi^k_x | \\le N^{-1\/\\gamma}$ for all $x \\notin J(k)\\cap \\mathbb{Z}$. \n\\end{Lemma}\n\n\n\\begin{proof}\nLet us first show that \n\\begin{align}\n\tP &:= \\mathsf P\\big( \\exists k \\in I(\\alpha), \\exists x,y\\in [1,N]\\cap \\mathbb{Z} : |x-y| \\ge N^\\gamma, |\\psi^k_x | \\ge N^{-1\/\\gamma},  |\\psi^k_y | \\ge N^{-1\/\\gamma}   \\big)  \n\t\\nonumber\\\\\n\t&\\le\\; \\mathrm{C}(\\alpha, \\gamma) \\mathrm{e}^{- N^{(\\gamma - 2\\alpha)\/2}}\n\t\\label{eq: proba 0 event localization}\n\\end{align}\nIndeed we compute\n\\begin{align*}\n\tP &\\le \\sum_{k\\in I(\\alpha)} \\sum_{x,y : |x-y| \\ge N^\\gamma} \\mathsf P \\big(  |\\psi^k_x | \\ge N^{-1\/\\gamma},  |\\psi^k_y | \\ge N^{-1\/\\gamma} \\big) \\\\\n\t&\\le \\sum_{k\\in I(\\alpha)}  \\sum_{x,y : |x-y| \\ge N^\\gamma} \\mathsf P \\big(  |\\psi^k_x\\psi^k_y | \\ge N^{-2\/\\gamma} \\big) \\\\\n\t&\\le \\sum_{k\\in I(\\alpha)}  \\sum_{x,y : |x-y| \\ge N^\\gamma} N^{2\/\\gamma} \\mathsf E ( |\\psi^k_x\\psi^k_y |) \\\\\n\t&\\le \\mathrm{C} N^{2\/\\gamma} \\sum_{x,y : |x-y| \\ge N^\\gamma} \\mathrm{e}^{-c|x-y|\/\\zeta(\\alpha)} \n\t\\le \\mathrm{C} N^{2\/\\gamma} N^{2} \\mathrm{e}^{- N^{\\gamma - 2 \\alpha}} \n\t\\le  \\mathrm{C}(\\alpha, \\gamma) \\mathrm{e}^{- N^{(\\gamma - 2\\alpha)\/2}}.\n\n\\end{align*}\nTherefore, there exists almost surely $N_0$ so that for for all $N\\ge N_0$, the event featuring in \\eqref{eq: proba 0 event localization} does not occur. \nHence in this case, for all $k \\in I(\\alpha)$ and all $|x - y|>N^{\\gamma}$, we must have either $|\\psi^k_x | \\le 1\/N^{1\/\\gamma}$ or $|\\psi^k_y | \\le 1\/N^{1\/\\gamma}$.\nThis means thus that for any $k\\in I(\\alpha)$  there exists an interval $J(k)$ with $|J(k)| \\le 2N^\\gamma$ such that $|\\psi^k_x | \\le N^{-1\/\\gamma}$ for all $x \\notin J(k)\\cap \\mathbb{Z}$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section::introduction}\n\nThe magnetotelluric (MT) method uses electromagnetic passive sources\nin the magnetosphere and ionosphere to estimate electrical resistivity\nin Earth's interior \\cite{chave1991electrical}. This method was\nintroduced almost 70 years ago by Cagniard\n\\cite{cagniard1953basic}. The passive sources excite a certain\nspectrum, that is, interval of frequencies. There is a long history of\nstudies based on some form of data fitting including a Bayesian\napproach \\cite{grandis1999bayesian}. It has been understood that high\nfrequency data enable determining the resistivity in the ``near''\nsubsurface, while low frequency ones enable obtaining an estimate of\nthe resistivity in the ``deep'' interior. The MT method has been used\nto study melt and hydration in Earth's crust and mantle\n\\cite{rokityansky2012geoelectromagnetic, feucht2017magnetotelluric,\n  evans1994electrical, rippe2013magnetotelluric}, and general\nelectrical structure in the lithosphere\n\\cite{malleswari2019magnetotelluric, thiel2008modelling} and below the\nocean floor \\cite{cox1971electromagnetic}. Moreover, the MT method has\nbeen applied to earthquake prediction \\cite{chouliaras1988application}.\n\n\\smallskip\n\nIn this paper, we analyze the inverse problem for the MT method. We\nlet $\\Omega \\subset \\R^3$ be an open bounded domain with $C^{1,1}$\nboundary and let $\\varepsilon, \\sigma, \\mu \\in C^2(\\overline\\Omega)$\nbe non-negative. We consider the time-harmonic Maxwell equations for\nelectromagnetic fields, $E$ and $H$,\n\\begin{equation}\\label{eqn::Maxwell}\n   \\nabla \\times E = i \\omega \\mu H\\quad\\text{and}\\quad\n   \\nabla \\times H = -i \\omega\n          \\Big(\\varepsilon + \\frac{i \\sigma}{\\omega}\\Big) E\n   \\quad\\text{in}\\quad\\Omega ,\n\\end{equation}\nwhere $\\omega > 0$ is a fixed frequency. The functions $\\varepsilon$,\n$\\sigma$ and $\\mu$ represent the material parameters, namely, electrical\npermittivity, electrical conductivity and magnetic permeability,\nrespectively. In the case of MT, one invokes the following\n\n\\begin{Assumption} \\label{MT assumption}\nThe electrical permittivity vanishes, $\\varepsilon = 0$, and the\nelectrical conductivity and magnetic permeability satisfy $\\sigma \\ge\n\\sigma_0$, $\\mu \\ge \\mu_0$ on $\\overline\\Omega$ for some constants\n$\\sigma_0, \\mu_0 > 0$.\n\\end{Assumption}\n\n\\noindent\nThis assumption has its origin in the low-frequency reduction of\nMaxwell's equations.\n\nWe let $\\nu$ be the outer unit normal to the boundary $\\p \\Omega$, and\ndefine the trace operator $\\bt: C^\\infty(\\overline\\Omega;\\C^3) \\to\nC^\\infty(\\p\\Omega;\\C^3)$ as\n$$\n   \\bt(u) := \\nu \\times u|_{\\p\\Omega}\\quad\\text{for}\\quad\n   u \\in C^\\infty(\\overline\\Omega;\\C^3) .\n$$\nThe trace $\\bt$ can be extended to a bounded linear operator from\n$H_{\\Div}^1(\\Omega)$ into $TH^{1\/2}_{\\Div}(\\p\\Omega)$, where\n$$\n   H^1_{\\Div}(\\Omega) := \\left\\{\n   u \\in H^1(\\Omega;\\C^3) : \\bt(u) \\in TH^{1\/2}_{\\Div}(\\p\\Omega)\\right\\}\n$$\nand\n$$\n   TH^{1\/2}_{\\Div}(\\p\\Omega) := \\left\\{f \\in H^{1\/2}(\\p \\Omega;\\C^3) :\n   \\nu \\cdot f = 0\\quad\\text{and}\\quad\n   \\Div(f)\\in H^{1\/2}(\\p \\Omega;\\C^3)\\right\\} ,\n$$\nand $\\Div$ denotes the surface divergence on $\\p\\Omega$. We refer the\nreader to \\cite{ola1993inverse} for more details. Under\nAssumption~\\ref{MT assumption}, when $\\omega > 0$ does not belong to a\ndiscrete set of magnetic resonant frequencies, the equation\n\\eqref{eqn::Maxwell} with the boundary condition $\\bt(H) = f \\in\nTH^{1\/2}_{\\Div}(\\p\\Omega)$ has a unique solution $(H,E) \\in\nH^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$; see\nSection~\\ref{section::well posedness}. Then the impedance map\n$Z^\\omega_{\\sigma,\\mu}: TH^{1\/2}_{\\Div}(\\p\\Omega)\\to\nTH^{1\/2}_{\\Div}(\\p\\Omega)$ is defined as\n$$\n   Z^\\omega_{\\sigma,\\mu}(f)\n           := \\bt(E),\\quad f\\in TH^{1\/2}_{\\Div}(\\p\\Omega) .\n$$\nThe inverse MT problem (IMTP) is to determine $\\sigma$ and $\\mu$ from\nthe knowledge $Z^\\omega_{\\sigma,\\mu}$. In the geophysics literature,\none commonly considers the resistivity, that is, the reciprocal of\nconductivity \\cite{weckmann2003images}.\n\n\\smallskip\n\n\\begin{Remark}\nIn practice, the data are represented as the graph of\n$Z^\\omega_{\\sigma,\\mu} : TH^{1\/2}_{\\Div}(\\p\\Omega) \\to\nTH^{1\/2}_{\\Div}(\\p\\Omega)$.\n\\end{Remark}\n\nOur first main result is\n\n\\begin{Theorem}\\label{main thm}\nLet $\\Omega\\subset \\R^3$ be a bounded domain with $C^{1,1}$ boundary\nand let $\\sigma_j,\\mu_j\\in C^2(\\overline\\Omega)$, $j=1,2$, be such\nthat $\\sigma_j \\ge \\sigma_0$ and $\\mu_j \\ge \\mu_0$ for some constants\n$\\sigma_0, \\mu_0 > 0$. Suppose that $\\omega > 0$ is not a resonant\nfrequency for $(\\sigma_1,\\mu_1)$ and $(\\sigma_2,\\mu_2)$ and that\n\\begin{equation}\\label{boundary assumption}\n   \\p^\\alpha \\sigma_1|_{\\p\\Omega} = \\p^\\alpha \\sigma_2|_{\\p\\Omega}\n   \\quad\\text{and}\\quad\n   \\p^\\alpha \\mu_1|_{\\p\\Omega} = \\p^\\alpha \\mu_2|_{\\p\\Omega}\n   \\quad\\text{for}\\quad|\\alpha| \\le 2 .\n\\end{equation}\nThen $Z_{\\sigma_1,\\mu_1}^\\omega = Z_{\\sigma_2,\\mu_2}^\\omega$ implies\nthat $\\sigma_1 = \\sigma_2$ and $\\mu_1 = \\mu_2$.\n\\end{Theorem}\n\n\\noindent\nCondition \\eqref{boundary assumption} is not important. We expect that\na suitable boundary determination result would allow to remove it as\nin \\cite{caro2019boundary, joshi2000total, mcdowall1997boundary}. See\nalso Remark~\\ref{remark in Appendix B}.\n\n\\smallskip\n\nThe inverse problem considered in the present paper formally looks\nlike a standard inverse electromagnetic problem (IEMP) proposed in\n\\cite{somersalo1992linearized}: Determine $\\varepsilon$, $\\mu$ and\n$\\sigma$, from the knowledge of the admittance map\n$\\Lambda^\\omega_{\\varepsilon,\\sigma,\\mu} : \\bt(E) \\mapsto \\bt(H)$ for\nall $(E,H) \\in H_{\\Div}^1(\\Omega) \\times H_{\\Div}^1(\\Omega)$ solving\n\\eqref{eqn::Maxwell}. However, the conditions in the IEMP do not allow\nthe vanishing of $\\varepsilon$ as in our Assumption~\\ref{MT\n  assumption}. To be precise, IEMP invokes\n\n\\begin{Assumption} \\label{EM assumption}\nThe electrical permittivity, electrical conductivity and magnetic\npermeability satisfy $\\varepsilon \\ge \\varepsilon_0$, $\\mu \\ge \\mu_0$\nand $\\sigma \\ge 0$ on $\\overline\\Omega$ for some constants\n$\\varepsilon_0, \\mu_0 > 0$.\n\\end{Assumption}\n\n\\noindent\nAs a consequence, the IMTP and IEMP problems are in fact\ndifferent. For comparison, we also mention the EIT, or inverse\nconductivity problem, also known as electrical resistivity tomography\n(ERT) in the geophysics literature, proposed in\n\\cite{calderon2006inverse}: Determine $\\sigma$, satisfying $\\sigma \\ge\n\\sigma_0$ for some constant $\\sigma_0 > 0$, from the\nDirichlet-to-Neumann map $u|_{\\p\\Omega} \\mapsto \\sigma\\p_\\nu\nu|_{\\p\\Omega}$ for all $u \\in H^1(\\Omega)$ solving the conductivity\nequation $\\nabla \\cdot (\\sigma \\nabla u) = 0$ in $\\p\\Omega$. A\nlow-frequency limit of IEMP as in \\cite{lassas1997impedance} will not\nconverge to EIT. However, one expects that a low-frequency limit of\nIMTP meaningfully relates to EIT.\n\n\\smallskip\n\nWe now give a brief overview of results pertaining to the IEMP and its\nhistory. The standard approach to solve this problem is to\nconstruct a family of exponentially growing solutions, also known as\n\\emph{complex geometric optics solutions}, following the celebrated\npaper~\\cite{sylvester1987global} on the inverse conductivity\nproblem. One of the main challenges in adopting the method of\n\\cite{sylvester1987global} is the fact that \\eqref{eqn::Maxwell} with\nAssumption~\\ref{EM assumption} is not elliptic. The linearized problem\nat constant material parameters was studied in\n\\cite{somersalo1992linearized}. For the nonlinear problem, a\nuniqueness result was given in \\cite{sun1992inverse} when the\nelectromagnetic parameters are close to constants. In this paper, to\nget ellipticity, equation \\eqref{eqn::Maxwell} with Assumption~\\ref{EM\n  assumption} was reduced to a system whose principal part is the\nLaplacian. However, this reduction gives some first order terms. For\nmaterial parameters that are nearly constant, the authors were able to\nmanage the first-order terms and introduce complex geometrical solutions\nfor \\eqref{eqn::Maxwell}. The first global uniqueness result was\nproven in~\\cite{ola1993inverse}. This proof was later simplified\nin~\\cite{ola1996electromagnetic}. The important point in the\nsimplified proof is to augment \\eqref{eqn::Maxwell} with\nAssumption~\\ref{EM assumption} to a certain $8 \\times 8$ Dirac\nequation and connect it via some other Dirac operator to an $8 \\times\n8$ system whose principal part is the Laplacian while its remainder\ninvolves only zeroth-order terms. This allowed the authors to\nconstruct complex geometric optics solutions for the latter system and\nconnect them to \\eqref{eqn::Maxwell} with Assumption~\\ref{EM\n  assumption} by applying the Dirac operator that was initially\nintroduced. This technique became popular in the subsequent works on\nvarious aspects of IEMP~\\cite{assylbekov2017kerrinverse,\n  caro2014global, kenig2011inverse}.\n\n\\smallskip\n\nIn the setting of the IMTP, however, one cannot simply employ the\ncomplex geometric optics solutions constructed in\n\\cite{ola1993inverse, ola1996electromagnetic} for the IEMP. Moreover,\nthe elliptization argument of \\cite{ola1996electromagnetic}, applied\nto \\eqref{eqn::Maxwell} with Assumption~\\ref{MT assumption}, does not\nhelp avoiding first-order terms. Instead of that, we\nfollow~\\cite{sun1992inverse} and reduce \\eqref{eqn::Maxwell} with\nAssumption~\\ref{MT assumption} to a system whose principal part is the\nLaplacian. We then introduce novel complex geometrical optics\nsolutions for the reduced system that are essentially solutions for\n\\eqref{eqn::Maxwell} with Assumption~\\ref{MT assumption}. Moreover,\nusing this reduction gives an integral identity with a clear relation\nto \\eqref{eqn::Maxwell}. To deal with the first-order terms, we use\nthe ideas from \\cite{colton1992uniqueness} with substantial\nmodifications since the latter paper assumes that $\\mu$ is constant.\n\n\\smallskip\n\nIn the MT method, performing measurements on the entire boundary (that\nis, the surface of the earth) is impossible. Therefore, the analysis\nof the inverse problem with local measurements is important. We can\nassume that the measurements are performed on a nonempty open subset\n$\\Gamma$ of $\\p\\Omega$ only and that the inaccessible part of the\nboundary $\\Gamma_0 = \\overline{\\p\\Omega \\setminus \\Gamma}$ is a part\nof a sphere (our planet's surface). Our second main result is the\nfollowing\n\n\\begin{Theorem}\\label{main thm 2}\nLet $\\Omega \\subset B_0$ be a bounded domain with $C^{1,1}$ boundary\nincluded in an open ball $B_0 \\subset \\R^3$ and let $\\Gamma_0 =\n\\p\\Omega \\cap \\p B_0$, $\\Gamma_0 \\neq \\p B_0$ and $\\Gamma =\n\\overline{\\p\\Omega \\setminus \\Gamma_0}$. Suppose that $\\sigma_j, \\mu_j\n\\in C^2(\\overline\\Omega)$, $j=1,2$, satisfy $\\sigma_j \\ge \\sigma_0$\nand $\\mu_j \\ge \\mu_0$, for some constants $\\sigma_0, \\mu_0 > 0$, and\n\\begin{equation}\\label{boundary assumption on Gamma}\n   \\p^\\alpha \\sigma_1|_{\\Gamma} = \\p^\\alpha \\sigma_2|_{\\Gamma}\n   \\quad\\text{and}\\quad\n   \\p^\\alpha \\mu_1|_{\\Gamma} = \\p^\\alpha \\mu_2|_{\\Gamma}\n   \\quad\\text{for}\\quad|\\alpha| \\le 2 .\n\\end{equation}\nIn addition, assume that $\\sigma_j$ and $\\mu_j$, $j=1,2$, can be\nextended to $\\R^3$ as $C^2$ functions which are invariant under\nreflection across $\\p B_0$. Suppose that $\\omega > 0$ is not a\nresonant frequency for $(\\sigma_1,\\mu_1)$ and $(\\sigma_2,\\mu_2)$. If\n$$\n   Z_{\\sigma_1,\\mu_1}^\\omega(f)|_{\\Gamma}\n             = Z_{\\sigma_2,\\mu_2}^\\omega(f)|_{\\Gamma}\n   \\quad\\text{for all}\\quad f \\in TH^{1\/2}_{\\Div}(\\p\\Omega)\n   \\quad\\text{with}\\quad \\supp(f)\\subset\\Gamma ,\n$$\nthen $\\sigma_1 = \\sigma_2$ and $\\mu_1 = \\mu_2$.\n\\end{Theorem}\n\n\\noindent\nFor the proof of Theorem~\\ref{main thm}, we follow Isakov's reflection\napproach \\cite{isakov2007uniqueness} which was originally proposed for\nthe inverse conductivity problem. An analogous result for IEMP was\nobtained in \\cite{caro2009inverse}.\n\n\\smallskip\n\nWe briefly describe a connection of our results to the land-based CSEM\n(controlled source electromagnetic) method in geophysical exploration\n\\cite{streich2016controlled}. Contrary to the MT method, the CSEM\nmethod employs active sources. In recent work by Schaller \\textit{et\n  al.}~\\cite{schaller2018land}, a land-based CSEM survey was designed\nand performed at the Schoonebeek oil field. The application of\nland-based CSEM for low-cost ${\\rm CO}_2$ monitoring was studied in\n\\cite{mcaliley2019analysis}. The marine CSEM method, which was\nintroduced by Cox \\textit{et al.}~\\cite{cox1971electromagnetic}, would\nrequire a careful incorporation of the ocean layer in the analysis,\nwhich we do not pursue in this paper. It has successfull applications\nin direct identification of hydrocarbons \\cite{chave1991electrical,\n  eidesmo2002sea}, and the study of the oceanic lithosphere and active\nspreading centers \\cite{chave1990some, constable1996marine,\n  cox1986controlled, evans1994electrical, macgregor2002use,\n  young1981electromagnetic}. For a more detailed exposition of\nprogress made on the marine CSEM, we refer to a review paper by\nConstable \\cite{constable2010ten}. Various basic data fitting\napproaches have been developed for CSEM \\cite{abubakar20082,\n  gribenko2011joint, li20072dp1, li20072dp2}. From a mathematical\npoint of view, the data for the land-based CSEM is modeled by point\nsource measurements. More precisely, for an arbitrary unit vector\n$\\alpha$ and $y \\in \\p\\Omega$, consider the equation\n$$\n   \\nabla_x \\times E_\\alpha(x,y) = i \\omega \\mu(x) H_\\alpha(x,y)\n   + \\delta(x-y) \\nu(y) \\times \\alpha\n   \\quad\\text{and}\\quad\n   \\nabla_x \\times H_\\alpha(x,y) = \\sigma(x) E_\\alpha(x,y)\n   \\quad\\text{in}\\quad\\R^3 ,\n$$\nwith the outgoing radiation condition. The equation governs the\nelectromagnetic field of a magnetic dipole (active source) tangential\nto the boundary $\\p\\Omega$. Then the inverse problem for the\nland-based CSEM is to determine $\\sigma$ and $\\mu$ from\n$$\n   \\mathcal A_{\\sigma,\\mu} := \\left\\{\\big(\\nu(x) \\times H_{e_j}(x,y) ,\\\n   \\nu(x) \\times E_{e_j}(x,y)\\big) : x, y \\in\\p\\Omega ,\\quad\n   x \\neq y ,\\quad j=1,2,3\\right\\} ,\n$$\nwhere $e_j$, $j=1,2,3$, denote the Cartesian coordinate vectors. We\nexpect that following the arguments similar to\n\\cite{ola1996electromagnetic}, one can show that to the knowledge of\n$\\mathcal A_{\\sigma,\\mu}$ is equivalent the knowledge of the graph of\n$Z^\\omega_{\\sigma,\\mu}$ via layer potentials. Then the land-based CSEM\nand MT would concern the same inverse problem with boundary\ndata.\n\n\\smallskip\n\nThe paper is organized as follows. In Section~\\ref{section::well\n  posedness}, we prove the well-posedness of the direct problem using\nstandard arguments. In Section~\\ref{section::CGOs}, we first rewrite\n\\eqref{eqn::Maxwell} as the curl-curl equation and then construct\ncomplex geometric optics solutions for it. We use these solutions to\nprove Theorem~\\ref{main thm} in Section~\\ref{section::proof of main\n  thm}. Next, in Section~\\ref{section::reflection approach} we perform\nthe reflection approach of Isakov \\cite{isakov2007uniqueness} and\nprove an analog of Theorem~\\ref{main thm 2} but in the case when the\npart of the boundary inaccessible for measurements is a subset of the\nplane $\\{x\\in\\R^3 : x_3=0\\}$. Theorem~\\ref{main thm 2} is then proved\nin Section~\\ref{section::proof of thm 2} by analyzing the behavior of\n\\eqref{eqn::Maxwell} under the Kelvin\ntransform. Appendix~\\ref{section::pullbacks} contains properties of\npullbacks used in the main text. Finally, in\nAppendix~\\ref{section::Appendix B} we show that the impedance map\n$Z^\\omega_{\\sigma,\\mu}$ is a pseudodifferential operator of order $1$\nif $\\sigma, \\mu \\in C^\\infty(\\overline\\Omega)$. Using this fact, we\ngain insight in the notion of apparent resistivity used in geophysics\nfrom a mathematical point of view.\n\n\\section{Well-posedness of the direct problem}\n\\label{section::well posedness}\n\nLet $\\Omega\\subset\\R^3$ be a bounded domain with $C^{1,1}$ boundary as\nbefore, and let $\\sigma,\\mu\\in C^1(\\overline\\Omega)$ be such that\n$\\sigma \\ge \\sigma_0$ and $\\mu \\ge \\mu_0$ for some constants\n$\\sigma_0, \\mu_0 > 0$. Consider the following system of equations for\nelectromagnetic fields $E$ and $H$:\n\\begin{equation}\\label{eqn::Maxwell homogenous in appendix}\n\\nabla\\times E=i\\omega\\mu H\\quad\\text{and}\\quad \\nabla\\times H=\\sigma E \\quad\\text{in}\\quad\\Omega,\n\\end{equation}\nwith the tangential boundary condition $\\mathbf{t}(H)=f$, where\n$\\omega$ is a complex number. The main result of the present section is\n\n\\begin{Theorem}\\label{thm::well posedness new version homogeneous}\nLet $\\Omega\\subset\\R^3$ be a bounded domain with $C^{1,1}$ boundary and let $\\sigma,\\mu\\in C^1(\\overline\\Omega)$ be such that $\\sigma \\ge \\sigma_0$ and $\\mu \\ge \\mu_0$ for some constants $\\sigma_0, \\mu_0 > 0$. There is a discrete subset $\\Sigma$ of $\\C$ such that for all $\\omega\\notin \\Sigma$ and for a given $f\\in TH_{\\Div}^{1\/2}(\\p \\Omega)$ the system \\eqref{eqn::Maxwell homogenous in appendix} with $\\mathbf t(H)=f$ has a unique solution $(E,H)\\in H_{\\Div}^1(\\Omega)\\times H_{\\Div}^1(\\Omega)$ satisfying\n$$\n\\|E\\|_{H_{\\Div}^1(\\Omega)}+\\|H\\|_{H_{\\Div}^1(\\Omega)}\\le C\\|f\\|_{TH_{\\Div}^{1\/2}(\\p \\Omega)}\n$$\nfor some constant $C>0$ independent of $f$.\n\\end{Theorem}\n\nFor $\\omega>0$ with $\\omega\\notin \\Sigma$, we define the \\emph{impedance map} $Z_{\\sigma,\\mu}^\\omega$ as\n$$\nZ_{\\sigma,\\mu}^\\omega(f) := \\mathbf t(E),\\quad f\\in TH_{\\Div}^{1\/2}(\\p \\Omega),\n$$\nwhere $(E,H)\\in H_{\\Div}^1(\\Omega)\\times H_{\\Div}^1(\\Omega)$ is the unique solution of the system \\eqref{eqn::Maxwell homogenous in appendix} with $\\mathbf t(H)=f$, guaranteed by Theorem~\\ref{thm::well posedness new version homogeneous}. Moreover, the estimate provided in Theorem~\\ref{thm::well posedness new version homogeneous} implies that the impedance map is a well-defined and bounded operator $Z_{\\sigma,\\mu}^\\omega: TH_{\\Div}^{1\/2}(\\p \\Omega)\\to TH_{\\Div}^{1\/2}(\\p \\Omega)$.\\smallskip\n\nTo prove Theorem~\\ref{thm::well posedness new version homogeneous}, we\nconsider the following non-homogeneous problem. Let $J_e$ and $J_m$ be\nvector fields defined in $\\Omega$ representing current sources. We\nconsider the non-homogenous time-harmonic Maxwell equations,\n\\begin{equation}\\label{eqn::Maxwell in appendix}\n\\nabla\\times E=i\\omega\\mu H + J_m\\quad\\text{and}\\quad \\nabla\\times H=\\sigma E + J_e \\quad\\text{in}\\quad\\Omega.\n\\end{equation}\nWe have\n\n\\begin{Theorem}\\label{thm::well posedness new version}\nLet $\\Omega\\subset\\R^3$ be a bounded domain with $C^{1,1}$ boundary and let $\\sigma,\\mu\\in C^1(\\overline\\Omega)$ be such that $\\sigma \\ge \\sigma_0$ and $\\mu \\ge \\mu_0$ for some constants $\\sigma_0, \\mu_0 > 0$. Suppose that $J_e,J_m\\in L^2(\\Omega;\\C^3)$ such that $\\nabla\\cdot J_e,\\nabla\\cdot J_m\\in L^2(\\Omega;\\C^3)$ and $\\nu\\cdot J_e|_{\\p\\Omega}, \\nu\\cdot J_m|_{\\p\\Omega}\\in H^{1\/2}(\\p\\Omega)$. Then there is a discrete subset $\\Sigma$ of $\\C$ such that for all $\\omega\\notin \\Sigma$ the boundary value problem \\eqref{eqn::Maxwell in appendix} with $\\mathbf t(H) = 0$ has a unique solution $(E,H)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$ satisfying\n$$\n\\|E\\|_{H_{\\Div}^1(\\Omega)}+\\|H\\|_{H_{\\Div}^1(\\Omega)}\\le C\\big(\\|J_e\\|_{L^2(\\Omega;\\C^3)}+\\|J_m\\|_{L^2(\\Omega;\\C^3)} + \\|\\nabla\\cdot J_e\\|_{L^2(\\Omega)}+\\|\\nabla\\cdot J_m\\|_{L^2(\\Omega)} + \\|\\nu\\cdot J_e\\|_{H^{1\/2}(\\p\\Omega)}+\\|\\nu\\cdot J_m\\|_{H^{1\/2}(\\p\\Omega)}\\big)\n$$\nfor some constant $C>0$ independent of $J_e$ and $J_m$\n\\end{Theorem}\n\n\\noindent\nWe first prove Theorem~\\ref{thm::well posedness new version} and then\nshow that it can be used to prove Theorem~\\ref{thm::well posedness new\n  version homogeneous}.\n\n\\subsection{Proof of Theorem~\\ref{thm::well posedness new version}}\n\nWe introduce some notion that will be used for the proof. We work with the following Hilbert space which is the largest domain of $\\nabla\\times$:\n$$\nH(\\curl;\\Omega):=\\{w\\in L^2(\\Omega;\\C^3):\\nabla\\times w\\in L^2(\\Omega;\\C^3)\\}\n$$\nendowed with the norm $\\|w\\|_{H(\\curl;\\Omega)}:=\\|w\\|_{L^2(\\Omega;\\C^3)} + \\|\\nabla\\times w\\|_{L^2(\\Omega;\\C^3)}$. Then the tangential trace operator has its extensions to bounded operators $\\mathbf{t}:H(\\curl; \\Omega)\\to H^{-1\/2}(\\p \\Omega;\\C^3)$. We also work with the space of vector fields in $H(\\curl;\\Omega)$ having zero tangential trace\n$$\nH(\\curl, 0; \\Omega):=\\{w\\in H(\\curl;\\Omega):\\mathbf{t}(w)=0\\}.\n$$\nFor the short proof, we follow the standard variational methods used in \\cite{kirsch2016mathematical, monk2003finite}. Substituting the first equation of \\eqref{eqn::Maxwell in appendix} into the second one, we obtain\n\\begin{equation}\\label{eqn::second order equation}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H) - i\\omega\\mu H = J_m +\\nabla\\times(\\sigma^{-1} J_e)\\quad\\text{in}\\quad\\Omega.\n\\end{equation}\nOur first step is to find a unique solution $H\\in H(\\curl, 0;\\Omega)$ of this equation satisfying\n\\begin{equation}\\label{eqn::estimate for H}\n\\|H\\|_{H(\\curl;\\Omega)}\\le C(\\|J_e\\|_{L^2(\\Omega;\\C^3)}+\\|J_m\\|_{L^2(\\Omega;\\C^3)}).\n\\end{equation}\nBy Helmholtz type decompositions in \\cite[Section~4.1.3]{kirsch2016mathematical} or \\cite[Section~3.7]{monk2003finite}, we can uniquely decompose\n\\begin{align*}\nH &= H_0+\\nabla h,\\quad H_0\\in H(\\curl, 0;\\Omega)_{0,\\mu} := \\{w\\in H(\\curl, 0; \\Omega): \\nabla\\cdot(\\mu w) = 0\\},\\quad h\\in H^1_0(\\Omega;\\C),\\\\\n\\mu^{-1}J_m &= J_{m,0}+\\nabla j_m,\\quad J_{m,0}\\in L^2(\\Omega;\\C^3)_{0,\\mu} := \\{w\\in L^2(\\Omega;\\C^3): \\nabla\\cdot(\\mu w) = 0\\},\\quad j_m\\in H^1_0(\\Omega;\\C).\n\\end{align*}\nWe note here that\n\\begin{equation}\\label{eqn::estimate for j_e in H^1 norm}\n\\|j_m\\|_{H^1(\\Omega;\\C)}\\le C\\|J_m\\|_{L^2(\\Omega;\\C^3)}.\n\\end{equation}\nUsing these decompositions, \\eqref{eqn::second order equation} can be rewritten as\n\\begin{equation}\\label{eqn::second order equation in a weak form -- rewritten}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H_0) - i\\omega\\mu H_0 - i\\omega\\mu \\nabla h = \\mu J_{m,0} + \\mu\\nabla j_m +\\nabla\\times(\\sigma^{-1} J_e)\\quad\\text{in}\\quad\\Omega.\n\\end{equation}\nTo extract $h$ from \\eqref{eqn::second order equation in a weak form -- rewritten}, we simply set $h=-(i\\omega)^{-1}j_m$. Thus, we need to find a unique $H_0\\in H(\\curl, 0;\\Omega)$ with $\\nabla\\cdot(\\mu H_0) = 0$ satisfying\n\\begin{equation}\\label{eqn::second order equation -- rewritten}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H_0) - i\\omega\\mu H_0 = \\mu J_{m,0} + \\nabla\\times(\\sigma^{-1} J_e)\\quad\\text{in}\\quad\\Omega.\n\\end{equation}\nTo solve this equation, we need the following result on existence of a solution operator\n\\begin{Proposition}\\label{prop::resonances}\nThere exist a constant $\\lambda>0$ and a bounded linear map $T_\\lambda : H(\\curl, 0;\\Omega)'\\to H(\\curl, 0;\\Omega)$ such that\n\\begin{equation}\\label{eqn::T_lambda is an inverse of a certain PDoperator}\n\\nabla \\times (\\sigma^{-1} \\nabla \\times T_\\lambda u) + \\lambda \\mu T_\\lambda u = u,\\quad u\\in H(\\curl, 0;\\Omega)'\n\\end{equation}\nand\n$$\nT_\\lambda(\\nabla\\times(\\sigma^{-1}\\nabla\\times e)+\\lambda\\mu e)=e,\\quad e\\in H(\\curl, 0;\\Omega).\n$$\nFurthermore, if $\\nabla\\cdot u = 0$, then $T_\\lambda u\\in H(\\curl, 0;\\Omega)_{0,\\mu}$.\n\\end{Proposition}\n\nHere and in what follows, $\\langle\\cdot,\\cdot\\rangle_\\Omega$ is the duality between $H(\\curl, 0;\\Omega)'$ and $H(\\curl, 0;\\Omega)$ naturally extending the inner product of $L^2(\\Omega;\\C^3)$.\n\n\\begin{proof}\nThe proof is similar to that of \\cite[Proposition~5.1]{assylbekov2016note} using Lax-Milgram's lemma.\n\\end{proof}\n\n\n\n\n\nThen $H_0\\in H(\\curl, 0;\\Omega)$ with $\\nabla\\cdot(\\mu H_0) = 0$ solves \\eqref{eqn::second order equation -- rewritten} if and only if\n\\begin{equation}\\label{eqn::second order equation -- rewritten in terms of T_lambda operators}\nH_0-(i\\omega+\\lambda)\\widetilde T_{\\lambda}H_0=T_\\lambda\\left(\\mu J_{m,0}+\\nabla\\times(\\sigma^{-1}J_e)\\right)\n\\end{equation}\nwhere $\\widetilde T_{\\lambda}=T_\\lambda \\circ m_\\mu\\circ P_\\mu$, $m_\\mu$ is multiplication by $\\mu$, and $P_\\mu$ is the bounded orthogonal projection of $L^2(\\Omega;\\C^3)$ onto $L^2(\\Omega;\\C^3)_{0,\\mu}$ constructed in \\cite[Section~4.1.3]{kirsch2016mathematical}. Since $J_{m,0}\\in L^2(\\Omega;\\C^3)_{0,\\mu}$ and $\\nabla\\cdot\\nabla\\times = 0$, we then have $\\nabla\\cdot\\left(\\mu J_{m,0}+\\nabla\\times(\\sigma^{-1}J_e)\\right) = 0$. Therefore, by the second part of Proposition~\\ref{prop::resonances}, this implies that $T_\\lambda\\left(\\mu J_{m,0}+\\nabla\\times(\\sigma^{-1}J_e)\\right)$ belongs to $H(\\curl, 0;\\Omega)_{0,\\mu}$. The second part of Proposition~\\ref{prop::resonances} implies also that $\\widetilde T_{\\lambda}$ can be considered as a bounded linear operator\n$$\n\\widetilde T_{\\lambda} : L^2(\\Omega;\\C^3)_{0,\\mu}\\overset{m_\\mu}{\\longrightarrow} L^2(\\Omega;\\C^3)_{0,1}\\overset{T_\\lambda}{\\longrightarrow} H(\\curl, 0;\\Omega)_{0,\\mu}\\hookrightarrow L^2(\\Omega;\\C^3)\\overset{P_\\mu}{\\longrightarrow} L^2(\\Omega;\\C^3)_{0,\\mu}.\n$$\nUsing the compactness of the inclusion $H(\\curl, 0;\\Omega)_{0,\\mu}\\hookrightarrow L^2(\\Omega;\\C^3)$ \\cite{weber1980local} and following similar reasoning as at the end of \\cite[Section~5]{assylbekov2016note}, one can show that  for any $\\omega\\notin \\Sigma$, where $\\Sigma:=\\{\\omega\\in \\C\\setminus\\{\\pm i\\lambda\\}:(i\\omega+\\lambda)^{-1}\\in \\Spec(\\widetilde T_{\\lambda})\\}$ which is discrete, \\eqref{eqn::second order equation -- rewritten in terms of T_lambda operators} has a unique solution $H_0\\in H(\\curl, 0;\\Omega)_{0,\\mu}$ satisfying\n$$\n\\|H_0\\|_{H(\\curl;\\Omega)}\\le C(\\|J_e\\|_{L^2(\\Omega;\\C^3)}+\\|J_{m,0}\\|_{L^2(\\Omega;\\C^3)}).\n$$\nNext, setting $H = H_0-(i\\omega)^{-1}j_m$, we obtain a unique $H(\\curl, 0;\\Omega)$ solution for \\eqref{eqn::second order equation} satisfying \\eqref{eqn::estimate for H} thanks to \\eqref{eqn::estimate for j_e in H^1 norm}. Defining $E := \\sigma^{-1}(\\nabla\\times H - J_e)$ we obtain a unique $(E,H)\\in H(\\curl, 0;\\Omega)\\times H(\\curl;\\Omega)$ solving \\eqref{eqn::Maxwell in appendix} and satisfying\n$$\n\\|E\\|_{H(\\curl;\\Omega)} + \\|H\\|_{H(\\curl;\\Omega)}\\le C(\\|J_e\\|_{L^2(\\Omega;\\C^3)}+\\|J_m\\|_{L^2(\\Omega;\\C^3)}).\n$$\nTo prove that $(E,H)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$, apply $\\nabla\\cdot$ to \\eqref{eqn::Maxwell in appendix} and get $\\nabla\\cdot(i\\omega\\mu H) =-\\nabla\\cdot J_m$ and $\\nabla\\cdot(\\sigma E) = -\\nabla\\cdot J_e$. Hence $\\nabla\\cdot E, \\nabla\\cdot H\\in L^2(\\Omega)$, since $\\nabla\\cdot J_e, \\nabla\\cdot J_m\\in L^2(\\Omega)$ by assumption. Then $\\bt(H)=0$ and the results in \\cite{costabel1990remark} imply that $H\\in H^1_{\\Div}(\\Omega)$ and\n$$\n\\|H\\|_{H^1_{\\Div}(\\Omega)} \\le C\\big(\\|H\\|_{H(\\curl;\\Omega)} + \\|J_e\\|_{L^2(\\Omega;\\C^3)} + \\|J_m\\|_{L^2(\\Omega;\\C^3)} + \\|\\nabla\\cdot J_e\\|_{L^2(\\Omega)} + \\|\\nabla\\cdot J_m\\|_{L^2(\\Omega)}\\big).\n$$\nTo show that $E\\in H^1(\\Omega;\\C^3)$, observe that $\\nu\\cdot (\\nabla\\times H)|_{\\p\\Omega} = - \\Div(\\bt(H))=0$ by \\cite[Corollary~A.20]{kirsch2016mathematical}. Then by \\eqref{eqn::Maxwell in appendix}, $\\nu\\cdot E|_{\\p\\Omega} = \\sigma^{-1} \\nu\\cdot (\\nabla\\times H)|_{\\p\\Omega} - \\sigma^{-1} \\nu\\cdot J_e|_{\\p\\Omega} = - \\sigma^{-1} \\nu\\cdot J_e|_{\\p\\Omega} \\in H^{1\/2}(\\p\\Omega)$. According to the results in \\cite{costabel1990remark}, this implies that $E\\in H^1(\\Omega;\\C^3)$ and\n$$\n\\|E\\|_{H^1_{\\Div}(\\Omega)} \\le C\\big(\\|E\\|_{H(\\curl;\\Omega)} + \\|J_e\\|_{L^2(\\Omega;\\C^3)} + \\|\\nabla\\cdot J_e\\|_{L^2(\\Omega)} + \\|\\nu\\cdot J_e\\|_{H^{1\/2}(\\p\\Omega)}\\big).\n$$\nNext, using \\cite[Corollary~A.20]{kirsch2016mathematical} and \\eqref{eqn::Maxwell in appendix}, we can show $\\Div(\\bt(E)) = - \\nu\\cdot (\\nabla\\times E)|_{\\p\\Omega} = - i\\omega\\mu \\nu\\cdot H|_{\\p\\Omega} + \\nu\\cdot J_m|_{\\p\\Omega} \\in H^{1\/2}(\\p\\Omega)$. Thus, $E\\in H^1_{\\Div}(\\Omega)$. Finally, the estimate in the statement of the theorem follows by combining all the above estimates. The proof of Theorem~\\ref{thm::well posedness new version} is thus complete.\n\n\\subsection{Proof of Theorem~\\ref{thm::well posedness new version homogeneous}}\n\nFirst prove the uniqueness of the solution. For a fixed $\\omega\\in\n\\C$, suppose that $(E_j,H_j)\\in H^1_{\\Div}(\\Omega)\\times\nH^1_{\\Div}(\\Omega)$, $j=1,2$, solve \\eqref{eqn::Maxwell homogenous in\n  appendix} and satisfy $\\mathbf{t}(H_1)=\\mathbf{t}(H_2)$. Then\n$(E,H)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$ also solve\n\\eqref{eqn::Maxwell homogenous in appendix} and satisfy\n$\\mathbf{t}(H)=0$, where $E:=E_1-E_2$ and $H:=H_1-H_2$. The uniqueness\npart of Theorem~\\ref{thm::well posedness new version} (with\n$J_e=J_m=0$) gives that $E=0$ and $H=0$.\\smallskip\n\nNext, we prove existence of a solution. For a given $f\\in TH^{1\/2}_{\\Div}(\\p\\Omega)$, there is $H'\\in H^1_{\\Div}(\\Omega)$ such that $\\mathbf{t}(H')=f$ and $\\|H'\\|_{H^1_{\\Div}(\\Omega)}\\le C\\|f\\|_{TH^{1\/2}_{\\Div}(\\p\\Omega)}$. Applying Theorem~\\ref{thm::well posedness new version} with $J_e=-\\nabla\\times H'$ and $J_m=i\\omega\\mu H'$, we obtain a unique $(E_0,H_0)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$ solving\n$$\n\\nabla\\times E_0=i\\omega\\mu H_0+i\\omega\\mu H',\\quad \\nabla\\times H_0=\\sigma E_0 - \\nabla\\times H',\\quad \\bt(H_0)=0\n$$\nand satisfying $\\|E_0\\|_{H_{\\Div}^1(\\Omega)}+\\|H_0\\|_{H_{\\Div}^1(\\Omega)}\\le C \\|f\\|_{TH^{1\/2}_{\\Div}(\\p\\Omega)}$. Here, we used the fact that $\\nu\\cdot (\\nabla\\times H')|_{\\p\\Omega} = - \\Div(\\bt(H'))\\in H^{1\/2}(\\p\\Omega)$ by \\cite[Corollary~A.20]{kirsch2016mathematical}. Then $(E,H)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$ solves \\eqref{eqn::Maxwell homogenous in appendix} with $\\mathbf{t}(E)=f$, where $E:=E_0+E'$ and $H:=H_0$. The proof is complete.\n\n\n\n\\section{Construction of complex geometric optics solutions}\\label{section::CGOs}\n\nThroughout this section, we assume that $\\sigma$ and $\\mu$ can be extended to the whole $\\R^3$ so that $\\sigma\\ge \\sigma_0$, $\\mu\\ge \\mu_0$ and\n\\begin{equation}\\label{sigma and mu are constants outside of a compact set}\n\\sigma - \\sigma_0,\\quad \\mu - \\mu_0 \\in C^2_0(\\R^3)\n\\end{equation}\nfor some constants $\\sigma_0, \\mu_0 > 0$. We also let $R>0$ be large enough (but fixed) so that $B_R(0)$ contains both $\\supp(\\sigma - \\sigma_0)$ and $\\supp(\\mu - \\mu_0)$.\\smallskip\n\nSubstituting the first equation of \\eqref{eqn::Maxwell} into the second one, we obtain the following second-order equation\n\\begin{equation}\\label{eqn::curl-curl equation}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H) - i\\omega \\mu H = 0\\quad\\text{in}\\quad \\Omega\n\\end{equation}\nThe aim of the present section is to construct a complex geometric optics solution in $H\\in H_{\\Div}^1(\\Omega)$ for the above equation. Instead of working in $\\Omega$, we conduct our analysis in the whole $\\R^3$. Therefore, we consider\n\\begin{equation}\\label{eqn::curl-curl equation R^3}\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H) - i\\omega \\mu H = 0\\quad\\text{in}\\quad \\R^3\n\\end{equation}\n\nTaking the divergence of \\eqref{eqn::curl-curl equation R^3}, it straightforwardedly follows that $\\nabla\\cdot(\\mu H) = 0$ in $\\R^3$. Therefore, we obtain\n$$\n\\nabla\\times\\nabla\\times H =  - \\Delta H - \\nabla(\\nabla\\beta \\cdot H)\\quad\\text{in}\\quad \\R^3,\n$$\nwhere $\\beta := \\log\\mu$. Then, we use the latter identity in \\eqref{eqn::curl-curl equation R^3} to show that this equation is equivalent to the system\n\\begin{align}\nL_{\\sigma, \\mu} H := -\\Delta H - \\nabla(\\nabla\\beta \\cdot H) - \\nabla\\alpha \\times \\nabla\\times H - i\\omega \\sigma \\mu H = 0\\quad\\text{in}\\quad \\R^3, \\label{eqn3-1}\\\\\n\\nabla\\cdot(\\mu H) = 0\\quad\\text{in}\\quad \\R^3, \\label{eqn3-2}\n\\end{align}\nwhere $\\alpha := \\log\\sigma$. We note that the derivatives $\\p^\\kappa \\alpha$ and $\\p^\\kappa \\beta$ are uniformly continuous on $\\R^3$ for $|\\kappa|=0, 1, 2$.\\smallskip\n\n\n\nThe complex geometric optics solutions we aim to construct are of the form\n$$\nH(x; \\zeta) = e^{i \\zeta\\cdot x}(a(x; \\zeta) + r(x; \\zeta)),\n$$\nwhere $\\zeta\\in \\C^3\\setminus\\{0\\}$ such that $\\zeta\\cdot\\zeta = i\\omega \\sigma_0 \\mu_0$, $a$ is a specific complex-valued smooth vector field on $\\R^3$ and $r$ is the correction term. Then \\eqref{eqn3-1} is equivalent to\n\\begin{equation} \\label{eq:Lsm}\ne^{-i \\zeta\\cdot x}L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}r) = - f,\\quad f := e^{-i \\zeta\\cdot x} L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}a).\n\\end{equation}\n\n\\subsection{Solution operator}\n\nFor $\\zeta\\in \\C^3\\setminus\\{0\\}$ such that $\\zeta\\cdot\\zeta = i\\omega \\sigma_0 \\mu_0$, we define the operators\n$$\n\\nabla_\\zeta := \\nabla + i\\zeta,\\quad \\Delta_\\zeta := \\Delta + 2i \\zeta\\cdot \\nabla.\n$$\nThen\n\\begin{equation}\\label{eqn:: conj grad and laplacian}\ne^{-i \\zeta\\cdot x}\\circ \\nabla\\circ e^{i \\zeta\\cdot x} = \\nabla_\\zeta\\quad\\text{and}\\quad e^{-i \\zeta\\cdot x}\\circ \\Delta\\circ e^{i \\zeta\\cdot x} = \\Delta_\\zeta - i\\omega \\sigma_0 \\mu_0.\n\\end{equation}\nFor $\\delta\\in\\R$, we define the $L^2$-based weighted space on $\\R^3$\n$$\nL^2_\\delta := \\left\\{f: \\R^3 \\to \\C^3: \\|f\\|_{L^2_\\delta}:=\\Big(\\int_{\\R^3} (1+|x|^2)^{\\delta}|f(x)|^2\\,dx\\Big)^{1\/2} < \\infty\\right\\}\n$$\nand\n$$\nH_\\delta^1 := \\left\\{f \\in L^2_\\delta: \\|f\\|_{H^1_\\delta}:=\\|f\\|_{L^2_\\delta} + \\sum_{j=1}^3\\|\\p_j f\\|_{L^2_\\delta} < \\infty\\right\\}.\n$$\n\n\\begin{Proposition}\\label{prop unique solvability}\nFor $k\\in \\C$, suppose $\\zeta\\in \\C^3$ with $\\zeta \\cdot \\zeta = k$, $-1 < \\delta < 0$. Assume that $\\gamma\\in C^2(\\R^3)$ is positive. Then for $f\\in L^2_{\\delta+1}$ there is a unique  $u\\in L^2_\\delta$ solving\n\\begin{equation}\\label{eqn elliptic}\n(-\\Delta_\\zeta - \\nabla\\log\\gamma \\cdot \\nabla_\\zeta) u = f\\quad\\text{in}\\quad\\R^3\n\\end{equation}\nsuch that\n$$\n\\|u\\|_{L^2_\\delta} \\le \\frac{C}{|\\zeta|}\\|f\\|_{L^2_{\\delta+1}}\n$$\nfor some constant $C>0$. Furthermore, $u$ belongs to $H^1_\\delta$.\n\\end{Proposition}\n\n\\begin{proof}\nIt follows from the identity\n$$\n(-\\Delta_\\zeta - \\nabla\\log\\gamma \\cdot \\nabla_\\zeta) u = \\gamma^{-1\/2}(-\\Delta_\\zeta + q) (\\gamma^{1\/2}u),\\quad q:=\\gamma^{-1\/2}\\Delta\\gamma^{1\/2}\\in C_0(\\R^3),\n$$\nthat solving \\eqref{eqn elliptic} is equivalent to solving\n$$\n(-\\Delta_\\zeta + q) \\tilde u = \\gamma^{1\/2} f\\quad\\text{in}\\quad\\R^3,\n$$\nwhere $\\tilde u := \\gamma^{1\/2} u$. By \\cite[Theorem~1.6]{sylvester1987global} there is a unique $\\tilde u\\in L^2_\\delta$  solving the above equation and satisfying\n$$\n\\|\\tilde u\\|_{L^2_\\delta} \\le \\frac{C}{|\\zeta|}\\|\\gamma^{1\/2}f\\|_{L^2_{\\delta+1}}.\n$$\nNext, \\cite[Lemma~1.15]{sun1992inverse} implies that $\\tilde u \\in H^1_\\delta$.\nThe result now follows immediately by setting $u = \\gamma^{-1\/2} \\tilde u$.\n\\end{proof}\n\nAccording to Proposition~\\ref{prop unique solvability}, for sufficiently large $|\\zeta|$, there is a bounded inverse $G_{\\zeta, \\gamma}: L^2_{\\delta + 1} \\to L^2_\\delta$ of $-\\Delta_\\zeta - \\nabla\\log\\gamma \\cdot \\nabla_\\zeta$ such that\n$$\n\\|G_{\\zeta,\\gamma}\\|_{L^2_{\\delta + 1} ; L^2_\\delta} = \\mathcal O\\Big(\\frac{1}{|\\zeta|}\\Big)\\quad\\text{as}\\quad |\\zeta|\\to\\infty\n$$\nMoreover, $G_{\\zeta, \\gamma}$ maps $L^2_{\\delta + 1}$ into $H^1_\\delta$.\n\n\\subsection{Mollified $\\sigma$ and $\\mu$}\nLet $\\Phi\\in C^\\infty_0(\\R^3)$ with $0 \\le \\Phi \\le 1$ and $\\int_{\\R^3} \\Phi(x)\\,dx=1$. For a fixed $\\epsilon$ with $0 < \\epsilon < 1\/8$, we consider\n$$\n\\Phi_\\tau(x) := \\Big(\\frac{1}{\\tau^{-\\epsilon}}\\Big)^3 \\Phi\\Big(\\frac{x}{\\tau^{-\\epsilon}}\\Big)\\quad\\text{for large } \\tau>0.\n$$\nWe define \n$$\n\\alpha^\\sharp(x;\\tau) := \\alpha * \\Phi_\\tau (x),\\quad \\beta^\\sharp(x;\\tau) := \\beta * \\Phi_\\tau(x)\\quad\\text{for}\\quad x\\in \\R^3.\n$$\nThen $\\alpha^\\sharp(\\cdot;\\tau), \\beta^\\sharp(\\cdot;\\tau) \\in C^\\infty(\\R^3)$. From\n$\\p^\\kappa \\alpha^\\sharp = (\\p^\\kappa \\alpha) * \\Phi_\\tau$ and $\\p^\\kappa \\beta^\\sharp = (\\p^\\kappa \\beta) * \\Phi_\\tau$,\nit follows that\n\\begin{equation}\\label{supports of derivatives of alpha and beta}\n\\supp(\\p^\\kappa \\alpha^\\sharp) ,\\ \\supp(\\p^\\kappa \\beta^\\sharp) \\subset B_{R+2\\tau^{-\\epsilon}}(0),\\quad |\\kappa| = 1, 2.\n\\end{equation}\nWe also have\n\\begin{equation}\\label{L^infty norms of derivatives of alpha and beta errors}\n\\|\\alpha -  \\alpha^\\sharp\\|_{W^{2,\\infty}(\\R^3)} = o(1), \\quad \\| \\beta - \\beta^\\sharp\\|_{W^{2,\\infty}(\\R^3)} = o(1)\\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{equation}\nIndeed,\n$$\n\\p^\\kappa \\alpha(x) - \\p^\\kappa \\alpha^\\sharp(x;\\tau) = \\int_{\\R^3} \\Phi(y) [\\p^\\kappa\\alpha(x) - \\p^\\kappa\\alpha(x - \\tau^{-\\epsilon} y)]\\,dy,\\quad |\\kappa|=0, 1, 2,\n$$\nand uniform continuity of $\\p^\\kappa\\alpha$ on $\\R^3$ gives the desired estimate using \\cite[Theorem~0.13]{folland1995introduction}. A similar argument can be used for $\\beta$.\\smallskip\n\nFinally, using that $\\p^\\kappa \\alpha^\\sharp = \\alpha * (\\p^\\kappa\\Phi_\\tau)$ and $\\p^\\kappa \\beta^\\sharp = \\beta * (\\p^\\kappa\\Phi_\\tau)$, a direct calculation shows that\n$$\n\\|\\p^\\kappa\\alpha^\\sharp\\|_{L^{\\infty}(\\R^3)},\\ \\|\\p^\\kappa\\beta^\\sharp\\|_{L^{\\infty}(\\R^3)} = \\mathcal O_\\kappa(\\tau^{|\\kappa|\\epsilon})\\quad\\text{for}\\quad |\\kappa|\\ge 0\\quad\\text{as}\\quad \\tau\\to \\infty,\n$$\nwhich implies that\n\\begin{equation}\\label{eqn::L^infty norms of derivatives of alpha & beta}\n\\|\\alpha^\\sharp\\|_{W^{k,\\infty}(\\R^3)},\\ \\|\\beta^\\sharp\\|_{W^{k,\\infty}(\\R^3)} = \\mathcal O_k(\\tau^{k\\epsilon})\\quad\\text{for}\\quad k = 0,1,2,\\ldots\\quad\\text{as}\\quad \\tau\\to \\infty.\n\\end{equation}\nWe note that stronger estimates follow from \\eqref{L^infty norms of derivatives of alpha and beta errors}\n\\begin{equation}\\label{eqn::W^2,infty norms of alpha & beta sharps}\n\\|\\alpha^\\sharp\\|_{W^{2,\\infty}(\\R^3)},\\ \\|\\beta^\\sharp\\|_{W^{2,\\infty}(\\R^3)} = \\mathcal O(1)\\quad\\text{as}\\quad \\tau\\to \\infty.\n\\end{equation}\n\n\\subsection{Transport equation}\n\nSince $\\zeta \\cdot \\zeta = i\\omega \\sigma_0 \\mu_0$, using \\eqref{eqn:: conj grad and laplacian}, we obtain (cf.~(\\ref{eq:Lsm}))\n\\begin{align*}\nf &= -\\Delta_\\zeta a - \\nabla_\\zeta(\\nabla\\beta\\cdot a) - \\nabla\\alpha\\times \\nabla_\\zeta\\times a  - i\\omega\\sigma\\mu a + i\\omega \\sigma_0 \\mu_0 a \\\\\n& = -\\Delta a - \\nabla(\\nabla\\beta\\cdot a) - \\nabla\\alpha\\times \\nabla\\times a  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) a\n- 2i \\zeta\\cdot \\nabla a - (\\nabla\\beta\\cdot a) i\\zeta - \\nabla\\alpha \\times (i\\zeta \\times a).\n\\end{align*}\nWe shall consider $\\zeta$ of the form $\\zeta = \\tau \\rho + \\zeta_1$ where $\\tau > 0$ is a large parameter, $\\rho \\in \\C^3$  is independent of $\\tau$ and satisfies $\\Re\\rho \\cdot \\Im\\rho = 0$ and $|\\Re\\rho| = |\\Im\\rho| = 1$, and $\\zeta_1 = \\mathcal O(1)$ as $\\tau \\to \\infty$. Then\n\\begin{align*}\nf &= -\\Delta a - \\nabla(\\nabla\\beta\\cdot a) - \\nabla\\alpha\\times \\nabla\\times a  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) a\n- 2i \\zeta_1\\cdot \\nabla a - (\\nabla\\beta\\cdot a) i\\zeta_1 - \\nabla\\alpha \\times (i\\zeta_1 \\times a)\\\\\n& \\quad - i\\tau ((\\nabla(\\beta-\\beta^\\sharp)\\cdot a) \\rho + \\nabla(\\alpha-\\alpha^\\sharp) \\times (\\rho \\times a))\n - i\\tau (2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)).\n\\end{align*}\nIn order to get $\\|f \\|_{L^2_{\\delta+1}} = o(\\tau)$ as $\\tau \\to\n\\infty$, for $-1 < \\delta < 0$, we should construct $a$ satisfying the\ntransport equation, that is,\n\\begin{equation}\\label{transport eqn}\n2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a) = o(1)\\quad\\text{in}\\quad L^2_{\\delta+1}\\quad\\text{as}\\quad\\tau\\to\\infty.\n\\end{equation}\nWe\nget\n\\begin{equation}\\label{transport eqn rewritten}\n2\\rho \\cdot \\nabla a + (\\nabla\\beta^\\sharp \\cdot a) \\rho + (\\nabla\\alpha^\\sharp \\cdot a) \\rho - (\\rho \\cdot \\nabla\\alpha^\\sharp) a = o(1)\\quad\\text{in}\\quad L^2_{\\delta+1}\\quad\\text{as}\\quad\\tau\\to\\infty.\n\\end{equation}\nFor arbitrary $s_0\\in \\R$, we seek solutions of \\eqref{transport eqn rewritten} in the form\n$$\na = e^{-\\alpha^\\sharp \/ 2} \\rho + s_0 e^{\\alpha^\\sharp \/ 2}e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho},\n$$\nwhere $\\chi_\\tau(x) := \\chi(\\tau^{-\\theta}x)$, $0< \\theta < 1\/2$ with $\\chi\\in C^\\infty_0(\\R^3)$ such that $\\chi(x) \\equiv 1$ for $|x| < 1\/2$ and $\\chi(x) \\equiv 0$ for $|x| \\ge 1$. Then\n\\begin{equation}\\label{nabla a}\n\\p_j a = - \\frac{1}{2} \\p_j \\alpha^\\sharp e^{-\\alpha^\\sharp \/ 2} \\rho + \\frac{s_0}{2}\\p_j \\alpha^\\sharp e^{\\alpha^\\sharp \/ 2}e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho}\n + \\frac{s_0}{\\tau^\\theta} (\\p_j\\chi)(\\theta^{-\\theta} x)\\Psi^\\sharp e^{\\alpha^\\sharp \/ 2}e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho} + s_0 \\chi_\\tau \\p_j \\Psi^\\sharp e^{\\alpha^\\sharp \/ 2}e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho}.\n\\end{equation}\nSubstituting these into \\eqref{transport eqn rewritten}, we come to\n$$\ns_0 \\Big\\{ \\tau^{-\\theta}\\Psi^\\sharp 2\\rho\\cdot(\\nabla \\chi)(\\tau^{-\\theta}\\,\\cdot\\,) + \\chi_\\tau 2\\rho \\cdot \\nabla \\Psi^\\sharp + \\overline{\\rho} \\cdot \\nabla(\\alpha^\\sharp + \\beta^\\sharp) \\Big\\} e^{\\alpha^\\sharp \/ 2} e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho} = o(1)\\quad\\text{in}\\quad L^2_{\\delta+1}\\quad\n\\text{as $\\tau\\to\\infty$}. \n$$\nWe observe that by \\eqref{supports of derivatives of alpha and beta}, $\\nabla(\\alpha^\\sharp + \\beta^\\sharp) = \\chi_\\tau \\nabla(\\alpha^\\sharp + \\beta^\\sharp)$ for large enough $\\tau>0$. Therefore, it is enough to find $\\Psi^\\sharp$ in $C^\\infty(\\R^3)$ with a nice control on $\\|\\Psi^\\sharp\\|_{L^\\infty(\\R^3)}$ and satisfying\n\\begin{equation}\\label{eqns for Phi and Psi}\n2\\rho \\cdot \\nabla \\Psi^\\sharp + \\overline{\\rho} \\cdot \\nabla(\\alpha^\\sharp + \\beta^\\sharp) = 0 \\quad\\text{in}\\quad \\R^3.\n\\end{equation}\nSince $\\rho\\cdot \\rho = 0$ and $\\Re\\rho \\cdot \\Im\\rho = 0$, the operator $N_\\rho := \\rho \\cdot \\nabla$ is just the $\\overline\\p$-operator in certain linear coordinates. Its inverse is defined as\n$$\nN_\\rho^{-1} f(x) := \\frac{1}{2\\pi} \\int_{\\R^2} \\frac{f(x - y_1 \\Re\\rho - y_2 \\Im\\rho)}{y_1 + i y_2}\\,dy,\\quad f\\in C_0(\\R^3).\n$$\nThen by \\cite[Lemma~4.6]{salo2004inverse},\n$$\n\\Psi^\\sharp(x,\\rho;\\tau) := - \\frac{1}{2} N_\\rho^{-1} \\{\\overline{\\rho} \\cdot \\nabla(\\alpha^\\sharp + \\beta^\\sharp)\\} \\in C^{\\infty}(\\R^3)\n$$\nsatisfies equation \\eqref{eqns for Phi and Psi}. It also follows from \\cite[Lemma~4.6]{salo2004inverse} and \\eqref{L^infty norms of derivatives of alpha and beta errors} that\n\\begin{equation}\\label{estimate W^1,infty of Psi^sharp}\n\\|\\Psi^\\sharp\\|_{W^{1,\\infty}(\\R^3)} \\le \\mathcal O(1) \\big\\{\\|\\alpha^\\sharp\\|_{W^{2,\\infty}(\\R^3)} + \\|\\beta^\\sharp\\|_{W^{2,\\infty}(\\R^3)} \\big\\} = \\mathcal O(1)\\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{equation}\nFurthermore, \\cite[Lemma~4.6]{salo2004inverse} and \\eqref{eqn::L^infty norms of derivatives of alpha & beta} imply that\n\\begin{equation}\\label{estimate W^k,infty of Psi^sharp}\n\\begin{aligned}\n\\|\\Psi^\\sharp\\|_{W^{k,\\infty}(\\R^3)} &\\le \\mathcal O_k(1) \\big\\{\\|\\alpha^\\sharp\\|_{W^{k+1,\\infty}(\\R^3)} + \\|\\beta^\\sharp\\|_{W^{k+1,\\infty}(\\R^3)} \\big\\}\\\\\n&= \\mathcal O_k(\\tau^{k\\epsilon+\\epsilon})\\quad\\text{for}\\quad k = 0,1,\\ldots \\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{aligned}\n\\end{equation}\nWe set\n$$\n\\Psi(x,\\rho) := -\\frac{1}{2} N_\\rho^{-1} \\{\\overline{\\rho} \\cdot \\nabla(\\alpha + \\beta)\\} \\in L^\\infty(\\R^3).\n$$\nThen \\cite[Lemma~3.1]{sylvester1987global} together with \\eqref{supports of derivatives of alpha and beta} and \\eqref{L^infty norms of derivatives of alpha and beta errors} implies that\n$$\n\\|\\Psi^\\sharp(\\cdot,\\rho;\\tau) - \\Psi(\\cdot,\\rho)\\|_{L^2_{\\rm loc}(\\R^3)} = o(1)\\quad\\text{as}\\quad \\tau\\to \\infty.\n$$\nFinally, by \\cite[Lemma~3.1]{salo2006semiclassical} together with \\eqref{eqn::L^infty norms of derivatives of alpha & beta} and \\eqref{eqn::W^2,infty norms of alpha & beta sharps},\n\\begin{equation}\\label{estimate pointwise derivatives of Psi^sharp}\n\\begin{aligned}\n|\\p^\\kappa \\Psi^\\sharp(x,\\rho;\\tau)| &\\le \\mathcal O_\\kappa(1) \\begin{cases}\n(1 + |x_T|^2)^{-1\/2}\\chi_{B(0,R)}(x_\\perp),&\\text{if } |\\kappa| = 0, 1,\\\\\n\\tau^{|\\kappa|\\epsilon + \\epsilon}(1 + |x_T|^2)^{-1\/2}\\chi_{B(0,R)}(x_\\perp),&\\text{otherwise}\n\\end{cases}\\\\\n&\\quad\\,\\,\\text{as}\\quad \\tau\\to\\infty,\n\\end{aligned}\n\\end{equation}\nwhere $x_T$ is the projection of $x$ onto $\\Span\\{\\rho_1,\\rho_2\\}$ and $x_\\perp = x - x_T$, and $R>0$ is such that $B(0,R)$ contains $\\supp(\\sigma - \\sigma_0)$ and $\\supp(\\mu - \\mu_0)$.\\smallskip\n\nIt follows that\n\\begin{equation}\\label{integral Japanese symbol estimate}\n\\int_{|x_T| \\le \\tau^\\theta,\\, |x_\\perp| \\le R} (1 + |x_T|^2)^{-1} (1 + |x|^2)^{\\delta+1}\\, dx\n\\le \\mathcal O(1) \\int_{|x_T| \\le \\tau^\\theta} (1 + |x_T|^2)^{\\delta}\\, dx_T = \\mathcal O(\\tau^{2(\\delta+1)\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{equation}\nSimilarly,\n\\begin{equation}\\label{integral Japanese symbol estimate -j}\n\\int_{|x_T| \\le \\tau^\\theta,\\, |x_\\perp| \\le R} (1 + |x_T|^2)^{-j} (1 + |x|^2)^{\\delta+1}\\, dx\n\\le \\mathcal O(1) \\Big(1-\\frac{1}{(1+\\tau^{2\\theta})^{j-2-\\delta}}\\Big) = \\mathcal O(1)\\quad\\text{as}\\quad\\tau \\to \\infty\\quad\\text{for}\\quad j\\ge 2.\n\\end{equation}\nWe have\n\n\\begin{Lemma}\nLet $\\chi_\\tau$ and $\\Psi^\\sharp$ be as above. Then\n\\begin{align}\n\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= \\mathcal O(\\tau^{(\\delta+1)\\theta}),\\label{lemma tech 1}\\\\\n\\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= \\mathcal O(1),\\label{lemma tech 2}\\\\\n\\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= \\mathcal O(1),\\label{lemma tech 3}\\\\\n\\|\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= o(\\tau),\\label{lemma tech 4}\\\\\n\\|\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= \\mathcal O(\\tau^{3\\epsilon}),\\\\\n\\|\\p_l\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &= o(\\tau^{1+\\epsilon})\\label{lemma tech 5}\n\\end{align}\nas $\\tau \\to \\infty$.\n\\end{Lemma}\n\n\\begin{proof}\nUsing \\eqref{estimate pointwise derivatives of Psi^sharp} and \\eqref{integral Japanese symbol estimate}, we obtain\n\\begin{align*}\n\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} &\\le \\frac{1}{\\tau^\\theta}\\|\\p_j\\chi(\\tau^{-\\theta}\\,\\cdot\\,) \\Psi^\\sharp\\|_{L^2_{\\delta+1}} + \\|\\chi_\\tau \\p_j\\Psi^\\sharp\\|_{L^2_{\\delta+1}} \\\\\n&\\le \\mathcal O\\Big(\\frac{1}{\\tau^\\theta} + 1\\Big) \\int_{|x_T| \\le \\tau^\\theta,\\, |x_\\perp| \\le R} (1 + |x_T|^2)^{-1} (1 + |x|^2)^{\\delta+1}\\, dx\\\\\n& = \\mathcal O(\\tau^{(\\delta+1)\\theta}) \\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{align*}\nThe other estimates follow readily.\n\\end{proof}\n\n\\subsection{Estimating $\\|f\\|_{L^2_{\\delta+1}}$}\\label{section on L^2 norm of f}\n\nWith our choice of $a$, we have\n\\begin{equation}\\label{transport equation part}\n2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a) = s_0 \\tau^{-\\theta} 2\\rho\\cdot(\\nabla \\chi)(\\tau^{-\\theta}\\,\\cdot\\,) \\Psi^\\sharp e^{\\alpha^\\sharp \/ 2} e^{\\chi_\\tau \\Psi^\\sharp} \\overline{\\rho}.\n\\end{equation}\nThen, as in the proof of \\eqref{lemma tech 1}, we use \\eqref{eqn::W^2,infty norms of alpha & beta sharps}, \\eqref{estimate W^1,infty of Psi^sharp}, \\eqref{estimate pointwise derivatives of Psi^sharp} and \\eqref{integral Japanese symbol estimate} to obtain\n\\begin{multline*}\n\\| i\\tau (2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)) \\|_{L^2_{\\delta+1}} \\\\\n\\le \\mathcal O(\\tau^{1-\\theta}) \\Big(\\int_{|x_T| \\le \\tau^\\theta,\\, |x_\\perp| \\le R} (1 + |x_T|^2)^{-1} (1 + |x|^2)^{\\delta+1}\\, dx \\Big)^{1\/2} = \\mathcal O(\\tau^{1+\\delta\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{multline*}\nUsing \\eqref{sigma and mu are constants outside of a compact set},\n\\eqref{supports of derivatives of alpha and beta},\n\\eqref{L^infty norms of derivatives of alpha and beta errors},\n\\eqref{eqn::W^2,infty norms of alpha & beta sharps} and\n\\eqref{estimate W^1,infty of Psi^sharp}, it is then straightforward to show\nthat\n\\begin{align*}\n\\| i\\tau ((\\nabla(\\beta-\\beta^\\sharp)\\cdot a) \\rho + \\nabla(\\alpha-\\alpha^\\sharp) \\times (\\rho \\times a)) \\|_{L^2_{\\delta+1}} &= o(1)\\ \\quad\\text{as}\\quad\\tau \\to \\infty,\\\\[0.05cm]\n\\| (\\nabla\\beta\\cdot a) i\\zeta_1 + \\nabla\\alpha \\times (i\\zeta_1 \\times a) \\|_{L^2_{\\delta+1}} &= \\mathcal O(1)\\quad\\text{as}\\quad\\tau \\to \\infty,\\\\\n\\|i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) a\\|_{L^2_{\\delta+1}} &= \\mathcal O(1)\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{align*}\nNow, using expressions \\eqref{eqn::W^2,infty norms of alpha & beta sharps} and \\eqref{estimate W^1,infty of Psi^sharp}, we find that\n$$\n\\|\\p_j a \\|_{L^2_{\\delta+1}} \\le \\mathcal O(1)\\|\\p_j\\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n$$\nThus with \\eqref{supports of derivatives of alpha and beta}, \\eqref{eqn::W^2,infty norms of alpha & beta sharps} and \\eqref{lemma tech 1}, we obtain\n\\begin{equation}\\label{L2 norm of der a}\n   \\|\\p_j a \\|_{L^2_{\\delta+1}}\n   = o(\\tau^{\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty,\n\\end{equation}\nand therefore,\n$$\n\\| 2i \\zeta_1\\cdot \\nabla a \\|_{L^2_{\\delta+1}} = o(\\tau^{\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty,\n$$\n$$\n\\|\\nabla\\alpha\\times \\nabla\\times a\\|_{L^2_{\\delta+1}} \\le \\mathcal O(1) \\|\\nabla\\times a\\|_{L^2_{\\delta+1}} \\le \\mathcal O(1)\\sum_{j=1}^3\\|\\p_j a \\|_{L^2_{\\delta+1}} = o(\\tau^{\\theta})\\quad\\text{as}\\quad\\tau \\to \\infty\n$$\nand\n$$\n\\|\\nabla(\\nabla\\beta\\cdot a)\\|_{L^2_{\\delta+1}} \\le \\|(\\nabla\\nabla\\beta) a\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\sum_{j=1}^3 \\|\\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^\\theta)\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nIn the last step, we used \\eqref{sigma and mu are constants outside of a compact set} and that $\\supp(\\nabla\\nabla\\beta)$ is compact.\nFinally, by \\eqref{eqn::W^2,infty norms of alpha & beta sharps} and\n\\eqref{estimate W^1,infty of Psi^sharp}\n\\begin{multline*}\n\\|\\p_j\\p_k a\\|_{L^2_{\\delta+1}} \\le \\mathcal O(1) \\|\\p_j\\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_j\\alpha^\\sharp\\|_{L^2_{\\delta+1}}\\\\\n+ \\mathcal O(1)\\|\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\\\\\n+ \\mathcal O(1)\\|\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\|\\p_j(\\chi_\\tau \\Psi^\\sharp)\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n\\end{multline*}\nwhence, by \\eqref{supports of derivatives of alpha and beta}, \\eqref{eqn::W^2,infty norms of alpha & beta sharps}, \\eqref{lemma tech 1}, \\eqref{lemma tech 2} and \\eqref{lemma tech 4},\n\\begin{equation}\\label{L2 norm of second der of a}\n   \\|\\p_j\\p_k a\\|_{L^2_{\\delta+1}}\n   = o(\\tau)\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{equation}\nThus,\n$$\n\\|\\Delta a\\|_{L^2_{\\delta+1}} \\le \\sum_{j=1}^3 \\|\\p_j^2 a\\|_{L^2_{\\delta+1}} = o(\\tau)\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nCombining all of the above estimates, we come to\n$$\n\\|f\\|_{L^2_{\\delta+1}} = o(\\tau)\\quad\\text{as}\\quad \\tau\\to\\infty.\n$$\n\n\\subsection{Estimating $\\|\\nabla_\\zeta \\times f\\|_{L^2_{\\delta+1}}$}\n\\label{section on L^2 norm of weighted curl of f}\n\nSince\n$$\n\\nabla_\\zeta\\times f = \\nabla\\times f + i\\zeta\\times f\\quad\\text{and}\\quad \\|i\\zeta\\times f\\|_{L^2_{\\delta+1}} = o(\\tau^2),\n$$\nwe need to estimate $\\|\\nabla\\times f\\|_{L^2_{\\delta+1}}$. By straightforward calculations,\n\\begin{align*}\n  \\nabla\\times f &= -\\nabla\\times \\Delta a - \\nabla\\times(\\nabla\\alpha\\times \\nabla\\times a)  - i\\omega \\nabla\\times\\big((\\sigma\\mu - \\sigma_0 \\mu_0) a \\big)\n  - 2i \\nabla\\times(\\zeta_1\\cdot \\nabla a) - \\nabla (\\nabla\\beta\\cdot a)\\times i\\zeta_1 - \\nabla\\times\\big(\\nabla\\alpha \\times (i\\zeta_1 \\times a)\\big)\\\\\n& \\quad - i\\tau \\nabla(\\nabla(\\beta-\\beta^\\sharp)\\cdot a)\\times \\rho - i\\tau\\nabla\\times\\big(\\nabla(\\alpha-\\alpha^\\sharp) \\times (\\rho \\times a)\\big)\n - i\\tau \\nabla\\times \\big(2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)\\big).\n\\end{align*}\nBy \\eqref{transport equation part},\n$$\n\\nabla\\times \\big(2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)\\big) = \\frac{s_0}{\\tau^{\\theta}} \\nabla\\Big(2\\rho\\cdot(\\nabla \\chi)(\\tau^{-\\theta}\\,\\cdot\\,) \\Psi^\\sharp e^{\\alpha^\\sharp \/ 2} e^{\\chi_\\tau \\Psi^\\sharp}\\Big) \\times \\overline{\\rho}.\n$$\nThen, as in the proof of \\eqref{lemma tech 4}, we use\n\\eqref{eqn::W^2,infty norms of alpha & beta sharps} and\n\\eqref{estimate W^1,infty of Psi^sharp},\n\\begin{align*}\n\\|i\\tau \\nabla &\\times \\big(2\\rho\\cdot \\nabla a + (\\nabla\\beta^\\sharp\\cdot a) \\rho + \\nabla\\alpha^\\sharp \\times (\\rho \\times a)\\big)\\|_{L^2_{\\delta+1}}\n\\le \\mathcal O(\\tau^{1-\\theta})\\sum_{j,k=1}^3\\|\\p_j(\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\Psi^\\sharp e^{\\alpha^\\sharp \/ 2} e^{\\chi_\\tau \\Psi^\\sharp})\\|_{L^2_{\\delta+1}}\\\\\n&\\le \\mathcal O(\\tau^{1-2\\theta})\\sum_{j,k=1}^3\\|\\p_j\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\Psi^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(\\tau^{1-\\theta})\\sum_{j,k=1}^3\\|\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\p_j\\Psi^\\sharp\\|_{L^2_{\\delta+1}}\n+ \\mathcal O(\\tau^{1-\\theta})\\sum_{j,k=1}^3\\|\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\p_j\\alpha^\\sharp \\Psi^\\sharp\\|_{L^2_{\\delta+1}} \\\\\n&\\quad+ \\mathcal O(\\tau^{1-2\\theta})\\sum_{j,k=1}^3\\|\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\p_j\\chi(\\tau^{-\\theta}\\,\\cdot\\,) \\Psi^\\sharp{}^2\\|_{L^2_{\\delta+1}}\n+ \\mathcal O(\\tau^{1-\\theta})\\sum_{j,k=1}^3\\|\\p_k\\chi(\\tau^{-\\theta}\\,\\cdot\\,)\\p_j\\Psi^\\sharp \\chi_\\tau \\Psi^\\sharp\\|_{L^2_{\\delta+1}}\n= o(\\tau)\n   \\quad\\text{as}\\quad \\tau\\to\\infty,\n\\end{align*}\nwhere in the last step, we also used \\eqref{supports of derivatives of\n  alpha and beta}, \\eqref{estimate pointwise derivatives of\n  Psi^sharp}, \\eqref{integral Japanese symbol estimate} and\n\\eqref{integral Japanese symbol estimate -j}. Next, using \\eqref{sigma\n  and mu are constants outside of a compact set}, \\eqref{supports of\n  derivatives of alpha and beta}, \\eqref{L^infty norms of derivatives\n  of alpha and beta errors}, \\eqref{eqn::W^2,infty norms of alpha &\n  beta sharps}, \\eqref{estimate W^1,infty of Psi^sharp} and \\eqref{L2\n  norm of der a}, we obtain\n\\begin{equation*}\n\\| i\\tau \\nabla(\\nabla(\\beta-\\beta^\\sharp)\\cdot a)\\times \\rho\\|_{L^2_{\\delta+1}}\\le \\mathcal O(\\tau)\\sum_{j,k} \\|\\p_j\\p_k (\\beta - \\beta^\\sharp)a\\|_{L^2_{\\delta+1}} + \\mathcal O(\\tau)\\sum_{j,k}\\|\\p_k (\\beta - \\beta^\\sharp) \\p_j a\\|_{L^2_{\\delta+1}}\n= o(\\tau^{1+\\theta})\\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{equation*}\nSimilarly,\n\\begin{equation*}\n\\| i\\tau \\nabla\\times\\big(\\nabla(\\alpha-\\alpha^\\sharp) \\times (\\rho \\times a)\\big) \\|_{L^2_{\\delta+1}}\\le \\mathcal O(\\tau)\\sum_{j,k} \\|\\p_j\\p_k (\\alpha-\\alpha^\\sharp)a\\|_{L^2_{\\delta+1}} + \\mathcal O(\\tau)\\sum_{j,k}\\|\\p_k (\\alpha-\\alpha^\\sharp) \\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^{1+\\theta}),\n\\end{equation*}\n\\begin{equation*}\n\\|\\nabla (\\nabla\\beta\\cdot a) \\times i\\zeta_1 \\|_{L^2_{\\delta+1}}\\le \\mathcal O(1)\\sum_{j,k} \\|\\p_j\\p_k \\beta\\, a\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\sum_{j,k}\\|\\p_k \\beta\\, \\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^{\\theta}),\n\\end{equation*}\n\\begin{equation*}\n\\|\\nabla\\times\\big(\\nabla\\alpha \\times (i\\zeta_1 \\times a)\\big) \\|_{L^2_{\\delta+1}}\\le \\mathcal O(1)\\sum_{j,k} \\|\\p_j\\p_k \\alpha\\, a\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\sum_{j,k}\\|\\p_k \\alpha\\, \\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^{\\theta}),\n\\end{equation*}\nand\n\\begin{equation*}\n\\|i\\omega\\nabla\\times\\big((\\sigma\\mu - \\sigma_0 \\mu_0) a\\big)\\|_{L^2_{\\delta+1}}\\le \\mathcal O(1)\\sum_{j} \\|\\p_j(\\sigma\\mu) a\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\sum_{j,k}\\|(\\sigma\\mu - \\sigma_0\\mu_0) \\p_j a\\|_{L^2_{\\delta+1}} = o(\\tau^\\theta)\n\\end{equation*}\nas $\\tau \\to \\infty$. Using \\eqref{L2 norm of second der of a} we find that\n$$\n\\|2i\\nabla\\times(\\zeta_1\\cdot \\nabla a)\\|_{L^2_{\\delta+1}} \\le \\mathcal O(1) \\sum_{j,k} \\|\\p_j\\p_k a\\|_{L^2_{\\delta+1}} = o(\\tau)\n$$\nand\n$$\n\\|\\nabla\\times(\\nabla\\alpha\\times \\nabla\\times a)\\|_{L^2_{\\delta+1}}\\le \\mathcal O(1)\\sum_{j,k,l} \\|\\p_j\\p_k \\alpha\\, \\p_l a\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\sum_{j,k,l}\\|\\p_k \\alpha\\, \\p_j\\p_l a\\|_{L^2_{\\delta+1}} = o(\\tau)\n$$\nas $\\tau \\to \\infty$.\n\nUsing \\eqref{eqn::W^2,infty norms of alpha & beta sharps} and \\eqref{estimate W^1,infty of Psi^sharp}, we estimate\n\\begin{align*}\n\\|\\p_l\\p_j\\p_k a\\|_{L^2_{\\delta+1}} &\\le \\mathcal O(1)\\|\\p_l\\p_j\\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1)\\|\\p_l\\p_j \\alpha^\\sharp \\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n+ \\mathcal O(1)\\|\\p_j \\alpha^\\sharp \\p_l\\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1)\\|\\p_j\\p_k \\alpha^\\sharp \\p_l \\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1)\\|\\p_l \\alpha^\\sharp \\p_j \\alpha^\\sharp \\p_k \\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_l\\p_j\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_l\\p_j\\alpha^\\sharp \\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_l\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k\\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_l\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_l\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_l\\p_k\\alpha^\\sharp\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_l\\p_k(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_j\\p_k\\alpha^\\sharp \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}} + \\mathcal O(1) \\|\\p_j \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l\\alpha^\\sharp\\|_{L^2_{\\delta+1}}+ \\mathcal O(1) \\|\\p_j \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_k\\alpha^\\sharp \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l\\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_j\\alpha^\\sharp \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n \\\\ &\\\n + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k\\alpha^\\sharp \\p_l\\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k\\alpha^\\sharp \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\n + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l\\alpha^\\sharp\\|_{L^2_{\\delta+1}}\n \\\\ &\\hspace*{4.55cm}\n + \\mathcal O(1) \\|\\p_j(\\chi_\\tau \\Psi^\\sharp) \\p_k(\\chi_\\tau \\Psi^\\sharp) \\p_l(\\chi_\\tau \\Psi^\\sharp)\\|_{L^2_{\\delta+1}}\\quad\\text{as}\\quad\\tau \\to \\infty\n\\end{align*}\nand conclude that\n\\[\n   \\|\\p_l\\p_j\\p_k a\\|_{L^2_{\\delta+1}}\n       = o(\\tau^{1+\\epsilon})\\quad\\text{as}\\quad\\tau \\to \\infty.\\ \\\n\\]\nThis implies that\n$$\n\\|\\nabla\\times \\Delta a\\|_{L^2_{\\delta+1}} = o(\\tau^{1+\\epsilon})\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nCombining all of these, we finally come to $\\|\\nabla\\times\nf\\|_{L^2_{\\delta+1}} = o(\\tau^{1+\\epsilon})$ and, hence,\n$\\|\\nabla_\\zeta\\times f\\|_{L^2_{\\delta+1}} =\no(\\tau^2)\\quad\\text{as}\\quad\\tau \\to \\infty$.\n\n\\subsection{Construction of complex geometric optics solutions} Now, we are ready to construct complex geometric optics solutions for the system \\eqref{eqn3-1} and \\eqref{eqn3-2} which is equivalent to\n\\begin{align}\ne^{-i \\zeta\\cdot x}L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}r) & = - f, \\label{eqn LE=0 rewritten}\\\\\n\\nabla_\\zeta \\cdot r + \\nabla\\log\\mu \\cdot r & = - \\nabla_\\zeta \\cdot a + \\nabla\\log\\mu \\cdot a. \\label{eqn div(sigma E)=0 rewritten}\n\\end{align}\nAccording to the discussion in Section~\\ref{section on L^2 norm of f}, we have $\\|f\\|_{L^2_{\\delta+1}} = o(\\tau)$ as $\\tau \\to \\infty$.\\smallskip\n\nFirst, we need to show that for sufficiently large $|\\zeta|$ there is $r\\in L^2_\\delta$ solving \\eqref{eqn LE=0 rewritten}. Using \\eqref{eqn:: conj grad and laplacian},\n\\begin{equation}\\label{another pde for r}\ne^{-i \\zeta\\cdot x} L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}r) = -\\Delta_\\zeta r - \\nabla_\\zeta(\\nabla\\log\\mu\\cdot r) - \\nabla\\log\\sigma\\times \\nabla_\\zeta\\times r  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) r.\n\\end{equation}\nWe have\n$$\n\\nabla_\\zeta(\\nabla\\log\\mu\\cdot r) = \\nabla\\log\\mu \\times\n\\nabla_\\zeta\\times r + \\nabla\\log\\mu \\cdot \\nabla_\\zeta r +\n\\nabla\\nabla(\\log\\mu)r\n$$\nTherefore, \\eqref{eqn LE=0 rewritten} can be written as\n\\begin{equation}\\label{pre final pde for r}\n-\\Delta_\\zeta r - \\nabla\\log\\mu \\cdot \\nabla_\\zeta r - \\nabla\\log(\\sigma\\mu) \\times \\nabla_\\zeta\\times r - V_1 r = - f,\n\\end{equation}\nwhere\n$$\nV_1 := \\nabla\\nabla(\\log\\mu) + i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0).\n$$\nWe need to deal with the third term on the left hand-side of\n\\eqref{pre final pde for r}. We define $Q := \\nabla_\\zeta\\times r$, and\nfind that\n$$\n\\Delta_\\zeta r = \\nabla_\\zeta \\nabla_\\zeta \\cdot r - \\nabla_\\zeta \\times \\nabla_\\zeta \\times r.\n$$\nAlso, it follows from \\eqref{eqn div(sigma E)=0 rewritten} that\n$$\n\\nabla_\\zeta \\nabla_\\zeta \\cdot r = - \\nabla_\\zeta \\nabla_\\zeta \\cdot a - \\nabla_\\zeta(\\nabla\\log\\mu \\cdot a) - \\nabla_\\zeta(\\nabla\\log\\mu \\cdot r).\n$$\nSubstituting these into \\eqref{another pde for r}, we come to\n$$\ne^{-i \\zeta\\cdot x} L_{\\sigma,\\mu}(e^{i \\zeta\\cdot x}r) = \\nabla_\\zeta \\times \\nabla_\\zeta \\times r + \\nabla_\\zeta \\nabla_\\zeta \\cdot a + \\nabla_\\zeta(\\nabla\\log\\mu \\cdot a) - \\nabla\\log\\sigma\\times \\nabla_\\zeta\\times r  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) r.\n$$\nHence, \\eqref{eqn LE=0 rewritten} implies\n$$\n\\nabla_\\zeta \\times Q + \\nabla_\\zeta \\nabla_\\zeta \\cdot a + \\nabla_\\zeta(\\nabla\\log\\mu \\cdot a) - \\nabla\\log\\sigma\\times Q  - i\\omega (\\sigma\\mu - \\sigma_0 \\mu_0) r = - f.\n$$\nApplying $\\nabla_\\zeta\\times$ to it, we get\n$$\n\\nabla_\\zeta \\times \\nabla_\\zeta \\times Q - \\nabla_\\zeta\\times (\\nabla\\log\\sigma\\times Q) - i\\omega \\nabla(\\sigma \\mu) \\times r - i\\omega (\\sigma \\mu - \\sigma_0 \\mu_0) Q = - \\nabla_\\zeta\\times f\n$$\nas\n$$\n\\nabla_\\zeta\\times (\\nabla_\\zeta \\nabla_\\zeta \\cdot a) = 0\\quad\\text{and}\\quad \\nabla_\\zeta\\times (\\nabla_\\zeta(\\nabla\\log\\mu \\cdot a) )= 0.\n$$\nUsing\nthe fact that $\\nabla_\\zeta \\cdot Q = 0$, we can write\n$$\n\\nabla_\\zeta\\times (\\nabla\\log\\sigma\\times Q) = \\nabla\\nabla(\\log\\sigma) Q - (\\Delta\\log\\sigma)Q - \\nabla\\log\\sigma \\cdot \\nabla_\\zeta Q.\n$$\nThus, we come to\n\\begin{equation}\\label{final eqn for Q}\n-\\Delta_\\zeta Q - \\nabla\\log\\sigma \\cdot \\nabla_\\zeta Q - V_2 Q \\\\\n= i\\omega \\nabla(\\sigma\\mu)\\times r - \\nabla_\\zeta \\times f,\n\\end{equation}\nwhere\n$$\nV_2:= \\nabla\\nabla(\\log\\sigma) + i\\omega(\\sigma\\mu - \\sigma_0 \\mu_0) - \\Delta\\log\\sigma.\n$$\nAccording to the discussion in Section~\\ref{section on L^2 norm of weighted curl of f}, for sufficiently large $\\tau$, there are bounded inverses\n$$\nG_{\\zeta, \\mu}: L^2_{\\delta + 1} \\to L^2_\\delta \\quad\\text{and}\\quad G_{\\zeta, \\sigma}: L^2_{\\delta + 1} \\to L^2_\\delta\n$$\nof $-\\Delta_\\zeta - \\nabla\\log\\mu \\cdot \\nabla_\\zeta$ and $-\\Delta_\\zeta - \\nabla\\log\\sigma \\cdot \\nabla_\\zeta$, respectively, satisfying\n\\begin{equation}\\label{ineq::operator norms of G sigma}\n\\|G_{\\zeta,\\mu}\\|_{L^2_{\\delta + 1} ; L^2_\\delta} \\le \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\quad\\text{as}\\quad\\tau \\to \\infty\\end{equation}\nand\n\\begin{equation}\\label{ineq::operator norms of G mu}\n\\|G_{\\zeta,\\sigma}\\|_{L^2_{\\delta + 1} ; L^2_\\delta} \\le \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\quad\\text{as}\\quad\\tau \\to \\infty.\\end{equation}\nFurthermore, $G_{\\zeta,\\mu}$ and $G_{\\zeta,\\sigma}$ map the space $L^2_{\\delta + 1}$ into $H^1_\\delta$.\\smallskip\n\nIf $\\tau$ is large enough, we apply $G_{\\zeta, \\sigma}$ to \\eqref{final eqn for Q} and obtain the following identity\n\\begin{equation}\\label{int identity for Q}\n(\\id - G_{\\zeta, \\sigma}V_2) Q = G_{\\zeta, \\sigma}\\big\\{i\\omega\\nabla(\\sigma\\mu)\\times r - \\nabla_\\zeta \\times f \\big\\}.\n\\end{equation}\nSince the support of $V_2$ is compact in $\\R^3$, the operator $\\id - G_{\\zeta, \\sigma}V_2$ is invertible in $L^2_\\delta$ for large enough $\\tau$. Also, if $r\\in L^2_\n\\delta$, then $\\nabla(\\sigma\\mu)\\times r\\in L^2_{\\delta+1}$ by \\eqref{sigma and mu are constants outside of a compact set}. Thus, one can solve the above identity for $Q\\in L^2_\\delta$,\n\\begin{equation}\\label{solution for Q}\nQ = (\\id - G_{\\zeta, \\sigma} V_2)^{-1} G_{\\zeta, \\sigma}\\big\\{i\\omega\\nabla(\\sigma\\mu)\\times r - \\nabla_\\zeta \\times f \\big\\}.\n\\end{equation}\nSubstituting this solution into \\eqref{pre final pde for r}, we obtain\n\\begin{equation}\\label{final integro pde for r}\n-\\Delta_\\zeta r - \\nabla\\log\\mu \\cdot \\nabla_\\zeta r - i\\omega W r - V_1 r = - F\n\\end{equation}\nwhere\n\\begin{align*}\nW &:=  \\nabla\\log(\\sigma\\mu) \\times (\\id - G_{\\zeta, \\sigma} V_1)^{-1}\\circ G_{\\zeta, \\sigma}\\circ \\nabla(\\sigma\\mu)\\times\\ ,\\\\\nF &:= f + \\nabla\\log(\\sigma\\mu) \\times (\\id - G_{\\zeta, \\sigma} V_1)^{-1}\\circ G_{\\zeta, \\sigma}\\circ \\nabla_\\zeta \\times f.\n\\end{align*}\nIt follows from \\eqref{ineq::operator norms of G mu} and \\eqref{sigma and mu are constants outside of a compact set} that\n\\begin{equation}\\label{operator norm of W}\n\\|W\\|_{L^2_{\\delta} ; L^2_{\\delta}} = \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\quad\\text{as}\\quad\\tau \\to \\infty.\n\\end{equation}\nand\n$$\n \\|F\\|_{L^2_{\\delta+1}} \\le  \\|f\\|_{L^2_{\\delta+1}} + \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\| \\nabla_\\zeta \\times f\\|_{L^2_{\\delta+1}} = o(\\tau)\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nApplying $G_{\\zeta, \\mu}$ to \\eqref{final integro pde for r}, we get\n\\begin{equation}\\label{int identity for r}\n(\\id - i\\omega G_{\\zeta, \\mu} W - G_{\\zeta, \\mu} V_1) r = - G_{\\zeta, \\mu} F.\n\\end{equation}\nSince $V_1$ is compactly supported and $W$ satisfies \\eqref{operator norm of W}, the operator $\\id - i\\omega G_{\\zeta, \\mu} W - G_{\\zeta, \\mu} V_1$ is invertible in $L^2_\\delta$ for $\\tau$ sufficiently large. Therefore, one can solve the above identity for $r\\in L^2_\\delta$ by\n$$\nr = - (\\id - i\\omega G_{\\zeta, \\mu} W - G_{\\zeta, \\mu} V_1)^{-1} G_{\\zeta, \\mu}F.\n$$\nFinally, by \\eqref{ineq::operator norms of G mu} and \\eqref{solution for Q}, we can show that\n$$\n\\|r\\|_{L^2_{\\delta}} \\le \\mathcal O\\Big(\\frac{1}{\\tau}\\Big)\\|F\\|_{L^2_{\\delta+1}} = o(1)\\quad\\text{as}\\quad\\tau \\to \\infty\n$$\nand\n$$\n\\|\\nabla_\\zeta\\times r\\|_{L^2_{\\delta}} \\le \\mathcal O\\Big(\\frac{1}{\\tau}\\Big) \\Big\\{\\|r\\|_{L^2_{\\delta}} + \\|\\nabla_\\zeta\\times f\\|_{L^2_{\\delta+1}}\\Big\\} = o(\\tau)\\quad\\text{as}\\quad\\tau \\to \\infty.\n$$\nIt follows from \\eqref{int identity for Q} and \\eqref{int identity for r} that\n\\begin{align*}\nr &= i\\omega G_{\\zeta, \\mu} (Wr) + G_{\\zeta, \\mu} (V_1 r) - G_{\\zeta, \\mu} F,\\\\\nQ &= G_{\\zeta, \\sigma}(V_2 Q) + G_{\\zeta, \\sigma}\\big\\{i\\omega\\nabla(\\sigma\\mu)\\times r - \\nabla_\\zeta \\times f \\big\\}.\n\\end{align*}\nSince $V_1$, $V_2$ and $W$ are compactly supported and $G_{\\zeta,\\mu}$ and $G_{\\zeta,\\sigma}$ map the space $L^2_{\\delta + 1}$ into $H^1_\\delta$, this implies that $r,\\nabla_\\zeta\\times r \\in H^1_\\delta$. Thus, we have constructed the following complex geometric optics solution for \\eqref{eqn3-1}\n$$\nH(x;\\zeta) = e^{i\\zeta\\cdot x}\\big\\{e^{-\\alpha^\\sharp(x;\\tau)\/2}\\rho + s_0 b e^{\\alpha^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp(x;\\rho)}\\overline\\rho + r(x;\\zeta)\\big\\}.\n$$\nOur next step is to show that $\\nabla\\cdot(\\mu H)=0$.\nWe observe that \\eqref{eqn3-1} is equivalent to\n$$\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H) - \\sigma^{-1}\\nabla(\\mu^{-1}\\nabla\\cdot(\\mu H)) - i\\omega\\mu H = 0.\n$$\nApplying the divergence to this identity and setting $v = \\mu^{-1}\\nabla\\cdot(\\mu H)$, we get\n$$\n- \\nabla\\cdot(\\sigma^{-1}\\nabla v) - i\\omega\\mu v = 0\n$$\nwhich can be written as\n$$\n- \\Delta \\tilde v + \\tilde q \\tilde v = 0,\\quad \\tilde v := \\sigma^{-1\/2} v,\\quad \\tilde q := \\sigma^{1\/2} \\Delta \\sigma^{-1\/2} - i\\omega \\sigma \\mu.\n$$\nStraightforward calculations give\n$$\n\\tilde v = e^{i\\zeta\\cdot x} u,\\quad u:= \\sigma^{-1\/2}\\mu^{-1}\\nabla_\\zeta\\cdot\\big(\\sigma(a+r)\\big).\n$$\nThen, by \\eqref{eqn:: conj grad and laplacian}, $u$ satisfies\n$$\n-\\Delta_\\zeta u= q u,\\quad q := - \\sigma^{1\/2} \\Delta \\sigma^{-1\/2} + i\\omega ( \\sigma \\mu - \\sigma_0 \\mu_0 ).\n$$\nBy \\cite[Theorem~1.6]{sylvester1987global}, again, there is a bounded inverse, $G_\\zeta : L^2_{\\delta+1} \\to L^2_{\\delta}$ of $-\\Delta_\\zeta$ such that $\\|G_\\zeta\\|_{L^2_{\\delta+1} ; L^2_{\\delta}} \\le \\mathcal O(|\\zeta|^{-1})$ as $|\\zeta|\\to\\infty$. Since $u = G_\\zeta (qu)$ for large enough $|\\zeta|$ and $q$ is compactly supported,\n$$\n\\|u\\|_{L^2_{\\delta}} \\le \\|G_\\zeta(qu)\\|_{L^2_{\\delta}} \\le \\mathcal O\\Big(\\frac{1}{|\\zeta|}\\Big)\\|qu\\|_{L^2_{\\delta+1}} \\le \\mathcal O\\Big(\\frac{1}{|\\zeta|}\\Big)\\|u\\|_{L^2_{\\delta}}\\quad\\text{as}\\quad |\\zeta|\\to\\infty.\n$$\nThis implies that $u=0$ and hence $\\nabla\\cdot(\\mu H)=0$ for sufficiently large $|\\zeta|$.\nThus, restricting $H$ onto $\\Omega$, we get $H\\in H^1(\\Omega;\\C^3)$ solving \\eqref{eqn::curl-curl equation} and satisfying $\\nabla\\times H\\in H^1(\\Omega;\\C^3)$. Clearly, $\\nu\\times H|_{\\p\\Omega}\\in H^{1\/2}(\\p\\Omega;\\C^3)$. Then by \\cite[Corollary~A.20.]{kirsch2016mathematical},\n$$\n\\Div(\\nu\\times H|_{\\p\\Omega}) = - \\nu\\cdot (\\nabla\\times H)|_{\\p\\Omega} \\in H^{1\/2}(\\p\\Omega;\\C^3).\n$$\nTherefore, $\\nu\\times H|_{\\p\\Omega}\\in TH^{1\/2}_{\\Div}(\\p\\Omega)$ and hence $H\\in H^1_{\\Div}(\\Omega)$. In a similar way, also using the fact that $H$ is a solution of \\eqref{eqn::curl-curl equation}, one can easily show that $\\nu\\times (\\nabla\\times H)|_{\\p\\Omega} \\in TH^{1\/2}_{\\Div}(\\p\\Omega)$ and hence $\\nabla \\times H\\in H^1_{\\Div}(\\Omega)$.\nThus, we proved\n\n\\begin{Proposition}\\label{prop CGO}\nLet $\\Omega\\subset \\R^3$ be an open bounded set with $C^{1,1}$ boundary and $\\sigma,\\mu\\in C^2(\\overline\\Omega)$ with $\\sigma \\ge \\sigma_0$, $\\mu \\ge \\mu_0$ for some $\\sigma_0,\\mu_0>0$. Assume that $\\sigma$ and $\\mu$ can be extended positively to $\\R^3$ so that $\\sigma - \\sigma_0, \\mu - \\mu_0 \\in C^2_0(\\R^3)$. Let $\\zeta\\in \\C^3$ be such that $\\zeta\\cdot\\zeta=i\\omega\\sigma_0\\mu_0$, $\\zeta = \\tau \\rho + \\zeta_1$ where $\\tau > 0$ is a large parameter, $\\rho \\in \\C^3$ is independent of $\\tau$ and satisfies $\\Re\\rho \\cdot \\Im\\rho = 0$ and $|\\Re\\rho| = |\\Im\\rho| = 1$, and $\\zeta_1 = \\mathcal O(1)$ as $\\tau \\to \\infty$. Then, for any $s_0 \\in \\R$, there is a solution $H\\in H^1_{\\Div}(\\Omega)$ for \\eqref{eqn::curl-curl equation} of the form\n$$\nH(x;\\zeta) = e^{i\\zeta\\cdot x}\\big\\{e^{-\\alpha^\\sharp(x;\\tau)\/2}\\rho + s_0 b e^{\\alpha^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp(x;\\rho)}\\overline\\rho + r(x;\\zeta)\\big\\}.\n$$\nFurthermore, $\\nabla\\times H\\in H^1_{\\Div}(\\Omega)$. The function\n$\\Psi^\\sharp(\\cdot, \\rho; \\tau)\\in C^\\infty(\\R^3)$ satisfies\n$\\|\\Psi^\\sharp\\|_{W^{1,\\infty}(\\R^3)} = \\mathcal O(1)$ as\n$\\tau\\to\\infty$ and converges to $\\Psi(\\cdot, \\rho) := - N_\\rho^{-1}\n\\{\\overline{\\rho} \\cdot \\nabla\\log(\\sigma\\mu)^{1\/2}\\} \\in\nL^\\infty(\\R^3)$ in $L^2_{\\rm loc}(\\R^3)$ as $\\tau\\to\\infty$. The\nfunction $\\alpha^\\sharp(\\,\\cdot\\,; \\tau)\\in C^\\infty(\\R^3)$ satisfies\n$\\| \\alpha^\\sharp \\|_{W^{2,\\infty}(\\R^3)} = \\mathcal O(1)$ as\n$\\tau\\to\\infty$ and $\\| \\alpha^\\sharp - \\log\\sigma\n\\|_{W^{2,\\infty}(\\R^3)} = o(1)$ as $\\tau\\to\\infty$. The correction\nterm, $r$, satisfies $\\|r\\|_{L^2(\\Omega;\\C^3)} = o(1)$ and\n$\\|\\nabla_\\zeta\\times r\\|_{L^2(\\Omega;\\C^3)} = o(\\tau)$ as\n$\\tau\\to\\infty$.\n\\end{Proposition}\n\n\\section{Proof of Theorem~\\ref{main thm}}\\label{section::proof of main thm}\n\nSince we assume that $\\p^\\alpha \\sigma_1|_{\\p\\Omega} = \\p^\\alpha\n\\sigma_2|_{\\p\\Omega}$ and $\\p^\\alpha \\mu_1|_{\\p\\Omega} = \\p^\\alpha\n\\mu_2|_{\\p\\Omega}$ for $|\\alpha| \\le 2$, we can extend $\\sigma_j$ and\n$\\mu_j$, $j=1,2$, to $C^2$ functions defined on $\\R^3$, still denoted\nby $\\sigma_j$ and $\\mu_j$, such that $\\sigma_j \\ge \\sigma_0$, $\\mu_j\n\\ge \\mu_0$ on $\\R^3$, $\\sigma_j - \\sigma_0, \\mu_j - \\mu_0 \\in\nC^2_0(\\R^3)$ and $\\sigma_1=\\sigma_2$ and $\\mu_1=\\mu_2$ on\n$\\R^3\\setminus\\overline\\Omega$. These kind of extensions (of Whitney\ntype) hold for all functions defined on any closed subset of $\\R^3$\nthat can be approximated by certain polynomials. The argument to prove\nthe existence of such polynomials is similar to the one in\n\\cite[Section~2]{caro2013stability} for $C^{1,\\varepsilon}$ functions\non $\\overline\\Omega$. The only difference, here, is that the authors\nof \\cite{caro2013stability} refer to \\cite[Section~2 of\n  Chapter~VI]{stein1970singular}, while we refer to \\cite[Section~4.7\n  of Chapter~VI]{stein1970singular}; see also\n\\cite[Section~3]{caro2014global}.\n\n\\begin{Proposition}\nLet $\\Omega\\subset \\R^3$ be a bounded domain with $C^{1,1}$ boundary and $\\sigma_j,\\mu_j\\in C^2(\\overline\\Omega)$, $j=1,2$, with $\\sigma_j \\ge \\sigma_0$, $\\mu_j \\ge \\mu_0$ for some $\\sigma_0,\\mu_0>0$. Suppose that $Z_{\\sigma_1,\\mu_1}^\\omega = Z_{\\sigma_2,\\mu_2}^\\omega$; then\n\\begin{equation}\\label{main integral identity}\n\\int_\\Omega (\\mu_1 - \\mu_2) H_1\\cdot H_2\\,dx + \\frac{1}{i\\omega} \\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\nabla\\times H_1\\cdot \\nabla\\times H_2\\,dx = 0\n\\end{equation}\nfor all $H_j\\in H_{\\Div}^1(\\Omega)$ with $\\nabla\\times H_j\\in H_{\\Div}^1(\\Omega)$ solving\n$$\n\\nabla\\times(\\sigma_j^{-1}\\nabla\\times H_j) - i\\omega \\mu_j H_j = 0\\quad\\text{in}\\quad\\Omega,\\quad j=1,2.\n$$\n\\end{Proposition}\n\\begin{proof}\nDefine\n$$\nE_j := \\sigma_j^{-1} \\nabla\\times H_j,\\quad j=1,2.\n$$\nThen $E_j\\in H_{\\Div}^1(\\Omega)$ and $\\nabla\\times E_j = i\\omega\\mu_j H_j$. Hence $(H_j, E_j) \\in H^1_{\\Div}(\\Omega) \\times H^1_{\\Div}(\\Omega)$, $j=1,2$, solve\n$$\n\\nabla\\times E_j=i\\omega\\mu_j H_j\\quad\\text{and}\\quad \\nabla\\times H_j=\\sigma_j E_j \\quad\\text{in}\\quad\\Omega,\\quad j=1,2.\n$$\nThen the assumption $Z_{\\sigma_1,\\mu_1}^\\omega = Z_{\\sigma_2,\\mu_2}^\\omega$ implies existence of $(H', E')\\in H^1_{\\Div}(\\Omega) \\times H^1_{\\Div}(\\Omega)$ satisfying\n$$\n\\nabla\\times E'=i\\omega\\mu_2 H'\\quad\\text{and}\\quad \\nabla\\times H'=\\sigma_2 E'\\quad\\text{in}\\quad\\Omega\n$$\nand\n$$\n\\bt(H') = \\bt(H_1)\\quad\\text{and}\\quad \\bt(E') = \\bt(E_1)\\quad\\text{on}\\quad\\p \\Omega.\n$$\nIntegrating by parts,\n\\begin{align*}\n\\int_\\Omega \\nabla\\times(H'-H_1)\\cdot E_2\\,dx - \\int_\\Omega i\\omega\\mu_2 (H'-H_1)\\cdot H_2\\,dx&= \\int_\\Omega \\nabla\\times(H'-H_1)\\cdot E_2\\,dx - \\int_\\Omega (H'-H_1)\\cdot \\nabla\\times E_2\\,dx\\\\\n&= \\int_{\\p\\Omega} \\bt (H'-H_1)\\cdot E_2\\,dS(x) = 0,\n\\end{align*}\nwhere $dS$ is the surface measure on $\\p \\Omega$. Similarly,\n$$\n\\int_\\Omega \\nabla\\times(E'-E_1)\\cdot H_2\\,dx - \\int_\\Omega \\sigma_2 (E'-E_1)\\cdot E_2\\,dx = 0.\n$$\nAdding these two identities, we obtain\n$$\n\\int_\\Omega [\\nabla\\times(H'-H_1) - \\sigma_2 (E'-E_1)]\\cdot E_2\\,dx + \\int_\\Omega [\\nabla\\times(E'-E_1) - i\\omega\\mu_2 (H'-H_1)]\\cdot H_2\\,dx = 0.\n$$\nIt is easy to show that\n$$\n\\nabla\\times(H'-H_1) - \\sigma_2 (E'-E_1) = (\\sigma_2 - \\sigma_1) E_1,\\quad \\nabla\\times(E'-E_1) - i\\omega\\mu_2 (H'-H_1) = i\\omega(\\mu_2 - \\mu_1) H_1.\n$$\nSubstituting these into the latter integral identity, we come to\n$$\n\\int_\\Omega (\\sigma_2 - \\sigma_1) E_1\\cdot E_2\\,dx + \\int_\\Omega i\\omega(\\mu_2 - \\mu_1) H_1\\cdot H_2\\,dx = 0.\n$$\nThis implies \\eqref{main integral identity}.\n\\end{proof}\n\nLet $\\xi,\\rho_1,\\rho_2\\in \\R^3$ be such that $|\\rho_1| = |\\rho_2| = 1$ and $\\rho_1\\cdot\\rho_2 = \\rho_1 \\cdot \\xi = \\rho_2 \\cdot \\xi =0$. Consider\n\\begin{align*}\n\\zeta^1 &= \\phantom{-}\\frac{\\xi}{2} + i \\tau \\rho_2 + \\tau \\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2} + \\frac{i\\omega\\sigma_0\\mu_0}{\\tau^2}} \\rho_1,\\\\\n\\zeta^2 &= -\\frac{\\xi}{2} + i \\tau \\rho_2 + \\tau \\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2} + \\frac{i\\omega\\sigma_0\\mu_0}{\\tau^2}} \\rho_1.\n\\end{align*}\nHere, by $\\sqrt{\\, \\cdot \\,}$ we mean its principal branch. Then\n$$\n  \\zeta^j = \\tau \\rho + \\zeta^j_1\\quad\\text{with}\\quad \\zeta^j_1 = \\mathcal O(1)\\quad\\text{as}\\ \\tau \\to \\infty\\quad\\text{and}\\quad\n  \\zeta^1 - \\zeta^2 = \\xi,\\quad \\zeta^j\\cdot\\zeta^j = i\\omega\\sigma_0\\mu_0,\n$$\nwhere $\\rho:=\\rho_1 + i\\rho_2$. By Proposition~\\ref{prop CGO}, there are complex geometric optics solutions $H_1,H_2\\in H^1_{\\Div}(\\Omega)$, with $\\nabla\\times H_1, \\nabla\\times H_2 \\in H^1_{\\Div}(\\Omega)$ satisfying\n$$\n\\nabla\\times(\\sigma_1^{-1}\\nabla\\times H_1) - i\\omega \\mu_1 H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma_2^{-1}\\nabla\\times H_2) - i\\omega \\mu_2 H_2 = 0 \\quad\\text{in}\\quad\\Omega,\n$$\nrespectively, which have the following forms\n$$\nH_1(x;\\zeta^1) = e^{i\\zeta^1\\cdot x} \\Big(a_1 \\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big),\\quad H_2(x;\\zeta^2) = e^{-i\\zeta^2\\cdot x} \\Big(-a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big),\n$$\nwhere\n\\begin{align*}\na_1 &= e^{-\\alpha_1^\\sharp(x;\\tau)\/2},\\quad b_1 = e^{\\alpha_1^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp_1(x, \\rho; \\tau)},\\\\\na_2 &= e^{-\\alpha_2^\\sharp(x;\\tau)\/2},\\quad b_2 = e^{\\alpha_2^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp_2(x, \\rho; \\tau)}.\n\\end{align*}\nThe functions $\\Psi_1^\\sharp(\\cdot, \\rho; \\tau),\\Psi_2^\\sharp(\\cdot, \\rho; \\tau)\\in C^\\infty(\\R^3)$ satisfy\n\\begin{align}\n\\|\\Psi_1^\\sharp\\|_{W^{1,\\infty}(\\R^3)}&,\\ \\|\\Psi_2^\\sharp\\|_{W^{1,\\infty}(\\R^3)} = \\mathcal O(1) \\quad \\text{as}\\quad\\tau\\to\\infty,\\label{eqn 4.2}\\\\\n\\|\\Psi_1^\\sharp - \\Psi_1\\|_{L^2_{\\rm loc}(\\R^3)}&,\\ \\|\\Psi_2^\\sharp - \\Psi_2\\|_{L^2_{\\rm loc}(\\R^3)} = o(1) \\quad\\text{as}\\quad\\tau\\to\\infty,\\label{eqn 4.3}\n\\end{align}\nwhere\n$$\n\\Psi_j(\\cdot, \\rho) := - N_\\rho^{-1} \\{\\overline{\\rho} \\cdot \\nabla\\log(\\sigma_j\\mu_j)^{1\/2}\\} \\in L^\\infty(\\R^3),\\quad j=1,2.\n$$\nFurthermore, the functions $\\alpha_1^\\sharp(\\,\\cdot\\,;\\tau),\\ \\alpha_2^\\sharp(\\,\\cdot\\,;\\tau) \\in C^\\infty(\\R^3)$ satisfy\n\\begin{align}\n\\| \\alpha_1^\\sharp \\|_{W^{2,\\infty}(\\R^3)}&,\\ \\| \\alpha_2^\\sharp \\|_{W^{2,\\infty}(\\R^3)} = \\mathcal O(1) \\quad\\text{as}\\quad\\tau\\to\\infty,\\label{eqn 4.4}\\\\\n\\| \\alpha_1^\\sharp - \\log\\sigma_1 \\|_{W^{2,\\infty}(\\R^3)}&,\\ \\| \\alpha_2^\\sharp - \\log\\sigma_2 \\|_{W^{2,\\infty}(\\R^3)} = o(1) \\quad\\text{as}\\quad\\tau\\to\\infty.\\label{eqn 4.5}\n\\end{align}\nThe correction terms, $r_1$, $r_2 \\in H_{\\Div}^1(\\Omega)$, satisfy\n\\begin{equation}\\label{eqn 4.6}\n\\|r_j\\|_{L^2(\\Omega)} = o(1)\\quad\\text{and}\\quad\\|\\nabla_{\\zeta^j}\\times r_j\\|_{L^2(\\Omega)} = o(\\tau) \\quad \\text{as}\\quad \\tau\\to\\infty,\\quad j=1,2.\n\\end{equation}\nThen, using that $\\zeta^j = \\tau\\rho + \\zeta^j_1$, we find that\n\\begin{align*}\n\\nabla\\times E_1 &= e^{i\\zeta^1\\cdot x}\\,\\nabla_{\\zeta^1}\\times\\Big(a_1 \\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big\n= e^{i\\zeta^1\\cdot x} \\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho + b_1 \\tau \\rho_1\\times \\rho_2 + \\nabla_{\\zeta^1}\\times r_1\\Big),\\\\%\\Big\\{e^{-\\beta^\\sharp\/2}\\Big(-\\frac{1}{2}\\nabla\\beta^\\sharp + i\\zeta^1_1\\Big)\\times \\rho + \\frac{1}{2}e^{\\beta^\\sharp\/2}e^{\\Psi^\\sharp_1}\\Big(\\frac{1}{2}\\nabla\\beta^\\sharp + \\nabla\\Psi^\\sharp_1 + i\\zeta^1_1\\Big)\\times \\overline{\\rho}\\\\\n\\nabla\\times E_2 &= e^{-i\\zeta^2\\cdot x} \\,\\nabla_{-\\zeta^2}\\times\\Big(- a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big\n= e^{-i\\zeta^2\\cdot x} \\Big(- \\nabla_{-\\zeta^2_1}a_2\\times \\rho - \\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho + b_2 \\tau \\rho_1\\times \\rho_2 + \\nabla_{-\\zeta^2}\\times r_2\\Big)\n\\end{align*}\nSubstituting $H_1$, $H_2$, $\\nabla\\times H_1$ and $\\nabla\\times H_2$ into \\eqref{main integral identity} and dividing the whole identity by $\\tau^2$, we obtain\n\\begin{equation}\\label{integral identity to find mu}\n\\begin{aligned}\n\\frac{1}{\\tau^2}\\int_\\Omega &(\\mu_1 - \\mu_2) e^{i\\xi\\cdot x}\\Big(a_1\\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big)\\cdot \\Big(- a_2\\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big)\\,dx\\\\\n&- \\frac{1}{\\tau^2}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\\\\n&+ \\frac{1}{\\tau}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot (b_2 \\rho_1\\times\\rho_2)\\,dx\\\\\n&- \\frac{1}{\\tau}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (b_1 \\rho_1\\times \\rho_2) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\\\\n&+ \\frac{1}{\\tau^2}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot (\\nabla_{-\\zeta^2}\\times r_2)\\,dx\\\\\n&- \\frac{1}{\\tau^2}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (\\nabla_{\\zeta^1}\\times r_1) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\\\\n&+ \\frac{1}{\\tau}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (b_1 \\rho_1\\times \\rho_2) \\cdot (\\nabla_{-\\zeta^2}\\times r_2)\\,dx\\\\\n&+ \\frac{1}{\\tau}\\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (\\nabla_{\\zeta^1}\\times r_1) \\cdot (b_2 \\rho_1\\times\\rho_2)\\,dx + \\int_\\Omega \\frac{1}{i\\omega}\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} b_1 b_2 \\,dx = 0.\n\\end{aligned}\n\\end{equation}\nFor the last term on the left-hand side, we also used the property $|\\rho_1\\times\\rho_2| = 1$ as $\\rho_1 \\cdot \\rho_2 = 0$ and $|\\rho_1| = |\\rho_2| = 1$. By the Cauchy-Schwartz inequality,\n\\begin{multline*}\n\\Big|\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x}\\Big(a_1\\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big)\\cdot \\Big(- a_2\\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big)\\,dx\\Big|\\\\\n\\le \\mathcal O(1) \\Big\\|a_1\\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big\\|_{L^2(\\Omega)} \\Big\\|- a_2\\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big\\|_{L^2(\\Omega)}\\quad\\text{as}\\quad \\tau\\to\\infty.\n\\end{multline*}\nThen, by \\eqref{eqn 4.2}, \\eqref{eqn 4.4} and \\eqref{eqn 4.6},\n$$\n\\Big|\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x}\\Big(a_1\\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big)\\cdot \\Big(- a_2\\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big)\\,dx\\Big| = \\mathcal O(1)\\quad\\text{as $\\tau\\to\\infty$}.\n$$\nIn a similar way, we obtain\n\\begin{align*}\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= \\mathcal O(1),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot (b_2 \\rho_1\\times\\rho_2)\\,dx\\Big| &= \\mathcal O(1),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (b_1 \\rho_1\\times \\rho_2) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= \\mathcal O(1),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x}\\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big)\\cdot (\\nabla_{-\\zeta^2}\\times r_2)\\,dx\\Big| &= o(\\tau),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (\\nabla_{\\zeta^1}\\times r_1) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho+\\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= o(\\tau),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (b_1 \\rho_1\\times \\rho_2) \\cdot (\\nabla_{-\\zeta^2}\\times r_2)\\,dx\\Big| &= o(\\tau),\\\\\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} (\\nabla_{\\zeta^1}\\times r_1) \\cdot (b_2 \\rho_1\\times\\rho_2)\\,dx\\Big| &= o(\\tau)\n\\end{align*}\nas $\\tau\\to\\infty$. Here, we used again that $\\zeta^j_1 = \\mathcal O(1)$ as $\\tau\\to\\infty$, $j=1,2$. Finally, we use \\eqref{eqn 4.2}-\\eqref{eqn 4.5}, to show that\n\\begin{align*}\n\\Big|\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} & e^{i\\xi\\cdot x} b_1 b_2 \\, dx - \\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} e^{i\\xi\\cdot x} \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi_1+\\Psi_2}\\,dx\\Big| \\\\\n&\\le \\mathcal O(1) \\big\\| e^{\\alpha_1^\\sharp\/2 + \\alpha_2^\\sharp\/2} e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} - \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi_1+\\Psi_2} \\big\\|_{L^2(\\Omega)}\\\\\n&\\le \\mathcal O(1) \\big\\| e^{\\alpha_1^\\sharp\/2 + \\alpha_2^\\sharp\/2} e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} - \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} \\big\\|_{L^2(\\Omega)}\n+ \\mathcal O(1) \\big\\| \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} - \\sigma_1^{1\/2}\\sigma_2^{1\/2} e^{\\Psi_1+\\Psi_2} \\big\\|_{L^2(\\Omega)}\\\\\n&\\le \\mathcal O(1) \\big\\| e^{\\alpha_1^\\sharp\/2 + \\alpha_2^\\sharp\/2} - e^{(\\log \\sigma_1)\/2 +(\\log\\sigma_2)\/2} \\big\\|_{L^2(\\Omega)}\n+ \\mathcal O(1) \\big\\| e^{\\Psi^\\sharp_1+\\Psi^\\sharp_2} - e^{\\Psi_1+\\Psi_2} \\big\\|_{L^2(\\Omega)}\\\\\n&\\le \\mathcal O(1) \\big\\| \\alpha_1^\\sharp- \\log \\sigma_1\\|_{L^2(\\Omega)} + \\mathcal O(1) \\|\\alpha_2^\\sharp  - \\log\\sigma_2 \\big\\|_{L^2(\\Omega)}\n + \\mathcal O(1) \\big\\| \\Psi^\\sharp_1 - \\Psi_1 \\|_{L^2(\\Omega)} + \\mathcal O(1) \\big\\| \\Psi^\\sharp_2 - \\Psi_2 \\big\\|_{L^2(\\Omega)}\\\\[0.15cm]\n&= o(1)\\quad\\text{as}\\quad \\tau\\to\\infty,\n\\end{align*}\nemploying the basic inequality, \n\\begin{equation}\\label{technical ineq from complex analysis}\n|e^z - e^w| \\le |z - w| e^{\\max(\\Re z,\\Re w)},\\quad z,w\\in \\C\n\\end{equation}\nfrom \\cite{krupchyk2014uniqueness}. According to these estimates, taking the limit as $\\tau \\to \\infty$ in \\eqref{integral identity to find mu}, we come to\n$$\n\\int_{\\R^3} e^{i\\xi\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1^{1\/2} \\sigma_2^{1\/2}} e^{\\Psi_1 + \\Psi_2}\\,dx = 0.\n$$\nNote that the integration is extended to all of $\\R^3$ since $\\sigma_1 - \\sigma_2 = 0$ on $\\R^3\\setminus\\overline\\Omega$. This implies that $\\sigma_1=\\sigma_2$.\\smallskip\n\nNext, we set $\\sigma = \\sigma_1=\\sigma_2$. By Proposition~\\ref{prop CGO}, there are complex geometric optics solutions $H_1,H_2\\in H^1_{\\Div}(\\Omega)$, with $\\nabla\\times H_1, \\nabla\\times H_2 \\in H^1_{\\Div}(\\Omega)$ satisfying\n$$\n\\nabla\\times(\\sigma^{-1}\\nabla\\times H_1) - i\\omega \\mu_1 H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma^{-1}\\nabla\\times H_2) - i\\omega \\mu_2 H_2 = 0 \\quad\\text{in}\\quad\\Omega,\n$$\nrespectively, which have the following forms\n$$\nH_1(x;\\zeta^1) = e^{i\\zeta^1\\cdot x} \\Big(a_1 \\rho + r_1\\Big),\\quad H_2(x;\\zeta^2) = e^{-i\\zeta^2\\cdot x} \\Big(-a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big),\n$$\nwhere\n$$\na_1 = e^{-\\alpha^\\sharp(x;\\tau)\/2},\\quad a_2 = e^{-\\alpha^\\sharp(x;\\tau)\/2},\\quad b_2 = e^{\\alpha^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp(x, \\rho; \\tau)}.\n$$\nThe function $\\Psi^\\sharp(\\cdot, \\rho; \\tau)\\in C^\\infty(\\R^3)$ satisfies\n\\begin{equation}\\label{eqn 4.7}\n\\|\\Psi^\\sharp\\|_{W^{1,\\infty}(\\R^3)} = \\mathcal O(1) \\quad \\text{and}\\quad  \\|\\Psi^\\sharp - \\Psi\\|_{L^2_{\\rm loc}(\\R^3)} = o(1) \\quad\\text{as}\\quad\\tau\\to\\infty,\n\\end{equation}\nwhere\n$$\n\\Psi(\\cdot, \\rho) := - N_\\rho^{-1} \\{\\overline{\\rho} \\cdot \\nabla\\log(\\sigma\\mu_2)^{1\/2}\\} \\in L^\\infty(\\R^3),\\quad j=1,2.\n$$\nFurthermore, the function $\\alpha^\\sharp(\\,\\cdot\\,;\\tau) \\in C^\\infty(\\R^3)$ satisfies\n\\begin{equation}\\label{eqn 4.8}\n\\| \\alpha^\\sharp \\|_{W^{2,\\infty}(\\R^3)} = \\mathcal O(1) \\quad\\text{and}\\quad  \\| \\alpha^\\sharp - \\log\\sigma \\|_{W^{2,\\infty}(\\R^3)} = o(1) \\quad\\text{as}\\quad\\tau\\to\\infty.\n\\end{equation}\nThe correction terms $r_1$, $r_2 \\in H_{\\Div}^1(\\Omega)$ satisfy\n\\begin{equation}\\label{eqn 4.9}\n\\|r_j\\|_{L^2(\\Omega)} = o(1)\\quad\\text{and}\\quad\\|\\nabla_{\\zeta^j}\\times r_j\\|_{L^2(\\Omega)} = o(\\tau) \\quad \\text{as}\\quad \\tau\\to\\infty,\\quad j=1,2.\n\\end{equation}\nSubstituting $H_1$, $H_2$ and $\\sigma = \\sigma_1=\\sigma_2$ into \\eqref{main integral identity}, we come to\n\\begin{align*}\n-\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} a_1 b_2\\,dx &+ \\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} a_1 \\rho\\cdot r_2\\,dx\\\\\n-&\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} r_1\\cdot \\Big(a_2\\rho+\\frac{1}{2}b_2\\overline\\rho\\Big)\\,dx + \\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} r_1\\cdot r_2\\,dx= 0.\n\\end{align*}\nBy the Cauchy-Schwartz inequality together with \\eqref{eqn 4.8} and \\eqref{eqn 4.9},\n$$\n\\Big|\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} a_1 \\rho\\cdot r_2\\,dx\\Big| \\le \\mathcal O(1) \\int_\\Omega | a_1 \\rho\\cdot r_2 |\\,dx \\le \\mathcal O(1) \\|a_1\\|_{L^2(\\Omega)} \\|r_2\\|_{L^2(\\Omega)} = o(1)\n$$\nas $\\tau\\to\\infty$. In a similar way, and also using \\eqref{eqn 4.7}, one can show that\n$$\n\\Big|\\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} r_1\\cdot \\Big(a_2\\rho+\\frac{1}{2}b_2\\overline\\rho\\Big)\\,dx\\Big| = o(1)\\quad\\text{and}\\quad \\Big| \\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} r_1\\cdot r_2\\,dx \\Big| = o(1)\\quad\\text{as}\\quad \\tau \\to\\infty.\n$$\nFinally, using \\eqref{eqn 4.7} and \\eqref{eqn 4.8},\n$$\n\\Big| \\int_\\Omega (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} a_1 b_2\\,dx - \\int_{\\R^3} (\\mu_1 - \\mu_2) e^{i\\xi\\cdot x} e^{\\Psi}\\,dx \\Big| \\le \\mathcal O(1) \\| e^{\\Psi^\\sharp} - e^{\\Psi}\\|_{L^2(\\Omega)} \\le \\mathcal O(1) \\|\\Psi^\\sharp - \\Psi\\|_{L^2(\\Omega)} = o(1)\\quad\\text{as}\\quad \\tau \\to\\infty.\n$$\nHere, we have again employed inequality \\eqref{technical ineq from complex analysis}. Thus, letting $\\tau\\to\\infty$, we obtain\n$$\n\\int_{\\R^3} e^{i\\xi\\cdot x} (\\mu_1 - \\mu_2) e^{\\Psi}\\,dx = 0.\n$$\nThe integration is extended to all of $\\R^3$ since $\\mu_1 - \\mu_2 = 0$ on $\\R^3\\setminus\\overline\\Omega$. This implies that $\\mu_1 = \\mu_2$ completing the proof of Theorem~\\ref{main thm}.\n\n\\section{Reflection approach} \\label{section::reflection approach}\n\nIn this section, we use Isakov's reflection approach \\cite{isakov2007uniqueness} to prove the following local uniqueness result where the region of the boundary that is inaccessible for measurements is a part of a plane. For a closed $\\Gamma\\subset\\p\\Omega$, define\n$$\nC_{\\Gamma}(\\sigma, \\mu; \\omega) := \\{(\\bt(H) |_{\\Gamma}, \\bt(E)|_{\\Gamma}): (H, E)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)\\text{ is a solution to \\eqref{eqn::Maxwell} with }\\supp(\\bt(H))\\subseteq\\Gamma\\}.\n$$\n\n\\begin{Theorem}\\label{main thm flat}\nLet $\\Omega\\subset \\{x\\in \\R^3: x_3 < 0 \\}$ be a bounded domain with $C^{1,1}$ boundary and let $\\Gamma_0 = \\p\\Omega\\cap \\{x\\in \\R^3: x_3 = 0\\}$ and $\\Gamma = \\overline{\\p\\Omega\\setminus \\Gamma_0}$. Suppose that $\\sigma_j,\\mu_j\\in C^2(\\overline\\Omega)$, $j=1,2$, satisfy $\\sigma_j \\ge \\sigma_0$ and $\\mu_j \\ge \\mu_0$, for some constants $\\sigma_0, \\mu_0 > 0$, and\n\\begin{equation}\\label{boundary assumption on Gamma flat}\n\\p^\\alpha \\sigma_1|_{\\Gamma} = \\p^\\alpha \\sigma_2|_{\\Gamma}\\quad\\text{and}\\quad \\p^\\alpha \\mu_1|_{\\Gamma} = \\p^\\alpha \\mu_2|_{\\Gamma}\\quad\\text{for}\\quad|\\alpha| \\le 2.\n\\end{equation}\nIn addition, assume that $\\sigma_j$ and $\\mu_j$, $j=1,2$, can be extended into $\\R^3$ as $C^2$ functions which are invariant under reflection across the plane $\\{x\\in \\R^3: x_3 = 0\\}$. Then $C_\\Gamma(\\sigma_1, \\mu_1; \\omega) = C_\\Gamma(\\sigma_2, \\mu_2; \\omega)$ implies $\\sigma_1=\\sigma_2$ and $\\mu_1=\\mu_2$.\n\\end{Theorem}\n\n\\noindent\nSimilar results were obtained for the inverse conductivity problem in\n\\cite{isakov2007uniqueness} and for the IEMP in\n\\cite{caro2009inverse}. Consider the reflected domain\n$$\n\\Omega^* := \\{(x_1, x_2, -x_3)\\in\\R^3 : (x_1, x_2, x_3)\\in\\Omega\\}\n$$\nand define\n$$\n\\mathcal U := \\Omega \\cap \\Gamma_0^{\\rm int} \\cap \\Omega^*.\n$$\nBy the assumptions in Theorem~\\ref{main thm flat}, we can extend the coefficients $\\sigma_j$ and $\\mu_j$ into $\\mathcal U$ as $C^2$ functions which are even with respect to $x_3$ for $j = 1,2$.  Next, by the assumption \\eqref{boundary assumption on Gamma flat}, we can extend $\\sigma_j$ and $\\mu_j$, $j=1,2$, to $C^2$ functions defined on $\\R^3$, still denoted by $\\sigma_j$ and $\\mu_j$, such that $\\sigma_j \\ge \\sigma_0$, $\\mu_j \\ge \\mu_0$ on $\\R^3$, $\\sigma_j - \\sigma_0, \\mu_j - \\mu_0 \\in C^2_0(\\R^3)$ and $\\sigma_1=\\sigma_2$ and $\\mu_1=\\mu_2$ on $\\R^3\\setminus\\overline{\\mathcal U}$.\n\n\\begin{Proposition}\\label{prop main integral identity local}\nLet $\\Omega\\subset \\{x\\in \\R^3: x_3 < 0 \\}$ be a bounded domain with $C^{1,1}$ boundary . Let $\\Gamma_0 := \\p\\Omega\\cap \\{x\\in\\R^3: x_3=0\\}$ and $\\Gamma:=\\overline{\\p\\Omega\\setminus \\Gamma_0}$. Suppose that\n\\begin{equation}\\label{local data assumption}\nZ_{\\sigma_1,\\mu_1}^\\omega(f)|_{\\Gamma} = Z_{\\sigma_2,\\mu_2}^\\omega(f)|_{\\Gamma}\\quad\\text{for all}\\quad f\\in TH^{1\/2}_{\\Div}(\\p\\Omega)\\quad\\text{with}\\quad \\supp(f)\\subset\\Gamma;\n\\end{equation}\nthen\n\\begin{equation}\\label{main integral identity local}\n\\int_\\Omega (\\mu_1 - \\mu_2) H_1\\cdot H_2\\,dx + \\frac{1}{i\\omega}\\int_\\Omega \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} (\\nabla\\times H_1)\\cdot (\\nabla\\times H_2)\\,dx = 0\n\\end{equation}\nfor all $H_j\\in H_{\\Div}^1(\\Omega)$ with $\\nabla\\times H_j\\in H_{\\Div}^1(\\Omega)$ solving\n$$\n\\nabla\\times(\\sigma_j^{-1}\\nabla\\times H_j) - i\\omega \\mu_j H_j = 0\\quad\\text{in}\\quad\\Omega,\\quad j=1,2.\n$$\nand satisfying $\\supp(\\bt(H_j))\\subseteq \\Gamma$.\n\\end{Proposition}\n\\begin{proof}\nSimilarly as in the proof of  \\eqref{main integral identity}, define\n$$\nE_j := \\sigma_j^{-1} \\nabla\\times H_j,\\quad j=1,2.\n$$\nThen $E_j\\in H_{\\Div}^1(\\Omega)$ and $\\nabla\\times E_j = i\\omega\\mu_j H_j$. Hence $(H_j, E_j) \\in H^1_{\\Div}(\\Omega) \\times H^1_{\\Div}(\\Omega)$, $j=1,2$, solve\n$$\n\\nabla\\times E_j=i\\omega\\mu_j H_j\\quad\\text{and}\\quad \\nabla\\times H_j=\\sigma_j E_j \\quad\\text{in}\\quad\\Omega,\\quad j=1,2.\n$$\nThen by the assumption \\eqref{local data assumption}, there is $(H', E')\\in H^1_{\\Div}(\\Omega) \\times H^1_{\\Div}(\\Omega)$ with $\\supp(\\bt(H'))\\subseteq\\Gamma$ satisfying\n$$\n\\nabla\\times E'=i\\omega\\mu_2 H'\\quad\\text{and}\\quad \\nabla\\times H'=\\sigma_2 E'\\quad\\text{in}\\quad\\Omega\n$$\nand\n$$\n\\bt(H')|_{\\Gamma} = \\bt(H_1)|_{\\Gamma}\\quad\\text{and}\\quad\\bt(E')|_{\\Gamma} = \\bt(E_1)|_{\\Gamma}.\n$$\nIntegrating by parts, leads to\n\\begin{align*}\n\\int_\\Omega \\nabla\\times(H'-H_1)\\cdot E_2\\,dx - \\int_\\Omega i\\omega\\mu_2 (H'-H_1)\\cdot H_2\\,dx&= \\int_\\Omega \\nabla\\times(H'-H_1)\\cdot E_2\\,dx - \\int_\\Omega (H'-H_1)\\cdot \\nabla\\times E_2\\,dx\\\\\n& = \\int_{\\p\\Omega} \\bt (H'-H_1)\\cdot E_2\\,dS(x) = \\int_{\\Gamma_0^{\\rm int}} \\bt (H'-H_1)\\cdot \\bt(E_2)\\,dS(x) = 0,\n\\end{align*}\nsince both $\\bt(H')$ and $\\bt(H_1)$ are supported on $\\Gamma$. Similarly,\n$$\n\\int_\\Omega \\nabla\\times(E'-E_1)\\cdot H_2\\,dx - \\int_\\Omega \\sigma_2 (E'-E_1)\\cdot E_2\\,dx = \\int_{\\Gamma_0^{\\rm int}} \\bt (E'-E_1)\\cdot \\bt(H_2)\\,dS(x) = 0,\n$$\nsince $\\bt(H_2)$ is supported on $\\Gamma$. The remainder of the proof of \\eqref{main integral identity local} is similar to that of \\eqref{main integral identity}.\n\\end{proof}\n\nFor $\\beta:\\R^3\\to \\C$ and $X:\\R^3\\to \\C^3$, we define the reflections as\n$$\n\\beta^*(x) := \\beta(x_1, x_2, - x_3),\\quad X^*(x) := \\big(X_1(x_1, x_2, - x_3), X_2(x_1, x_2, - x_3), - X_3(x_1, x_2, - x_3)\\big)\n$$\nwith the properties,\n\\begin{equation}\\label{eqn::reflection identities}\n\\nabla \\beta^* = (\\nabla \\beta)^*,\\quad (\\beta X)^* = \\beta^* X^*,\\quad \\nabla\\times X^* = - (\\nabla\\times X)^*\n\\end{equation}\n\n\\subsection* {Proof of Uniqueness}\n\nConsider $\\zeta^1$ and $\\zeta^2$ defined as in the proof of Theorem~\\ref{main thm}.\nThen, by Proposition~\\ref{prop CGO}, there are complex geometric optics solutions $\\tilde H_1, \\tilde H_2\\in H^1_{\\Div}(\\mathcal U)$, with $\\nabla\\times \\tilde H_1, \\nabla\\times \\tilde H_2 \\in H^1_{\\Div}(\\mathcal U)$, for\n$$\n\\nabla\\times(\\sigma_1^{-1}\\nabla\\times\\tilde H_1) - i\\omega \\mu_1\\tilde H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma_2^{-1}\\nabla\\times\\tilde H_2) - i\\omega \\mu_2\\tilde H_2 = 0 \\quad\\text{in}\\quad\\mathcal U,\n$$\nrespectively, which have the following forms\n$$\n\\tilde H_1(x;\\zeta^1) = e^{i\\zeta^1\\cdot x} \\Big(a_1 \\rho + \\frac{1}{2} b_1 \\overline\\rho + r_1\\Big),\\quad \\tilde H_2(x;\\zeta^2) = e^{-i\\zeta^2\\cdot x} \\Big(-a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big),\n$$\nwhere\n\\begin{align*}\na_1 &= e^{-\\alpha_1^\\sharp(x;\\tau)\/2},\\quad b_1 = e^{\\alpha_1^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp_1(x, \\rho; \\tau)},\\\\\na_2 &= e^{-\\alpha_2^\\sharp(x;\\tau)\/2},\\quad b_2 = e^{\\alpha_2^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp_2(x, \\rho; \\tau)}.\n\\end{align*}\nThe functions $\\Psi_1^\\sharp(\\cdot, \\rho; \\tau),\\Psi_2^\\sharp(\\cdot, \\rho; \\tau)\\in C^\\infty(\\R^3)$, $\\alpha_1^\\sharp(\\,\\cdot\\,;\\tau), \\alpha_2^\\sharp(\\,\\cdot\\,;\\tau) \\in C^\\infty(\\R^3)$ and the correction terms $r_1$, $r_2 \\in H^1_{\\Div}(\\mathcal U)$ satisfy \\eqref{eqn 4.3}-\\eqref{eqn 4.6}. Using \\eqref{eqn::reflection identities}, it follows that\n$$\n\\nabla\\times (\\sigma_j^{-1}\\nabla\\times \\tilde H_j^*) - i\\omega \\mu_j \\tilde H_j^* = - \\nabla\\times (\\sigma_j^{-1}\\nabla\\times\\tilde H_j)^* - (i\\omega \\mu_j \\tilde H_j)^* = \\big(\\nabla\\times (\\sigma_j^{-1}\\nabla\\times\\tilde H_j) - i\\omega \\mu_j\\tilde H_j\\big)^* = 0\\quad\\text{in}\\quad \\mathcal U.\n$$\nTherefore,\n$$\nH_1(x;\\rho) := \\tilde H_1(x;\\rho) - \\tilde H_1^*(x;\\rho),\\quad H_2(x;\\rho) := \\tilde H_2(x;\\rho) - \\tilde H_2^*(x;\\rho).\n$$\nalso satisfy\n$$\n\\nabla\\times(\\sigma_1^{-1}\\nabla\\times\\tilde H_1) - i\\omega \\mu_1\\tilde H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma_2^{-1}\\nabla\\times\\tilde H_2) - i\\omega \\mu_2\\tilde H_2 = 0 \\quad\\text{in}\\quad\\mathcal U.\n$$\nAs in the proof of Proposition~\\ref{prop CGO}, it is not difficult to see that $H_1|_\\Omega$ and $H_2|_\\Omega$, still denoted by $H_1$ and $H_2$, respectively, belong to $H^1_{\\Div}(\\Omega)$. Moreover, $\\nabla\\times H_1, \\nabla\\times H_2 \\in H^1_{\\Div}(\\Omega)$. Using the fact that $\\nu = (0, 0, 1)$ on $\\Gamma$, we have  $\\nu\\times H_1|_{\\Gamma}=\\nu\\times H_2|_{\\Gamma}=0$. Thus, $H_1$ and $H_2$ satisfy the hypotheses of Proposition~\\ref{prop main integral identity local}, and hence we can substitute $H_1$ and $H_2$ into \\eqref{main integral identity local}. To that end, we first compute $\\nabla\\times H_1$ and $\\nabla\\times H_2$. Using \\eqref{eqn::reflection identities},\n\\begin{align*}\n\\nabla\\times H_1 &= \\nabla\\times \\tilde H_1 + (\\nabla\\times \\tilde H_1)^* \\\\\n&= e^{i\\zeta^1\\cdot x} \\Big(\\nabla_{\\zeta^1_1}a_1\\times \\rho+\\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho + b_1 \\tau \\rho_1\\times \\rho_2 + \\nabla_{\\zeta^1}\\times r_1\\Big)\\\\\n&\\quad + e^{i{\\zeta^1}^*\\cdot x} \\Big((\\nabla_{\\zeta^1_1}a_1\\times \\rho)^*+\\frac{1}{2}(\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho)^* + b_1^* \\tau (\\rho_1\\times \\rho_2)^* + (\\nabla_{\\zeta^1}\\times r_1)^*\\Big).\n\\end{align*}\nSimilarly,\n\\begin{align*}\n\\nabla\\times H_2 &= e^{-i\\zeta^2\\cdot x} \\Big(- \\nabla_{-\\zeta^2_1}a_2\\times \\rho - \\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho + b_2 \\tau \\rho_1\\times \\rho_2 + \\nabla_{-\\zeta^2}\\times r_2\\Big)\\\\\n&\\quad + e^{-i{\\zeta^2}^*\\cdot x} \\Big(- (\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* - \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^* + b_2^* \\tau (\\rho_1\\times \\rho_2)^* + (\\nabla_{-\\zeta^2}\\times r_2)^*\\Big).\n\\end{align*}\nWe take a closer look at the phases of products of these vector fields. We have\n$$\ni(\\zeta^1 - \\zeta^2)\\cdot x = i\\xi\\cdot x,\\quad i({\\zeta^1}^* - {\\zeta^2}^*)\\cdot x = i\\xi^*\\cdot x,\n$$\n$$\ni(\\zeta^1 - {\\zeta^2}^*)\\cdot x = i\\tilde\\xi_{+}\\cdot x - 2\\tau\\rho_{2,3} x_3 - \\eta_{+},\\quad i({\\zeta^1}^* - \\zeta^2)\\cdot x = i\\tilde\\xi_{-}\\cdot x + 2\\tau\\rho_{2,3} x_3 - \\eta_{-},\n$$\nwhere\n$$\n\\tilde\\xi_{\\pm} = \\Bigg(\\xi', \\pm 2\\tau\\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2}}\\rho_{1,3}\\Bigg),\\quad |\\tilde\\xi_{\\pm}|\\to\\infty\\quad\\text{as}\\quad \\tau\\to \\infty\n$$\nand\n$$\n\\eta_{\\pm}:= \\pm\\frac{2\\omega \\sigma_{0} \\mu_{0} \\rho_{1,3} x_3}{\\tau\\Big(\\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2} + \\frac{i\\omega\\sigma_0\\mu_0}{\\tau^2}} + \\sqrt{1 - \\frac{|\\xi|^2}{4\\tau^2}}\\Big)} ,\\quad |\\eta_{\\pm}| = \\mathcal O(\\tau^{-1})\\quad\\text{as}\\quad \\tau\\to \\infty.\n$$\nFurthermore, we assume that $\\rho_{1,3} \\neq 0$ and $\\rho_{2,3} = 0$. Then\n\\begin{align*}\n\\int_\\Omega & e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(\\nabla_{\\zeta^1_1} a_1\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^*\\Big)\\,dx\\\\\n& = \\int_\\Omega e^{i\\tilde\\xi_{+}\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big\\{e^{- \\eta_{+}}\\Big(\\nabla_{\\zeta^1_1} a_1\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^*\\Big)\\\\\n&\\qquad\\qquad\\qquad\\qquad - \\Big(\\nabla_{\\zeta^1_1} \\mu_1^{1\/2}\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}(\\mu^{-1\/2}e^{\\Psi_1})\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}\\mu_2^{1\/2}\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}(\\mu_2^{-1\/2}e^{\\Psi_2})\\times \\overline\\rho)^*\\Big)\\Big\\}\\,dx\\\\\n&\\quad + \\int_\\Omega e^{i\\tilde\\xi_{+}\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(\\nabla_{\\zeta^1_1} \\mu_1^{1\/2}\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}(\\mu^{-1\/2}e^{\\Psi_1})\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}\\mu_2^{1\/2}\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}(\\mu_2^{-1\/2}e^{\\Psi_2})\\times \\overline\\rho)^*\\Big)\\,dx\n\\end{align*}\nThe first integral on the right-side goes to zero as $\\tau\\to\\infty$ according to estimates \\eqref{eqn 4.2}-\\eqref{eqn 4.5} and the fact that $|\\eta_{\\pm}| = \\mathcal O(\\tau^{-1})$ as $\\tau\\to \\infty$. The second integral goes to zero as $\\tau\\to\\infty$ by the Riemann-Lebesgue lemma. Therefore,\n$$\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(\\nabla_{\\zeta^1_1} a_1\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^*\\Big)\\,dx\\Big| = o(1)\\quad\\text{as}\\quad\\tau\\to\\infty.\n$$\nIn a similar way, one can show that\n\\begin{align*}\n\\Big|\\int_\\Omega e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big((\\nabla_{\\zeta^1_1}a_1\\times \\rho)^*+\\frac{1}{2}(\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho)^*\\Big) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho + \\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= o(1)\\\\\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(\\nabla_{\\zeta^1_1} a_1\\times \\rho + \\frac{1}{2}\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho\\Big) \\cdot \\Big(b_2^* \\tau (\\rho_1\\times \\rho_2)^*\\Big)\\,dx\\Big| &= o(\\tau)\\\\\n\\Big|\\int_\\Omega e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(b_1^* \\tau (\\rho_1\\times \\rho_2)^*\\Big) \\cdot \\Big(\\nabla_{-\\zeta^2_1}a_2\\times \\rho + \\frac{1}{2}\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho\\Big)\\,dx\\Big| &= o(\\tau)\\\\\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(b_1 \\tau \\rho_1\\times \\rho_2\\Big) \\cdot \\Big((\\nabla_{-\\zeta^2_1}a_2\\times \\rho)^* + \\frac{1}{2}(\\nabla_{-\\zeta^2_1}b_2\\times \\overline\\rho)^*\\Big)\\,dx\\Big| &= o(\\tau)\\\\\n\\Big|\\int_\\Omega e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big((\\nabla_{\\zeta^1_1}a_1\\times \\rho)^*+\\frac{1}{2}(\\nabla_{\\zeta^1_1}b_1\\times \\overline\\rho)^*\\Big) \\cdot \\Big(b_2\\tau \\rho_1\\times\\rho_2\\Big)\\,dx\\Big| &= o(\\tau)\\\\\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(b_1 \\tau \\rho_1\\times \\rho_2\\Big) \\cdot \\Big(b_2^*\\tau (\\rho_1\\times\\rho_2)^*\\Big)\\,dx\\Big| &= o(\\tau^2)\\\\\n\\Big|\\int_\\Omega  e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1\\sigma_2} \\Big(b_1^* \\tau (\\rho_1\\times \\rho_2)^*\\Big) \\cdot \\Big(b_2\\tau \\rho_1\\times\\rho_2\\Big)\\,dx\\Big| &= o(\\tau^2)\\\\\n\\Big|\\int_\\Omega(\\mu_1 - \\mu_2) e^{i\\tilde\\xi_{+}\\cdot x - \\eta_{+}} \\Big(a_1 \\rho + \\frac{1}{2} b_1 \\overline\\rho\\Big)\\cdot \\Big(a_2^* \\rho^* + \\frac{1}{2} b_2^* \\overline\\rho^*\\Big)\\,dx\\Big| &= o(1)\\\\\n\\Big|\\int_\\Omega(\\mu_1 - \\mu_2) e^{i\\tilde\\xi_{-}\\cdot x - \\eta_{-}} \\Big(a_1^* \\rho^* + \\frac{1}{2} b_1^* \\overline\\rho^*\\Big)\\cdot \\Big(a_2 \\rho + \\frac{1}{2} b_2 \\overline\\rho\\Big)\\,dx\\Big| &= o(1)\n\\end{align*}\nas $\\tau\\to\\infty$. Now we substitute $H_1$, $H_2$, $\\nabla\\times H_1$ and $\\nabla\\times H_2$ into \\eqref{main integral identity local} and divide the whole identity by $\\tau^2$. According to the estimates obtained above, the terms with phases $i\\tilde\\xi_{\\pm}\\cdot x - \\eta_{\\pm}$ that do not involve the correction terms $r_1$ and $r_2$ go to zero as $\\tau\\to\\infty$. The terms with phases $i\\tilde\\xi_{\\pm}\\cdot x - \\eta_{\\pm}$ that involve the correction terms $r_1$ and $r_2$ go to zero as $\\tau\\to\\infty$, because of the correction terms. Finally, the terms with phases $i\\xi\\cdot x$ or $i\\xi^*\\cdot x$ can be controlled exactly as in the proof of Theorem~\\ref{main thm} using \\eqref{eqn 4.2} - \\eqref{eqn 4.6} and \\eqref{technical ineq from complex analysis}. Thus, letting $\\tau\\to\\infty$, we obtain\n$$\n\\int_{\\Omega} e^{i\\xi\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1^{1\/2} \\sigma_2^{1\/2}}e^{\\Psi_1 + \\Psi_2}\\,dx + \\int_{\\Omega} e^{i\\xi^*\\cdot x} \\Bigg(\\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1^{1\/2} \\sigma_2^{1\/2}}e^{\\Psi_1 + \\Psi_2}\\Bigg)^*\\,dx = 0.\n$$\nMaking the change of the variables $(x_1, x_2, x_3)\\mapsto (x_1, x_2, - x_3)$ in the second integral, we come to\n$$\n\\int_{\\mathcal D} e^{i\\xi\\cdot x} \\frac{(\\sigma_1 - \\sigma_2)}{\\sigma_1^{1\/2} \\sigma_2^{1\/2}} e^{\\Psi_1 + \\Psi_2}\\,dx = 0.\n$$\nThis integral can be extended to all of $\\R^3$ since $\\sigma_1 - \\sigma_2 = 0$ on $\\R^3\\setminus\\overline{\\mathcal D}$. Then this will imply that $\\sigma_1=\\sigma_2$ in $\\R^3$.\\smallskip\n\nNext, we set $\\sigma = \\sigma_1 = \\sigma_2$. By Proposition~\\ref{prop CGO}, there are complex geometric optics solutions $\\tilde H_1, \\tilde H_2\\in H^1_{\\Div}(\\mathcal U)$, with $\\nabla\\times \\tilde H_1, \\nabla\\times\\tilde H_2 \\in H^1_{\\Div}(\\mathcal U)$, for\n$$\n\\nabla\\times(\\sigma^{-1}\\nabla\\times\\tilde H_1) - i\\omega \\mu_1\\tilde H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma^{-1}\\nabla\\times\\tilde H_2) - i\\omega \\mu_2\\tilde H_2 = 0 \\quad\\text{in}\\quad\\mathcal U,\n$$\nrespectively, which have the following forms\n$$\n\\tilde H_1(x;\\zeta^1) = e^{i\\zeta^1\\cdot x} (a_1 \\rho + r_1),\\quad \\tilde H_2(x;\\zeta^2) = e^{-i\\zeta^2\\cdot x} \\Big(-a_2 \\rho - \\frac{1}{2} b_2 \\overline\\rho + r_2\\Big),\n$$\nwhere\n$$\na_1 = e^{-\\alpha^\\sharp(x;\\tau)\/2},\\quad a_2 = e^{-\\alpha^\\sharp(x;\\tau)\/2},\\quad b_2 = e^{\\alpha^\\sharp(x;\\tau)\/2} e^{\\Psi^\\sharp(x, \\rho; \\tau)}.\n$$\nThe functions $\\Psi^\\sharp(\\cdot, \\rho; \\tau), \\alpha^\\sharp(\\,\\cdot\\,;\\tau)\\in C^\\infty(\\R^3)$ and the correction terms $r_1$, $r_2 \\in H_{\\Div}^1(\\mathcal U)$ satisfy \\eqref{eqn 4.7} - \\eqref{eqn 4.9}. In a similar way as before, one can show that\n$$\nH_1(x;\\rho) := \\tilde H_1(x;\\rho) - \\tilde H_1^*(x;\\rho),\\quad H_2(x;\\rho) := \\tilde H_2(x;\\rho) - \\tilde H_2^*(x;\\rho).\n$$\nsatisfy\n$$\n\\nabla\\times(\\sigma_1^{-1}\\nabla\\times\\tilde H_1) - i\\omega \\mu_1\\tilde H_1 = 0\\quad\\text{and}\\quad \\nabla\\times(\\sigma_2^{-1}\\nabla\\times\\tilde H_2) - i\\omega \\mu_2\\tilde H_2 = 0 \\quad\\text{in}\\quad\\mathcal U.\n$$\nAlso, the restrictions of $H_1$ and $H_2$ onto $\\Omega$, still denoted by $H_1$ and $H_2$, respectively, belong to $H^1_{\\Div}(\\Omega)$ and satisfy $\\nu\\times H_1|_{\\Gamma}=\\nu\\times H_2|_{\\Gamma}=0$. Thus, $H_1$ and $H_2$ satisfy the hypotheses of Proposition~\\ref{prop main integral identity local}. We substitude $H_1$, $H_2$ and $\\sigma = \\sigma_1=\\sigma_2$ into \\eqref{main integral identity local}. As before, we assume that $\\rho_{1,3} \\neq 0$ and $\\rho_{2,3} = 0$. Therefor, we obtain\n\\begin{align*}\n0 &= \\int_\\Omega (\\mu_1-\\mu_2) e^{i\\xi\\cdot x}(a_1\\rho + r_1)\\cdot \\Big(-a_2\\rho-\\frac{1}{2}b_2\\overline{\\rho}+r_2\\Big)\\,dx\\\\\n&\\quad+\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\xi^*\\cdot x}(a_1^*\\rho^*+r_1^*)\\cdot \\Big(-a_2^*\\rho^*-\\frac{1}{2}b_2^*\\overline{\\rho}^*+r_2^*\\Big)\\,dx\\\\\n&\\quad-\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\tilde\\xi_+\\cdot x-\\eta_+}(a_1\\rho+r_1)\\cdot \\Big(-a_2^*\\rho^*-\\frac{1}{2}b_2^*\\overline{\\rho}^*+r_2^*\\Big)\\,dx\\\\\n&\\quad-\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\tilde\\xi_-\\cdot x-\\eta_-}(a_1^*\\rho^*+r_1^*)\\cdot \\Big(-a_2\\rho-\\frac{1}{2}b_2\\overline{\\rho}+r_2\\Big)\\,dx.\n\\end{align*}\nAs before, we use \\eqref{eqn 4.7} - \\eqref{eqn 4.8}, the fact that $|\\eta_{\\pm}| = \\mathcal O(\\tau^{-1})$ as $\\tau\\to \\infty$ and the Riemann-Lebesgue lemma, to show that\n\\begin{align*}\n\\Big|\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\tilde\\xi_+\\cdot x-\\eta_+}a_1\\rho\\cdot \\Big(-a_2^*\\rho^*-\\frac{1}{2}b_2^*\\overline{\\rho}^*\\Big)\\,dx\\Big| &= o(1),\\\\\n\\Big|\\int_\\Omega (\\mu_1-\\mu_2) e^{i\\tilde\\xi_-\\cdot x-\\eta_-}a_1^*\\rho^*\\cdot \\Big(-a_2\\rho-\\frac{1}{2}b_2\\overline{\\rho}\\Big)\\,dx\\Big| &= o(1)\n\\end{align*}\nas $\\tau\\to\\infty$. These estimates guarantee that the terms with phases $i\\tilde\\xi_{\\pm}\\cdot x - \\eta_{\\pm}$ that do not involve the correction terms $r_1$ and $r_2$ go to zero as $\\tau\\to\\infty$. The terms with phases $i\\tilde\\xi_{\\pm}\\cdot x - \\eta_{\\pm}$ that involve the correction terms $r_1$ and $r_2$ also go to zero as $\\tau\\to\\infty$ by \\eqref{eqn 4.9}. Finally, the terms with phases $i\\xi\\cdot x$ or $i\\xi^*\\cdot x$ can be controlled exactly as in the proof of Theorem~\\ref{main thm} using \\eqref{eqn 4.7} - \\eqref{eqn 4.9} and \\eqref{technical ineq from complex analysis}. Thus, letting $\\tau\\to\\infty$, we obtain\n$$\n\\int_{\\Omega} e^{i\\xi\\cdot x} (\\mu_1 - \\mu_2) e^{\\Psi}\\,dx + \\int_{\\Omega} e^{i\\xi^*\\cdot x} \\Big((\\mu_1 - \\mu_2) e^{\\Psi}\\Big)^*\\,dx = 0.\n$$\nMaking the change of the variables $(x_1, x_2, x_3)\\mapsto (x_1, x_2, - x_3)$ in the second integral, we come to\n$$\n\\int_{\\mathcal U} e^{i\\xi\\cdot x} (\\mu_1 - \\mu_2) e^{\\Psi}\\,dx = 0\n$$\nThis integral can be extended to all of $\\R^3$ since $\\mu_1 - \\mu_2 = 0$ on $\\R^3\\setminus\\overline\\Omega$. This implies that $\\mu_1 = \\mu_2$ completing the proof of Theorem~\\ref{main thm flat}.\n\n\\section{Proof of Theorem~\\ref{main thm 2}}\\label{section::proof of thm 2}\n\nWithout loss of generality, we can assume that $B_0$ is the open ball of radius $1\/2$ centered at $x_0 = (0,0,1\/2)$ and that $0\\notin \\overline\\Omega$. We recall that\n$$\n\\Gamma_0 = \\p\\Omega\\cap \\p B_0,\\quad \\Gamma_0 \\neq \\p B_0\\quad\\text{and}\\quad\\Gamma = \\overline{\\p\\Omega\\setminus \\Gamma_0}.\n$$\nWe define the map\n$$\nK:\\Omega\\to \\R^3\\setminus\\{0\\},\\quad K(x) := |x|^{-2}x\n$$\nwhich is known as the Kelvin transform. One can easily verify that $K^{-1}(y) = |y|^{-2}y$ for $y\\in K(\\Omega)$. We let $DK$ and $DK^{-1}$ denote the Jacobi matrices of $K$ and $K^{-1}$, respectively.\n\n\\smallski\n\nNext, we define $\\widetilde\\Omega := \\{- y + x_0 : y\\in K(\\Omega)\\}$ and\n$$\n F : \\widetilde\\Omega\\to \\Omega,\\quad F(y) := - K^{-1}(y) + x_0,\\quad y\\in\\widetilde\\Omega.\n$$\nThen\n$$\nF^{-1}(x) = - K(x-x_0),\\quad x\\in \\Omega.\n$$\nIt is not difficult to verify that $\\widetilde\\Omega\\subset \\{x\\in\n\\R^3: x_3 < 0 \\}$ and $\\widetilde\\Gamma_0:=F^{-1}(\\Gamma_0)$ is a\nsubset of the plane $\\{x\\in \\R^3: x_3 = 0\\}$. We also write\n$\\widetilde\\Gamma:=F^{-1}(\\Gamma)$. Thus, we are in a situation when\nthe inaccessible part of the boundary is part of a plane. A direct\ncalculation gives\n$$\nDF^{-1} = - |x-x_0|^{-2} I + 2 |x-x_0|^{-4} (x-x_0)(x-x_0)^T ,\\quad DF = - |y|^{-2} I + 2 |y|^{-4} yy^T,\n$$\nwhere $x-x_0$ and $y$ are considered as column vectors and $I$ is the $3\\times 3$ identity matrix. These identities can be used to show that\n\\begin{equation}\\label{eqn::DFDF^T}\nDF=(DF)^T\\quad\\text{and}\\quad DF(DF)^T=|y|^{-4}I\n\\end{equation}\nand\n\\begin{equation}\\label{eqn::key identities for the Kelvin transform}\nDF^{-1}\\circ F = |y|^{4} DF,\\quad DF = (DF)^T \\quad\\text{and}\\quad \\det(DF) = |y|^{-6}.\n\\end{equation}\n\n\\begin{Lemma}\\label{invariance of maxwell equation under pullback}\nLet $(H_j, E_j)\\in H^1_{\\Div}(\\Omega)\\times H^1_{\\Div}(\\Omega)$, $j=1,2$. Consider their pullbacks onto $\\Omega$,\n$$\n\\widetilde H_j := F^* H_j,\\quad \\widetilde E_j := F^* E_j,\\quad\\widetilde\\sigma_j = \\sigma_j\\circ F,\\quad \\widetilde\\mu_j = \\mu_j\\circ F,\\quad j=1,2.\n$$\nThen\n$$\n\\nabla\\times E_j=i\\omega\\mu_j H_j\\quad\\text{and}\\quad \\nabla\\times H_j=\\sigma_j E_j \\quad\\text{in}\\quad\\Omega\n$$\nif and only if\n$$\n\\widetilde\\nabla\\times \\widetilde E_j=i\\omega|y|^{-2}\\widetilde\\mu_j \\widetilde H_j\\quad\\text{and}\\quad \\widetilde\\nabla\\times \\widetilde H_j=|y|^{-2}\\widetilde\\sigma_j\\widetilde E_j \\quad\\text{in}\\quad\\widetilde\\Omega.\n$$\n\\end{Lemma}\n\n\\noindent\nHere and in what follows, $\\widetilde\\nabla\\times$ denotes the curl operator with respect to the coordinates in $\\widetilde\\Omega$.\n\\begin{proof}\nThe claim of the lemma is easy to prove by straightforward calculations using \\eqref{eqn::key identities for the Kelvin transform} and the facts from Appendix~\\ref{section::pullbacks}.\n\\end{proof}\n\n\\begin{Lemma}\\label{impedance map under pullback}\nThe following holds true,\n\\[\n   Z_{|y|^{-2}\\widetilde\\sigma_j ,|y|^{-2}\\widetilde\\mu_j}^\\omega(F^*f)\n        = F^*\\big(Z_{\\sigma_j ,\\mu_j}^\\omega(f)\\big)\n\\]\nfor all $f \\in TH^{1\/2}_{\\Div}(\\p\\Omega)$, $j=1,2$.\n\\end{Lemma}\n\n\\begin{proof}\nConsider a $C^{1,1}$ boundary defining function $\\rho$ for $\\p\\Omega$. Then using \\eqref{eqn::DFDF^T},\n$$\n\\widetilde\\nu = - \\frac{\\nabla(\\rho\\circ F)}{|\\nabla(\\rho\\circ F)|}\\Bigg|_{\\p\\widetilde\\Omega} = - \\frac{(DF)^T (\\nabla \\rho)\\circ F}{|(DF)^T (\\nabla \\rho)\\circ F|}\\Bigg|_{\\p\\widetilde\\Omega} = - \\frac{|y|^2 F^*(\\nabla \\rho)}{|(\\nabla\\rho)\\circ F|}\\Bigg|_{\\p\\widetilde\\Omega} = |y|^2 F^*\\Bigg(-\\frac{\\nabla \\rho}{|\\nabla\\rho|}\\Bigg)\\Bigg|_{\\p\\widetilde\\Omega} = |y|^2 F^*\\nu.\n$$\nUsing Lemma~\\ref{lemma::pullback of the cross product} and \\eqref{eqn::key identities for the Kelvin transform},\n$$\n\\tilde\\nu\\times\\tilde H_j = |y|^2 F^*\\nu\\times F^*H_j = |y|^{-4} \\big((DF)^{-1}(\\nu\\times H_j)\\big)\\circ F = DF\\,\\big((\\nu\\times H_j)\\circ F\\big) = (DF)^T\\big((\\nu\\times H_j)\\circ F\\big) = F^*(\\nu\\times H_j).\n$$\nSimilarly, $\\tilde\\nu\\times\\tilde E_j =  F^*(\\nu\\times E_j)$.\n\\end{proof}\n\nWith $Z_{\\sigma_1 ,\\mu_1}^\\omega = Z_{\\sigma_2 ,\\mu_2}^\\omega$, it follows from these lemmas that\n$$\nZ_{|y|^{-2}\\widetilde\\sigma_1 ,|y|^{-2}\\widetilde\\mu_1}^\\omega = Z_{|y|^{-2}\\widetilde\\sigma_2 ,|y|^{-2}\\widetilde\\mu_2}^\\omega.\n$$\nFinally, by hypothesis, $\\sigma_j$ and $\\mu_j$, $j=1,2$, can be extended to $\\R^3$ as $C^2$ functions which are invariant under reflection across $\\p B_0$. This is equivalent to the invariance of such extensions of $\\sigma_j$ and $\\mu_j$, $j=1,2$, under the map $x\\mapsto F\\circ R\\circ F^{-1}(x)$, where $R(y_1,y_2,y_3)=(y_1,y_2,-y_3)$. Therefore, $\\tilde\\sigma_j$ and $\\tilde\\mu_j$ can be extended into $\\R^3$ as $C^2$ functions which are invariant under reflection across the plane $\\{x\\in\\R^3: x_3=0\\}$. Thus, by Theorem~\\ref{main thm flat}, we get $|y|^{-2}\\tilde\\sigma_1=|y|^{-2}\\tilde\\sigma_2$ and $|y|^{-2}\\tilde\\mu_1=|y|^{-2}\\tilde\\mu_2$ in $\\widetilde\\Omega$. Hence, $\\sigma_1=\\sigma_2$ and $\\mu_1=\\mu_2$ in $\\Omega$ as desired.\n\n\n\\section*{Ackkowledgments}\n\nYA would like to thank Total E \\& P Research \\& Technology USA, for\nfinancial support. MVdH was supported by the Simons Foundation under\nthe MATH + X program, the National Science Foundation under grant\nDMS-1815143, and the corporate members of the Geo-Mathematical Imaging\nGroup at Rice University.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe potential effect of technical networks on territories, and more particularly of transportation networks, have fed scientific debates which remain relatively open in the current state-of-the-art, such as the issue of identifying of structuring effects of infrastructures~\\citep{offner1993effets}. These can be observed on long time periods, but several works recall that caution is key regarding the contingency of each situation and the dangers of a political application of the concept~\\citep{espacegeo2014effets}.\n\n\nA relevant approach to this question, explored by \\cite{raimbault2018caracterisation} of which we do a short synthesis here, is to understand territories and transportation networks as co-evolving, i.e. exhibiting strongly coupled dynamics which are difficult to isolate~\\citep{bretagnolle:tel-00459720}. The construction of the concept of territory by \\cite{raimbault2018caracterisation} at the intersection of Raffestin's approach and Pumain's approach, i.e. by combining the concept of human territory~\\citep{raffestin1988reperes} with the one of territory as the space in which systems of cities are embedded~\\citep{pumain2018evolutionary}, shows that it implies potential relations between geographical objects, and by induction the emergence of technical networks through the realization of transactional projects~\\citep{dupuy1987vers}. Therefore, the interaction between networks and territories can be intrinsically understood as endogenous to the concept of territory itself. We go further by postulating the potential existence of a \\emph{co-evolution} between transportation networks and territories, that we will define below from an interdisciplinary point of view in a dedicated section. We study more particularly in our work the \\emph{transportation networks}, in particular because these are typically representative of potentialities to shape territories that are attributed to technical networks, more specifically through the effective transportation of flows~\\citep{bavoux2005geographie}.\n\n\nThe research work we summarize proposes to explore this perspective of a co-evolution by studying it through the lens of modeling and simulation, considering the model as an instrument of knowledge in itself~\\citep{banos2013pour} complementary to the theoretical and empirical aspects~\\citep{raimbault2017applied}, and which impact is amplified by the use of methods and tools for model exploration and intensive computation~\\citep{pumain2017urban}. The use of generative simulation models for the considered systems allows to construct an indirect knowledge on implied processes and for example to directly test and compare hypothesis in this virtual laboratory~\\citep{epstein1996growing}.\n\n\nFollowing \\cite{raimbault2017invisible} which establishes a map of scientific approaches to the question, domains which studied the modeling of interactions between transportation networks and territories are much varied, from geography to planning of urban economics, and more recently physics, but these seem to be highly isolated~\\citep{raimbault2017models}. According to a systematic litterature review and a modelography done by~\\cite{raimbault2018caracterisation}, and a typology of interaction processes, two scales appear as relevant to unveil the possible existence of co-evolution processes : the mesoscopic scale, which will typically be a metropolitan spatial scale, and the macroscopic scale, which corresponds to the scale of the system of cities.\n\n\nTwo complementary modeling directions corresponding to these two scales have thus been developed, extending the previous works modeling this co-evolution at the macroscopic~\\citep{baptistemodeling,schmitt2014modelisation} and mesoscopic~\\citep{raimbault2014hybrid} scale. These two axes correspond to different theoretical frameworks, namely the evolutive urban theory for the macroscopic \\citep{pumain2018evolutionary} and urban morphogenesis for the mesoscopic, which we understand as the emergent relation between form and function~\\citep{doursat2012morphogenetic}.\n\n\nThis contribution aims at giving a synthesis of these research works, and is organized the following way : we firstly define the concept of co-evolution from a multi-disciplinary viewpoint. We then detail the results obtained for the modeling at the macroscopic scale, and then the ones for the mesoscopic scale. We finally discuss the perspectives opened and the future developments in the context of modeling co-evolution between transportation networks and territories.\n\n\n\\section{Defining co-evolution}\n\n\nTransfer of concepts between disciplines is always difficult, and as co-evolution has been used by several disciplines, we propose here a definition inspired by the various ones in which it was developed. A multi-disciplinary review and a more general definition not specific to territorial systems is developed in \\cite{raimbault2018co}. The detailed description of the different approaches is given in this paper, of which we give here only a broad summary.\n\n\nOriginating in biology, the concept of evolution requires typical features for a system to exhibit it~\\citep{durham1991coevolution}, namely (i) transmission processes between agents; (ii) transformation processes; and (iii) isolation of sub-system such that differentiations emerge from the previous processes. Co-evolution then corresponds to entangled evolutionary changes in two species~\\citep{janzen1980coevolution}. It was generalized to diffuse co-evolution by taking into account the broader context of numerous species in the ecological interaction network and of the environment~\\citep{strauss2005toward}.\n\nThe concept was transferred to disciplines closer to social and human sciences, including the emerging field of cultural evolution \\cite{Mesoudi25072017} for which building bricks of culture are transmitted and mutated, possibly with an interplay with biological evolution itself~\\citep{bull2000meme}, but also sociology to interpret for example the interactions between social organizations as entities themselves~\\citep{volberda2003co}. In the frame of evolutionary economics~\\citep{nelson2009evolutionary}, the concept was also largely applied in economic geography~\\citep{schamp201020}, to investigate for example the link between economic clusters and knowledge spillovers~\\citep{doi:10.1080\/00343400802662658}, the link between territories and technological innovation~\\citep{colletis2010co}, or environmental economics issues~\\citep{kallis2007coevolution}. The concept was particularly developed in geography in the frame of the evolutive urban theory \\citep{pumain1997pour}. Its operational definition in that context taken for example by \\cite{paulus2004coevolution} or \\cite{schmitt2014modelisation} relies on systems of cities constituted by strongly coupled subsystems and entangled interactions.\n\n\n\n\n\nWe can observe that these various definitions of co-evolution mostly correspond to the theoretical framework introduced by \\cite{holland2012signals}, in which hierarchically imbricated subsystems correspond to ecological niches and are therefore containing co-evolutive dynamics.\n\n\n\nThe definition of co-evolution we construct for the particular case of transportation networks and territories is the following, similar to the definition of \\cite{raimbault2018co}:\n\n\\begin{enumerate}\n    \\item the evolutive processes are carried by the transformation of territorial components at different scales;\n   \n\t\\item a co-evolution may occur in the strong coupling of such evolutive processes, with different levels of strength, namely as circular causal relation (i) between individual entities; (ii) within populations of individual, i.e. with a certain statistical sense within a given territory; (iii) between most elements of the system with a difficulty to disentangle these relations.\n\n\t\\item these different level and the spatial isolation typically enhancing evolutionary drift imply the existence of \\emph{territorial niches}, i.e. niches of co-evolution, imbricated at different scales.\n\n\\end{enumerate}\n\n\nOur definition is strongly anchored within a dynamical vision of processes, within the initial spirit of the introduction of the concept in biology. The opening on the different levels to which the co-evolution can occur yields a generality but also a precision and the construction of empirical characterization methods, such as the one introduced for the second level (population co-evolution) by \\cite{raimbault2017identification}. Furthermore, the implicit integration of the concept of niche implies a suitability with territoriality and its declination at different scales within territorial subsystems both independent and interdependent. Our approach is more general than the notion of congruence proposed by \\cite{offner1993effets}, which remains fuzzy in the interdependency relations between the entities concerned, and could be similar to the third level of systemic interdependencies.\n\n\n\n\n\n\n\n\\section{Macroscopic scale}\n\n\nThe first modeling axis relates to the macroscopic scale and is based on the principles of the evolutionary urban theory~\\citep{pumain1997pour}. The family of Simpop models is mainly situated in corresponding ontologies and scales, i.e. elementary entities constituted by cities themselves, at the spatial scale of the system of cities (regional to continental) and on relatively long time scales (longer than half a century)~\\citep{pumain2012multi}.\n\n\n\\subsection{Network effects}\n\nA first model that can be interpreted as a control, in which the network is static but has a retroaction on cities, indirectly suggests network effects. This preliminary work is detailed by~\\cite{raimbault2018indirect} which details the model and applies it to the French system of cities on a long time scale (1830-1999). The model is based on expected populations and captures complexity through non-linear interactions between cities, which are carried by the network. Three processes are added to determine the growth rate of cities: (i) an endogenous growth fixed by a parameter, corresponding to the Gibrat model ; (ii) direct interaction processes described as a gravity potential which influences the growth rate ; (iii) a retroaction of flows circulating in the network on the cities traversed. The model is initialized with real populations at the start of a period, and then evaluated by comparison to simulated populations on the full period, on two objectives which allow to take into account the adjustment of the total population or of their logarithm. The production of Pareto fronts shows that it is not possible to uniformly adjust the model for the full spectrum of city sizes. We show that the addition of the network component provides an effective increase in the adjustment, what suggests network effects.\n\n\n\\subsection{Co-evolution model}\n\n\nThis model is then extended to a co-evolutive model by~\\cite{raimbault2018modeling}, in which cities and links of the transportation network are both dynamic and within a reciprocal dependency. This model is close to~\\cite{schmitt2014modelisation} for the rules of population evolution, and to~\\cite{baptistemodeling} for the evolution of the network. More precisely, it operates iteratively with the following steps : (i) population of cities evolve according the the specification of the static model described above ; (ii) the network evolves, following an abstract implementation such that distances between cities are updated with a self-reinforcement function depending on flows between each city, with a threshold parameter. This version of the model is strictly macroscopic and does not include the spatial form of the network since it updates the distance matrix only.\n\n\nThe systematic exploration of this model using the OpenMOLE software~\\citep{reuillon2013openmole} and the application of an empirical method to characterize co-evolution~\\citep{raimbault2017identification} allow us to show that it captures a large variety of coupled dynamics, including indeed co-evolutive dynamics: among the broad range of interaction regimes between network variables and population variables (in the sense of \\cite{raimbault2017identification}, by classifying lagged correlation patterns), more than half of the regimes are effectively co-evolutive, i.e. exhibit circular causalities. This aspect could appear as trivial for a model conceived to integrate a co-evolution, but one has to realize that processes integrated in models are at the microscopic scale whereas the co-evolution is quantified at the macroscopic scale: it is indeed a property of the model to make co-evolution emerge. For example in the case of the SimpopNet model~\\citep{schmitt2014modelisation}, \\cite{raimbault2018unveiling} shows that co-evolution regimes are much more rare and that this other model produces more often configurations of the type structuring effects or without any relation. Furthermore, the exploration unveils the existence of an optimal value for an interaction range parameter, at which the system exhibits a maximal complexity of city trajectories. This range correspond to the appearance of territorial niches, within which populations are co-evolving, corresponding to the third point of our definition.\n\n\nThe calibration on the French system of cities on the same time period than the static model, with population data and dynamical railway network data taken into account with dynamical distance matrices constructed from the database of \\cite{thevenin2013mapping}, unveils Pareto fronts between the objective on the distance between cities and population objective, suggesting an impossibility to simultaneously calibrate this type of models for the network component and for the territorial component. Moreover, the adjustment on populations is improved by this model for a certain number of periods, compared to the model with a static network, suggesting the relevance of taking into account co-evolutive dynamics. The evolution of the calibrated threshold parameter suggests that the model captures a ``High Speed Rail (TGV) effect'', i.e. an increase in the effective accessibility for the metropolitan areas concerned but a loss of speed for the territories left behind.\n\n\n\\section{Mesoscopic scale}\n\n\nThe second axis, at the mesoscopic scale, considers the approach through urban morphogenesis, understood as the simultaneous emergence of the form and the function of a system \\citep{doursat2012morphogenetic}. It allows to consider a more precise description of territories, at the scale of fine population grids (500m resolution) and of a vectorial representation of the network at the same scale.\n\n\n\\subsection{Morphogenesis by aggregation-diffusion}\n\nThe territorial systems produced are quantified with morphological indicators for the population \\citep{le2015forme}. A first preliminary model, integrating only the population, shows that aggregation and diffusion processes are sufficient to explain a large majority of urban forms existing in Europe \\citep{raimbault2018calibration}. This result suggests that taking into account the form only can be achieved in an autonomous way, but that functional processes will then not be taken into account in the core of dynamics. In order to grasp morphogenesis processes, i.e. the strong link between form and function during the emergence of these, we follow the idea of using the transportation network as a proxy of functional properties of territories, in particular through centrality properties. This leads us to consider a morphogenesis model by co-evolution at the mesoscopic scale.\n\n\n\\subsection{Morphogenesis by co-evolution}\n\n\nThe urban form indicators computed on moving windows of size 50km for the whole Europe are completed with structural network indicators for the road network, computed from OpenStreetMap data after application of a specific simplification algorithm which conserves topological properties \\citep{raimbault2018urban}. These indicators and their static spatial correlations are thus computed on windows of a similar size covering the whole Europe. We show through the study of the spatial study of these correlations the non-stationarity of interaction processes at the second order, confirming the relevance of the notion of niche as a consistent territorial sub-system.\n\nWe then introduce in \\cite{raimbault2018urban} a morphogenesis model capturing the co-evolution of the spatial distribution of population and of the road network. This model combines the logic of \\cite{raimbault2018calibration} for the complementarity of aggregation and diffusion processes to the one of \\cite{raimbault2014hybrid} for the hybrid grid and vectorial network structure and also the influence of local explicative variables on the territorial evolution. This model works the following way: (i) morphological and functional local properties, integrated as local normalized explicative variables (including population, distance to the network, betweenness centrality, closeness centrality, accessibility), determine the value of a utility function from which new population is added through preferential attachment, and a diffusion of population is achieved; (ii) the road network evolves following rules depending on different heuristics (multi-modeling approach), which include, after the addition of nodes preferentially to the new population and their direct connection to the existing network, an addition of links with an heuristic among random links, random potential breakdown \\citep{schmitt2014modelisation}, deterministic potential breakdown \\citep{raimbault2016generation}, biological self-organized network \\citep{tero2010rules}, cost-benefit compromises \\citep{louf2013emergence}.\n\n\nThe computation of topological indicators for the road network allows to calibrate the model, and \\cite{raimbault2018multi} shows that the different network growth processes which have been included following the multi-modeling procedure are complementary to reach a maximum a real network configurations. Furthermore, the model is simultaneously calibrated on morphological indicators, topological indicators, and their correlation matrices, and the relatively low distances to data for a non negligible number of points suggest that the model is able to reproduce outcomes of processes at the first order (indicators) but also at the second order (interactions between indicators). Regarding the causality regimes produced by the model, we obtain configurations corresponding to a co-evolution, but a much lower diversity than for the more simple model of \\cite{raimbault2014hybrid} which was used as a toy model to show the relevance of the method of causality regimes in \\cite{raimbault2017identification}, suggesting a tension between static performance (reproduction of outcomes of processes) and dynamical performance (reproduction of processes themselves) for this kind of models.\n \n \n\n\\subsection{Transportation governance}\n \nFinally, a last metropolitan model (Lutecia model) is described in \\cite{raimbault2018caracterisation}, extending the one proposed by \\citep{lenechet:halshs-01272236}. It explores the role of governance processes in the growth of the transportation network, within a co-evolution model. Here, the metropolitan scale indeed corresponds to the mesoscopic scale. The collaboration between local actors for the construction of transportation infrastructures is included using game theory paradigms. This allows to simulate the emergence of the transportation network and its interaction with the urban form quantified by spatial accessibility patterns. This model allows for example to show that co-evolutive dynamics can in some case lead to the inversion of the behavior of accessibility gains in comparison to a situation without evolution of land-use, i.e. to qualitatively change the regime of the metropolitan system. The calibration of this model on the stylized case of the mega-urban region of Pearl River Delta in China allows to extrapolate on governance processes, and suggests that the shape of the current network is more likely to be the consequence either of fully regional decisions or of fully local decisions, but never an intermediate configuration, contradicting the view of the Chinese political context as a multi-level decision system~\\citep{liao2017ouverture}.\n\n\n\n\\section{Perspectives}\n\nThis research therefore developed complementary approaches at different scales of interactions between transportation networks and territories by modeling their co-evolution. We finally detail some immediate development perspectives opened by this work.\n\n\n\\subsection{Developments at the macroscopic scale}\n\n\nOur macroscopic models have not been tested yet on other urban systems and other time spans, and future developments will have to study which conclusions obtained here are specific to the French urban system on these periods, and which are more general are could be more generic in systems of cities. The application of the model to other systems of cities also recalls the difficulty to define urban systems. In our case, a strong bias must be induced by the fact of considering France only, since the insertion of its urban system within an European system is a reality we had to neglect. The span and scale of such models is always a difficult subject. We rely here on the administrative consistence and on the consistence of the database \\citep{pumain1986fichier}, but the sensitivity to the definition of the system and to its spatial extent must still be tested.\n\n\nFurthermore, the calibration used only the railway network for the distances between cities. Considering a single transportation mode is naturally a reduction, and an immediate direction for developments is testing the model with real distance matrices for other types of networks, such as the freeway network which has undergone a significant growth in France in the second half of the 20th century. This application would necessitate the construction of a dynamical database for the freeway network spanning 1950-2015, since classical bases (IGN or OpenStreetMap) do not integrate the opening date of segments. A natural extension of the model would then consist in the implementation of a multilayer network, what is a typical approach to represent multi-modal transportation networks~\\citep{gallotti2014anatomy}. Each layer of the transportation network should have a co-evolutive dynamic with populations, possibly with the existence of inter-layer dynamics.\n\n\n\n\\subsection{Developments at the mesoscopic scale}\n\n\nThe issue of the generic character of the co-evolution morphogenesis model is also opened, i.e. if it would work similarly to reproduce urban forms on very different systems such as the United States or China. A first interesting development would be to test it on these systems and at slightly different scales (1km cell size for example).\n\n\nFurthermore, the Lutecia model is also a fundamental contribution towards the inclusion of more complex processes implied in co-evolution, such as the governance of the transportation system. This model paves the way to a new generation of models, that could potentially become operational in the case of regional systems with a very high evolution speed such as in the Chinese case.\n\n\n\\subsection{Towards multi-scale models}\n\n\nOur work finally opens perspectives for integrated approaches, towards multi-scale models of these interactions, which appear to be more and more necessary for the construction of operational models that can be applied to the design of sustainable planning policies \\citep{rozenblat2018conclusion}.\n\n\nA first contribution towards multi-scalar models would be to take into account in a finer way the physical network in macroscopic models, what is for example the object of~\\cite{mimeur:tel-01451164}, which produces interesting results regarding the influence of the centralization of network investment decision-making on final forms, but keeps static populations and does not produce a co-evolution model. Similarly, the choice of indicators to quantify the distance of the simulated network to a real network is a difficult question in that context: indicators such as the number of intersections taken by~\\cite{mimeur:tel-01451164} is associated to procedural modeling and does not reflect structural indicators. It is probably for the same reason that~\\cite{schmitt2014modelisation} is only interested in population trajectories and not in network indicators: the superposition and adjustment of population and network dynamics at different scales remains an open problem.\n\n\nA second entry consists in the integration of the mesoscopic morphogenesis model into population of models in interaction. This approach would allow to take into account the non-stationarity of territorial systems that we moreover showed empirically. Here at the mesoscopic scale, total population and growth rate are fixed by exogenous conditions due to processes at the macroscopic scale. It is in particular the aim of spatial growth models such as the macroscopic model we already introduced to determine such parameters through the relations between cities as agents. It would then be possible to condition the morphological development of each area to the values of parameters determined at the upper level. In that context, one must remain careful on the role of emergence: should the emergent urban form influence the macroscopic behavior in its turn ? Such complex multi-scalar models are promising but must be considered with caution for the level of complexity required and the way to couple scales.\n\n\n\\section{Conclusion}\n\nWe did here a synthesis of main results of \\cite{raimbault2018caracterisation}, confirming the relevance of the co-evolution approach to focus on interactions between transportation networks and territories, in particular to model these interactions. We developed a precise definition of co-evolution, moreover associated to an empirical characterization method. The models at different scales, and the perspective of multi-scale models, can become precious tools for territorial prospective in the context of sustainable transitions on long time, allowing to quantify the possible territorial systems and the one desirables regarding sustainability, taking into account scales and processes still relatively poorly tackled in the literature.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The shift property}\nWe consider in this paper a property of Schauder bases that has come up on several\noccasions since the first construction of a truly non-classical Banach space\nby B. S. Tsirelson in 1974 \\cite{tsirelson}. It is a weakening of the property of  perfect homogeneity, which\nreplaces the condition \n\\begin{quote}\n{\\em all normalised block bases are equivalent}\n\\end{quote}\nwith the weaker \n\\begin{quote}\n{\\em all normalised block bases with the same growth rate are equivalent,}\n\\end{quote}\nand is satisfied by bases constructed along the lines of the Tsirelson basis,\nincluding the standard bases for the Tsirelson space and its dual.\n\n\nTo motivate our study and in order to fix ideas,  in the following result\nwe sum up a number of conditions that have been studied at various occasions in the literature and that can all be seen to be reformulations of the aforementioned property. Though I know of no single reference for the proof of the equivalence, parts of it are implicit in J. Lindenstrauss and L. Tzafriri's paper \\cite{lindenstrauss} and the paper by P. G.  Casazza, W. B. Johnson and L. Tzafriri \\cite{casazza}. Moreover, any idea needed for the proof can be found in, e.g., the book by F. Albiac and N. J. Kalton \\cite{albiac} (see Lemma 9.4.1, Theorem 9.4.2. and Problem 9.1) and the statement should probably be considered folklore knowledge.\n\\begin{thm}\\label{equivalences}\nLet $(e_n)_{n=1}^\\infty$ be a normalised unconditional Schauder basis for a Banach space $X$. Then the following conditions are equivalent.\n\\begin{enumerate}\n\\item Any block subspace is complemented.\n\\item Any block subspace $[x_n]_{n=1}^\\infty$ is complemented by a projection $P$ such that \n$$\nPz=\\sum_{n=1}^\\infty x^*_n(z)x_n,\n$$\nwhere $x_n^*\\in X^*$ satisfy ${\\rm supp}\\; x^*_n\\subseteq {\\rm supp}\\; x_n$.\n\\item  If $(x_n)_{n=1}^\\infty$ and $(y_n)_{n=1}^\\infty$ are normalised block sequences of $(e_n)_{n=1}^\\infty$ with\n$$\nx_1<y_1<x_2<y_2<\\ldots,\n$$\nthen $(x_n)_{n=1}^\\infty\\sim (y_n)_{n=1}^\\infty$.\n\\item If $(x_n)_{n=1}^\\infty$ is a normalised block basis, then $(x_{n})_{n=1}^\\infty\\sim(x_{n+1})_{n=1}^\\infty$.\n\\item If $(x_i)_{i=1}^\\infty$ and $(y_i)_{i=1}^\\infty$ are normalised block sequences such that \n$$\n\\max({\\rm supp}\\; x_i\\cup {\\rm supp}\\; y_i)<\\min({\\rm supp}\\; x_{i+1}\\cup {\\rm supp}\\; y_{i+1})\n$$\nfor all $i$, then $(x_i)_{i=1}^\\infty\\sim(y_i)_{i=1}^\\infty$.\n\\item For all normalised block bases $(x_n)_{n=1}^\\infty$, if $k_n\\in {\\rm supp}\\; x_n$ for all $n$,\nthen $(e_{k_n})_{n=1}^\\infty\\sim (x_n)_{n=1}^\\infty$.\n\\end{enumerate}\nMoreover, if any of the above properties hold, then they do so uniformly, e.g.,\nin (4) there is a constant $C$ such that for all normalised block bases\n$(x_n)_{n=1}^\\infty$, we have $(x_n)_{n=1}^\\infty\\sim_C(x_{n+1})_{n=1}^\\infty$. \n\\end{thm}\n\n\nAn unconditional basis satisfying the above equivalent conditions will be said to have the {\\em shift property}. This is a natural weakening of {\\em perfect homogeneity}, i.e., that all normalised block bases are equivalent, which was shown to be just a reformulation of being equivalent to the standard unit vector basis of $c_0$ or $\\ell_p$, $1\\leqslant p<\\infty$, by M. Zippin \\cite{zippin}.  Let us also note that the shift property is stronger than what it is called the {\\em block property} in \\cite{kutzarova}, which is the requirement that every block sequence is equivalent with some subsequence of the basis. Finally, we remark that the shift property is obviously hereditary, that is, any normalised block basis of an unconditional basis with the shift property will itself have the shift property. \n\n\nMoreover, while the canonical bases of both Tsirelson's space and its dual have the shift property, only one of them contains a {\\em minimal} subspace, i.e., an infinite-dimensional subspace that embeds into all of its further infinite-dimensional subspaces. On the other hand, recall that a space $E$ is {\\em locally minimal} \\cite{minimal} if there is a constant $K$ such that for all finite-dimensional\n$F\\subseteq E$ and infinite-dimensional $X\\subseteq E$, $F\\sqsubseteq_KX$,\ni.e., $F$ embeds with constant $K$ into $X$. As was pointed out in \\cite{minimal} (Proposition 6.7), the proof of \nTheorem 14 in  \\cite{casazza} essentially shows that any locally minimal space with a basis having the shift property is minimal.\n\n\n\nThe goal of the present paper is not to study the shift property per se, but rather to characterise the separable reflexive spaces that embed into a Banach space having a Schauder basis with the shift property. This will require some rather sophisticated techniques developed by E. Odell and T. Schlumprecht in a series of papers (see, e.g., \\cite{haydon, odell}) and that we shall  summarise and slightly develop here. As a first application of their techniques, they characterised in \\cite{odell} the separable reflexive Banach spaces embedding into an $\\ell_p$-sum of finite-dimensional spaces for $1<p<\\infty$ and their result was further improved in \\cite{IAG} to the following statement.\n\n\\begin{thm}[see \\cite{odell, IAG}]\\label{OS}\nLet $E$ be a separable reflexive Banach space such that any normalised weakly null tree $T$ in $E$ has a  branch $(x_i)_{i=1}^\\infty\\in [T]$ equivalent with all its subsequences. Then $E$ embeds into an $\\ell_p$-sum, $1<p<\\infty$, of finite-dimensional spaces.\n\\end{thm} \n\n\nThe result we shall obtain here has a weaker, though similar sounding hypothesis, but its conclusion is perhaps more satisfactory, since it provides a  basis rather than a finite-dimensional decomposition.\n\n\\begin{thm}\\label{ros}\nLet $E$ be a separable reflexive Banach space such that any normalised weakly null tree $T$ in $E$ has a  branch $(x_i)_{i=1}^\\infty\\in [T]$ satisfying $(x_{2i-1})_{i=1}^\\infty\\sim(x_{2i})_{i=1}^\\infty$. Then $E$ embeds into a reflexive space having an unconditional basis with the shift property.\n\\end{thm} \n\nIf the reader is not familiar with the techniques of Odell and Schlumprecht, this  should not be a hindrance to understanding the present construction, as we shall take certain of their technical results as black boxes that are directly applicable in our situation.\n\n\nWithout further introduction, let us commence the technical part of the paper by proving Theorem \\ref{equivalences} for the record and the convenience of the reader.\n\n\\begin{proof}\nThe implication (1)$\\Rightarrow$(2) follows directly from Lemma 9.4.1 in \\cite{albiac}, so we shall not repeat the proof here.\n\n(2)$\\Rightarrow$(3): Suppose (2) holds and $(x_n)_{n=1}^\\infty$ and $(y_n)_{n=1}^\\infty$ are normalised block sequences satisfying\n$$\nx_1<y_1<x_2<y_2<\\ldots.\n$$\nAssume that $a_n$ are scalars such that $\\sum_{n=1}^\\infty a_nx_n$ converges and choose $s_n>0$ converging to $0$ such that also \n$\\sum_{n=1}^\\infty  \\frac{a_n}{s_n} x_n$ converges. Put $w_n=x_n+s_ny_n$ and find $w_n^*\\in X^*$ such that ${\\rm supp}\\;w_n^*\\subseteq {\\rm supp}\\;w_n$ and \n$$\nPz=\\sum_{n=1}^\\infty w_n^*(z)w_n\n$$\ndefines a bounded projection onto $[w_n]_{n=1}^\\infty$, whence $\\sup\\norm{w_n^*}<\\infty$. Then\n$$\nP\\big(\\sum_{n=1}^\\infty  \\frac{a_n}{s_n}x_n\\big)=\\sum_{n=1}^\\infty  \\frac{a_n}{s_n}P(x_n)=\\sum_{n=1}^\\infty  \\frac{a_n}{s_n}w_n^*(x_n)w_n=\\sum_{n=1}^\\infty  \\frac{a_n}{s_n}w_n^*(x_n)(x_n+s_ny_n)\n$$\nand so the last series is norm convergent. By unconditionality, it follows that the series $\\sum_{n=1}^\\infty a_nw_n^*(x_n)y_n$ is norm convergent too. Thus, as \n$$\nw_n^*(x_n)=w_n^*(w_n)-w^*_n(s_ny_n)=1-s_nw_n^*(y_n)\\Lim{n\\rightarrow \\infty}1,\n$$\nusing unconditionality again, we find that also $\\sum_{n=1}^\\infty a_ny_n$ is norm convergent. A symmetric argument shows that if  $\\sum_{n=1}^\\infty a_ny_n$ converges, then so does  $\\sum_{n=1}^\\infty a_nx_n$, whence $(x_n)_{n=1}^\\infty$ and $(y_n)_{n=1}^\\infty$ are equivalent.\n\n\n(3)$\\Rightarrow$(4): Assume that (3) holds and that $(x_n)_{n=1}^\\infty$ is a normalised block sequence. Then using (3) \n$$\n(x_{2n-1})_{n=1}^\\infty\\sim (x_{2n})_{n=1}^\\infty\\sim (x_{2n+1})_{n=1}^\\infty.\n$$\nBy unconditionality, it follows that the sequence $(x_n)_{n=1}^\\infty$, which is the disjoint union of the sequences $(x_{2n-1})_{n=1}^\\infty$ and  $(x_{2n})_{n=1}^\\infty$, is equivalent to the sequence $(x_{n+1})_{n=1}^\\infty$, which itself is the disjoint union of the sequences $(x_{2n})_{n=1}^\\infty$ and  $(x_{2n+1})_{n=1}^\\infty$.\n\n(4)$\\Rightarrow$(5): If $(x_i)_{i=1}^\\infty$ and $(y_i)_{i=1}^\\infty$ are normalised block sequences such that \n$$\n\\max({\\rm supp}\\; x_i\\cup {\\rm supp}\\; y_i)<\\min({\\rm supp}\\; x_{i+1}\\cup {\\rm supp}\\; y_{i+1}),\n$$\nthen both $x_1,y_2,x_3,y_4,\\ldots$ and $x_2,y_3,x_4,y_5,\\ldots$ are normalised block sequences, whence $(x_{2i-1})_{i=1}^\\infty\\sim(y_{2i})_{i=1}^\\infty$ and $(x_{2i})_{i=1}^\\infty\\sim(y_{2i+1})_{i=1}^\\infty$. By unconditionality, it follows that $(x_i)_{i=1}^\\infty\\sim(y_{i+1})_{i=1}^\\infty\\sim (y_i)_{i=1}^\\infty$.\n\n\n\n\n\n\n(5)$\\Rightarrow$(6): Trivial.\n\n(6)$\\Rightarrow$(1): If (6) holds, then it does so uniformly, that is, there is a constant $C$ such that $(x_n)_{n=1}^\\infty\\sim_C(e_{k_{n}})_{n=1}^\\infty$ whenever $(x_n)_{n=1}^\\infty$ is a normalised block basis and $k_n\\in {\\rm supp}\\;x_n$.\nThis can easily be seen, as otherwise one would be able to piece together finite bits of sequences  with worse and worse constants of equivalence to get a counter-example to (6). Let also $K_u$ be the constant of unconditionality of $(e_n)_{n=1}^\\infty$.\n\nSuppose $(x_n)_{n=1}^\\infty$ is a normalised block sequence and let $I_1<I_2<I_3<\\ldots$ be a partition of $\\N$ into successive finite intervals such that ${\\rm supp}\\;x_n\\subseteq I_n$. Find also norm $1$ functionals $x_n^*\\in X^*$ such that ${\\rm supp}\\; x_n^*\\subseteq {\\rm supp}\\; x_n$ and $x_n^*(x_n)=1$. We claim that\n$$\nP(z)=\\sum_{n=1}^\\infty x_n^*(z)x_n\n$$\ndefines a projection of norm $\\leqslant K_uC^2$ from $X$ onto $[x_n]_{n=1}^\\infty$. To see this, suppose $z\\in X$ and write $z=\\sum_{n=1}^\\infty a_nz_n$, where the $z_n$ are normalised block vectors such that ${\\rm supp}\\;z_n\\subseteq I_n$. Modulo, perturbing $x_n$ and $z_n$ ever so slightly to get ${\\rm supp}\\; x_n=I_n={\\rm supp}\\; z_n$ and picking $k_n\\in I_n$, we see that $(x_n)_{n=1}^\\infty\\sim_C(e_{k_n})_{n=1}^\\infty\\sim_C(z_n)_{n=1}^\\infty$. So $\\sum_{n=1}^\\infty a_n x_n$ converges and, by unconditionality, so does $\\sum_{n=1}^\\infty x_n^*(z_n)a_n x_n=\\sum_{n=1}^\\infty x_n^*(z) x_n$. Therefore, $P$ is defined and satisfies\n\\[\\begin{split}\n\\norm{P(z)}&=\\norm{\\sum_{n=1}^\\infty x_n^*(z) x_n}=\\norm{\\sum_{n=1}^\\infty x_n^*(z_n) a_nx_n}\\\\\n&\\leqslant K_u\\norm{\\sum_{n=1}^\\infty a_nx_n}\\leqslant K_uC^2\\norm{\\sum_{n=1}^\\infty a_nz_n}=K_uC^2\\norm{z},\n\\end{split}\\]\nproving the estimate on the norm.\n\\end{proof}\n\n\n\n\n\n\n\nFinally, let us also remark that unconditionality is already implied by conditions (4), (5) and (6) of Theorem \\ref{equivalences}. E.g., if a normalised basis  $(e_n)_{n=1}^\\infty$ satisfies (4) and\n$(\\theta_n)_{n=1}^\\infty\\in\\{-1,1\\}^\\infty$, then \n$$\n(e_1,\\theta_1\ne_2,\\theta_1\\theta_2e_3,\\theta_1\\theta_2\\theta_3e_4,\\ldots)\\sim (\\theta_1\ne_2,\\theta_1\\theta_2e_3,\\theta_1\\theta_2\\theta_3e_4,\\theta_1\\theta_2\\theta_3\\theta_4e_5,\\ldots),\n$$\nand therefore, multiplying both sides with $(\\theta_1,\\theta_1\\theta_2,\\theta_1\\theta_2\\theta_3,\\theta_1\\theta_2\\theta_3\\theta_4,\\ldots)$, we have\n$$\n(\\theta_1 e_1,\\theta_2e_2,\\theta_3e_3,\\theta_4e_4,\\ldots)\\sim (e_2,e_3,e_4,e_5,\\ldots)\\sim(e_1,e_2,e_3,e_4,\\ldots).\n$$\nSince $(\\theta_n)_{n=1}^\\infty\\in\\{-1,1\\}^\\infty$ is arbitrary, this shows that $(e_n)_{n=1}^\\infty$ is unconditional.\n\n\n\n\n\n\nBefore continuing with the proof of Theorem \\ref{ros}, let us note that, while Theorem \\ref{ros} characterises reflexive spaces embeddable into a space with a basis having the shift property, we do not know of any significant characterisation of the spaces containing a basic sequence with the shift property. Using W. T. Gowers' block Ramsey theorem from \\cite{gowers} and Lemma 6.4 of \\cite{minimal}, we can conclude that if $X$ is a Banach space with a Schauder basis $(e_n)_{n=1}^\\infty$, then $X$ contains a normalised block sequence $(y_n)_{n=1}^\\infty$ that either is unconditional and has the shift property or such that there is  a non-empty tree $T$ consisting of finite normalised block sequences of $(y_n)_{n=1}^\\infty$ with the following property:\n\\begin{itemize}\n\\item[(a)] if $(z_1,\\ldots,z_m)\\in T$ and $Z$ is a block subspace of $[y_n]_{n=1}^\\infty$, then there is $z\\in Z$ such that $(z_1,\\ldots,z_m, z)\\in T$, and\n\\item[(b)] if $(z_1,z_2,z_3,\\ldots)$ is an infinite branch of $T$, then $(z_{2n-1})_{n=1}^\\infty\\not \\sim(z_{2n})_{n=1}^\\infty$.\n\\end{itemize}\nHowever, it is not clear what can be concluded from the existence of such a tree $T$ and one would like to draw stronger or more informative consequences from this.\n\\begin{prob}\nFormulate and prove a dichotomy that characterises the Banach spaces containing a unconditional basis sequence with the shift property.\n\\end{prob}\n\n\n\n\n\n\n\\section{Subspaces of spaces with an F.D.D.}\\label{subspaces of\nspaces with FDD's}\n\nWe fix in the following Banach spaces $E\\subseteq F$ and an F.D.D. $(F_i)_{i=1}^\\infty$ of $F$. For each interval $I\\subseteq \\N$, we let $I(x)$  denote the canonical projection of  $x\\in F$\nonto the subspace $\\sum_{i\\in I}F_i$ and shall also sometimes write\n$[\\sum_{i\\in I}F_i](x)$ for $I(x)$ if there is any chance of confusion. So, if $K$ denotes the constant of the decomposition $(F_i)_{i=1}^\\infty$, then $\\norm{I}\\leqslant 2K$ for any interval $I\\subseteq \\N$.\n\nFixing notation, if $A$ is a set, we let $A^\\infty$ denote the set of all infinite sequences $(a_i)_{i=1}^\\infty$ of elements of $A$ and let $A^{<\\infty}$ denote the set of all finite sequences $(a_1,\\ldots,a_n)$ of elements of $A$, including the empty sequence $\\tom$. A {\\em tree} on $A$ is a subset $T\\subseteq A^{<\\infty}$ closed under initial segments, i.e., such that  $(a_1,\\ldots,a_n)\\in T$ implies that $(a_1,\\ldots,a_m)\\in T$ for all $m\\leqslant n$. When $T$ is a tree on $A$, we let $[T]$ denote the set of all {\\em infinite branches} of $T$, i.e., the set of all sequences $(a_i)_{i=1}^\\infty$ such that $(a_1,\\ldots,a_{n})\\in T$ for all $n$.\n\nTo simplify notation, if $\\Delta=(\\delta_i)_{i=1}^\\infty$  \nis a decreasing sequence of real numbers $\\delta_i>0$ tending to $0$, we will denote this simply by $\\Delta\\searrow0$.\nSimilarly, if $M=(m_i)_{i=1}^\\infty$ is a strictly increasing sequence of natural numbers, we shall denote this by $M\\nearrow\\infty$.\n\n\n\nIf $\\B\\subseteq S_E^\\infty$ is a set of normalised sequences in $E$, we let \n$$\n\\B_\\Delta=\\big\\{(x_i)_{i=1}^\\infty\\in S_E^\\infty\\del \\e (y_i)_{i=1}^\\infty\\in \\B\\; \\a i\\; \\norm{x_i-y_i}<\\delta_i\\big\\}\n$$\nand\n$$\n{\\rm Int}_\\Delta(\\B)=\\big\\{(x_i)_{i=1}^\\infty\\in S_E^\\infty\\del \\a (y_i)_{i=1}^\\infty\\in S_E^\\infty\\; \\big(\\a i\\; \\norm{x_i-y_i}<\\delta_i\\rightarrow (y_i)_{i=1}^\\infty\\in  \\B\\big)\\big\\},\n$$\nand note that ${\\rm Int}_\\Delta(\\B)=\\;\\sim\\!(\\sim \\B)_\\Delta$, where the complement is taken with respect to $S_E^\\infty$.\n\n\n\n\n\\begin{defi}\\label{delta-block}\nGiven $\\Delta\\searrow0$, a normalised sequence $(x_i)_{i=1}^\\infty\\in S_E^\\infty$ is said to be\na $\\Delta$-{\\em block sequence} if there are intervals $I_i\\subseteq \\N$ such that\n$$\nI_1<I_2<I_3<\\ldots\n$$\nand for every $i$,\n$$\n\\|{I_i}(x_i)-x_i\\|<\\delta_i.\n$$\nMoreover, if $M\\nearrow \\infty$, we say that $(x_i)_{i=1}^\\infty$ is {\\em $M$-separated} if the witnesses $I_i\\subseteq \\N$ can be chosen such that \n$$\nm_1<I_1\\quad \\&\\quad \\a i\\;\\e j\\;\\;\\; I_i<m_j<m_{j+1}<I_{i+1}.\n$$\n\nWe let ${\\rm bb}_{E,\\Delta}(F_i)$ denote the set of  $\\Delta$-block sequences in $E$ and let ${\\rm bb}_{E,\\Delta, M}(F_i)$ denote the set of $M$-separated $\\Delta$-block sequences in $E$.\n\\end{defi}\n\nWe notice that if $K$ is the constant of the decomposition $(F_i)_{i=1}^\\infty$ and $(x_i)_{i=1}^\\infty$ and $(y_i)_{i=1}^\\infty$ are normalised sequences such that $ \\|x_i-y_i\\|<\\delta_i$ for all $i$, then if $(x_i)_{i=1}^\\infty$\nis a $\\Delta$-block, $(y_i)_{i=1}^\\infty$ is a $4K\\Delta$-block (with the same sequence of witnesses $I_1<I_2<\\ldots$).\n\nAlso, since $\\Delta\\searrow0$ is a decreasing sequence, the sets  ${\\rm bb}_{E,\\Delta}(F_i)$ and  ${\\rm bb}_{E,\\Delta, M}(F_i)$ are closed under taking subsequences, that is, if $(x_i)_{i=1}^\\infty\\in {\\rm bb}_{E,\\Delta, M}(F_i)$, as witnessed by a sequence $(I_i)_{i=1}^\\infty$, and $A\\subseteq \\N$, then $(I_i)_{i\\in A}$ witnesses that $(x_i)_{i\\in A}\\in {\\rm bb}_{E,\\Delta, M}(F_i)$. Lemma \\ref{all blocks} below essentially improves this to closure under taking normalised block sequences.\n\n\n\n\n\n\\begin{lemme}\\label{interior}\nSuppose $E$ is a subspace of a space $F$ with an F.D.D. $(F_i)_{i=1}^\\infty$. Let\n$\\B\\subseteq  S_E^\\infty$ be a set of sequences invariant under equivalence.\nThen there is a $\\Delta\\searrow0$ such that\n$$\n\\B_{\\Delta}\\cap {\\rm bb}_{E,\\Delta}(F_i)\\subseteq {\\rm Int}_{\\Delta}(\\B).\n$$\n\\end{lemme}\n\n\n\n\\begin{proof}Pick a $\\Delta\\searrow0$ depending on the constant of the decomposition $(F_i)_{i=1}^\\infty$ such that if $(y_i)_{i=1}^\\infty$ is a normalised block sequence in $F$ and $(v_i)_{i=1}^\\infty$ is a sequence in $F$ satisfying $\\norm{v_i-y_i}<5\\delta_i$ for all $i$, then $(v_i)_{i=1}^\\infty\\sim (y_i)_{i=1}^\\infty$. Assume also that $\\delta_i>2\\delta_i^2$ for every $i$.\n\n\nNow, suppose $(x_i)_{i=1}^\\infty\\in\\B_{\\Delta}\\cap {\\rm bb}_{E,\\Delta}(F_i)$ and let $(u_i)_{i=1}^\\infty$  be a normalised sequence in $E$ such that $\\norm{x_i-u_i}<\\delta_i$ for all $i$. We must show that $(u_i)_{i=1}^\\infty\\in \\B$, which will imply that $(x_i)_{i=1}^\\infty\\in {\\rm Int}_\\Delta(\\B)$.\n\n\nBy assumption on $(x_i)_{i=1}^\\infty$, we can find $(z_i)_{i=1}^\\infty\\in \\B$ and intervals $I_1<I_1<\\ldots$ such that $\\norm{x_i-z_i}<\\delta_i$  and $\\norm {{I_i}(x_i)-x_i}<\\delta_i$ for all $i$. Letting $y_i=\\frac{{I_i}(x_i)}{\\norm{{I_i}(x_i)}}$, we see that $(y_i)_{i=1}^\\infty$ is a normalised block sequence in $F$ and a simple calculation using $\\delta_i>2\\delta_i^2$ gives $\\norm{x_i-y_i}<4\\delta_i$, whence $\\norm{u_i-y_i}<5\\delta_i$ and $\\norm{z_i-y_i}<5\\delta_i$. It follows that \n$(u_i)_{i=1}^\\infty\\sim (y_i)_{i=1}^\\infty\\sim (z_i)_{i=1}^\\infty\\in \\B$ and so also $(u_i)_{i=1}^\\infty\\in \\B$.\n\\end{proof}\n\n\n\n\n\n\\begin{lemme}\\label{all blocks}\nSuppose $E$ is a subspace of a space $F$ with an F.D.D. $(F_i)_{i=1}^\\infty$ and $\\Theta\\searrow0$. Then there is $\\Gamma\\searrow0$ such that for any $M\\nearrow0$ and $(x_i)_{i=1}^\\infty\\in  {\\rm bb}_{E,\\Gamma, M}(F_i)$,\n\\begin{enumerate}\n\\item $(x_i)_{i=1}^\\infty$ is a normalised basic sequence, and\n\\item any normalised block sequence $(z_i)_{i=1}^\\infty$ of $(x_i)_{i=1}^\\infty$ belongs to ${\\rm bb}_{E,\\Theta,M}(F_i)$.\n\\end{enumerate}\n\\end{lemme}\n\n\n\\begin{proof}Let $K$ be the constant of the decomposition $(F_i)_{i=1}^\\infty$. As in the proof of Lemma \\ref{interior}, there is some $\\Lambda\\searrow0$ such that if $(x_i)_{i=1}^\\infty\\in {\\rm bb}_{E,\\Lambda}(F_i)$, as witnessed by a sequence of intervals $(I_i)_{i=1}^\\infty$, then\n$$\n(x_i)_{i=1}^\\infty\\sim_2\\Big(\\frac{I_ix_i}{\\norm{I_ix_i}}\\Big)_{i=1}^\\infty.\n$$\nLet now $\\Gamma\\searrow0$ be chosen such  that\n$12K^2\\sum_{i=m}^\\infty\\gamma_i<\\theta_m$ and $\\gamma_m<\\lambda_m$ for all $m$.\n\nNow suppose $(x_i)_{i=1}^\\infty\\in  {\\rm bb}_{E,\\Gamma, M}(F_i)$ for some $M\\nearrow \\infty$, as witnessed by a sequence of intervals\n$(I_i)_{i=1}^\\infty$. Then $(x_i)_{i=1}^\\infty\\in  {\\rm bb}_{E,\\Lambda}(F_i)$ and hence is $2$-equivalent to the normalised block basis $\\Big(\\frac{I_ix_i}{\\norm{I_ix_i}}\\Big)_{i=1}^\\infty$, whence $(x_i)_{i=1}^\\infty$ is itself a basic sequence.\n\n\nSuppose also that $z=\\sum_{i=n}^ma_ix_i$ is a block vector. We claim\nthat if we let $J=[\\min I_n,\\max I_m]$, then\n$$\n\\norm{Jz-z}<\\theta_n\\norm{z},\n$$\nwhich is enough to obtain condition (2). To see this, notice first that for $i=n,\\ldots m$,\n\\begin{displaymath}\\begin{split}\n\\norm{Jx_i-x_i}\n=&\\Norm{[1,\\min I_n-1](x_i)+[\\max I_m+1,\\infty[(x_i)}\\\\\n=&\\Norm{[1,\\min I_n-1](x_i-I_ix_i)+[\\max I_m+1,\\infty[(x_i-I_ix_i)}\\\\\n\\leqslant&\\Norm{[1,\\min I_n-1](x_i-I_ix_i)}+\\Norm{[\\max I_m+1,\\infty[(x_i-I_ix_i)}\\\\\n\\leqslant& K\\norm{x_i-I_ix_i}+2K\\norm{x_i-I_ix_i}\\\\\n<&3K\\gamma_i.\n\\end{split}\\end{displaymath}\nSince $\\Norm{P_{I_i}}\\leqslant 2K$ and $(x_i)_{i=1}^\\infty$ is $2$-equivalent to $\\Big(\\frac{I_ix_i}{\\norm{I_ix_i}}\\Big)_{i=1}^\\infty$, we have\n$$\n\\sup_{n\\leqslant i\\leqslant m}|a_i|\n=\\sup_{n\\leqslant i\\leqslant m}\\Norm{a_i\\frac{I_ix_i}{\\norm{I_ix_i}}}\n\\leqslant 2K\\Norm{\\sum_{i=n}^ma_i\\frac{I_ix_i}{\\norm{I_ix_i}}}\n\\leqslant 4K\\Norm{\\sum_{i=n}^ma_ix_i},\n$$\nand therefore\n\\begin{displaymath}\\begin{split}\n\\Norm{J(\\sum_{i=n}^ma_ix_i)-(\\sum_{i=n}^ma_ix_i)}\n=&\\Norm{\\sum_{i=n}^ma_i(Jx_i-x_i)}\\\\\n\\leqslant&\\sum_{i=n}^m|a_i|\\;\\norm{Jx_i-x_i}\\\\\n<& \\sup_{n\\leqslant i\\leqslant m}|a_i|\\cdot \\sum_{i=n}^m3K\\gamma_i\\\\\n\\leqslant& 12K^2\\Norm{\\sum_{i=n}^ma_ix_i}\\sum_{i=n}^m\\gamma_i\\\\\n\\leqslant&\\theta_n\\|\\sum_{i=n}^ma_ix_i\\|,\n\\end{split}\\end{displaymath}\nwhich shows that $\\norm{Jz-z}<\\theta_n\\norm z$.\n\\end{proof}\n\n\n\n\n\n\\begin{defi}\nGiven $\\Delta\\searrow0$, a $\\Delta$-{\\em block tree} $T$ is a non-empty tree on $S_E$ such that for all\n$(x_1,\\ldots,x_{n})\\in T$ the set\n$$\n\\{y\\in S_E\\del (x_1,\\ldots,x_{n},y)\\in T\\}\n$$\ncan be written as $\\{y_i\\}_{i=0}^\\infty$, where for each $i$ there is an interval $I_i\\subseteq \\N$ satisfying\n\\begin{itemize}\n\\item $\\norm{I_iy_i-y_i}<\\delta_{n+1}$,\n\\item $\\min I_i\\Lim{i\\rightarrow \\infty}\\infty$.\n\\end{itemize}\n\\end{defi}\n\nNow, an easy inductive construction shows that any $\\Delta$-block tree $T$ contains a subtree $T'\\subseteq T$ such that any infinite branch in $T'$ is a $\\Delta$-block sequence, i.e., $[T']\\subseteq {\\rm bb}_{E,\\Delta}(F_i)$.\nSo, without loss of generality, we can always assume that any $\\Delta$-block tree satisfies this additional hypothesis.\n\n\n\n\n\nWe recall the following result from \\cite{IAG}, which is proved using infinite-dimensional Ramsey theory. A similar statement for closed sets was proved earlier by Odell and Schlumprecht in \\cite{odell}.\n\n\n\\begin{thm}\\label{strategy for I}\nLet $\\B\\subseteq S_E^\\infty$ be a  coanalytic set. Then the following\nare equivalent.\n\\begin{enumerate}\n\\item $\\e \\Delta\\searrow0\\;\\;\\; \\e M\\nearrow\\infty\\quad  {\\rm bb}_{E,\\Delta,M}(F_i)\\subseteq   {\\rm Int}_\\Delta(\\B)$,\n\\item $\\e \\Delta\\searrow0$  such that any $\\Delta$-block tree has a branch in ${\\rm Int}_\\Delta(\\B)$.\n\\end{enumerate}\n\\end{thm}\n\n\n\n\n\n\\begin{defi}\nA {\\em weakly null tree} is a tree $T$ on $S_E$ such that, for any $(x_1,\\ldots,x_{n})\\in T$, the set\n$$\n\\{y\\in S_E\\del (x_1,\\ldots,x_{n},y)\\in T\\}\n$$\ncan be written as $\\{y_i\\}_{i=1}^\\infty$ for some weakly null sequence $(y_i)_{i=1}^\\infty$.\n\\end{defi}\n\n\n\n\n\nWe recall also a statement from \\cite{IAG} that sums up some of the elements of the construction of Odell and Schluprecht from \\cite{odell}\nthat we shall use in the following.\n\n\n\n\n\\begin{prop}\\label{wbjohnson}\nLet $E$ be a separable reflexive Banach space. Then there is a reflexive Banach space\n$F\\supseteq E$ having an F.D.D. $(F_i)_{i=1}^\\infty$  and a constant $c>1$ such\nthat whenever $\\Delta\\searrow0$ and $T$ is a $\\Delta$-block\ntree in $S_E$ with respect to $(F_i)_{i=1}^\\infty$, there is a weakly null tree $S$ in $S_E$ such that\n$$\n[S]\\subseteq [T]_{\\Delta c}\\quad\\&\\quad[T]\\subseteq [S]_{\\Delta c}.\n$$\n\\end{prop}\n\n\n\nWe can now assemble the above results into the following general lemma.\n\n\\begin{lemme}\\label{basic}\nSuppose $E$ is a separable reflexive Banach space and $\\B\\subseteq S_E^\\infty$ is a coanalytic set, invariant under equivalence, such that any weakly null tree on $S_E$ has a branch in $\\B$. \nThen there are $\\Gamma\\searrow0$, $M\\nearrow \\infty$ and  a reflexive space  $F\\supseteq E$ with an F.D.D. $(F_i)_{i=1}^\\infty$ such that  any element of ${\\rm bb}_{E,\\Gamma, M}(F_i)$ is a  basic sequence all of whose normalised block sequences belong to  $\\B$.\n\\end{lemme}\n\n\n\n\\begin{proof}\nPick first, by Proposition \\ref{wbjohnson}, a space $F$\ncontaining $E$ with a shrinking F.D.D. $(F_i)_{i=1}^\\infty$ and a constant $c>1$ such that,\nfor any $\\Delta\\searrow0$ and $\\Delta$-block tree $T$ in $E$, \nthere is a weakly null tree $S$ in $E$ with\n\\begin{equation}\\label{close}\n[S]\\subseteq [T]_{\\Delta c}\\quad\\&\\quad[T]\\subseteq [S]_{\\Delta c}.\n\\end{equation}\nChoose also, by Lemma \\ref{interior}, some  $\\Delta\\searrow0$ such that\n$$\n\\B_{\\Delta c}\\cap {\\rm bb}_{E,\\Delta c}(F_i)\\subseteq {\\rm Int}_{\\Delta c}(\\B).\n$$\n\nWe claim that any $\\Delta$-block tree has a branch in ${\\rm Int}_\\Delta(\\B)$. To see this, suppose\n$T$ is a $\\Delta$-block tree and assume without loss of generality that  $[T]\\subseteq {\\rm bb}_{E,\\Delta}(F_i)\\subseteq {\\rm bb}_{E,\\Delta c}(F_i)$.\nPick also a  weakly null tree $S$ satisfying (\\ref{close}). Then, as $[S]\\cap \\B\\neq \\tom$, also \n$$\n\\tom\\neq [T]\\cap \\B_{\\Delta c}\\subseteq[T]\\cap  {\\rm bb}_{E,\\Delta c}(F_i)\\cap \\B_{\\Delta c}\\subseteq [T]\\cap {\\rm Int}_{\\Delta c}(\\B)\\subseteq [T]\\cap {\\rm Int}_{\\Delta}(\\B),\n$$\nshowing that $T$ has a branch in ${\\rm Int}_\\Delta(\\B)$.\n\n\nApplying Theorem \\ref{strategy for I}, we find some $\\Theta\\searrow0$ and $M\\nearrow\\infty$ such that ${\\rm bb}_{E,\\Theta,M}(F_i)\\subseteq   {\\rm Int}_\\Theta(\\B)\\subseteq \\B$ and, applying Lemma \\ref{all blocks}, the statement follows.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Killing the overlap}\n\n\n\nThe next proposition is Corollary 4.4 in \\cite{odell}, except that condition (5) is not listed in the statement of the corollary. However, it can easily be gotten from the proof, provided that one chooses, in the notation of the paper, $\\epsilon_i<\\delta_i$.\n\n\\begin{prop}\\label{overlap}\nSuppose $F$ is a reflexive space with an F.D.D. $(H_i)_{i=1}^\\infty$, $E\\subseteq F$ is a subspace and $\\Sigma\\searrow0$. Then there are integers $0=a_0<a_1<\\ldots$ such that for all $x\\in S_E$ there are a sequence $(x_i)_{i=1}^\\infty$ in $E$, a subset\n$D\\subseteq \\N$ and numbers $a_{i-1}<b_i\\leqslant a_i$, $b_0=0$, satisfying the following five conditions.\n\\begin{enumerate}\n  \\item $x=\\sum_{i=1}^\\infty x_i$,\n  \\item $\\a i\\notin D\\; \\norm{x_i}<\\sigma_i$,\n  \\item $\\a i\\in D\\;\\Norm{[H_{b_{i-1}+1}\\oplus\\ldots\\oplus H_{b_i-1}]x_i-x_i}<\\sigma_i\\|x_i\\|$,\n  \\item $\\a i\\;\\Norm{[H_{b_{i-1}+1}\\oplus\\ldots\\oplus H_{b_i-1}]x-x_i}<\\sigma_i$,\n  \\item $\\a i\\; \\norm{H_{b_i}x}<\\sigma_i$.\n\\end{enumerate}\n\\end{prop}\n\n\n\n\n\nCombining Lemma \\ref{basic} and Proposition \\ref{overlap}, we are now in a position to prove our main result, Theorem \\ref{ros}.\n\n\n\n\n\n\\begin{thm}\nSuppose that $E$ is a separable reflexive Banach space such that any weakly\nnull tree in $E$ has a branch $(x_i)_{i=1}^\\infty$ satisfying $(x_{2i-1})_{i=1}^\\infty\\sim(x_{2i})_{i=1}^\\infty$. Then $E$\nembeds into a reflexive space with an unconditional Schauder basis having the shift property.\n\\end{thm}\n\n\n\\begin{proof}Applying Lemma \\ref{basic} to the set \n$$\n\\B=\\big\\{(x_i)_{i=1}^\\infty\\in S_E^\\infty\\del (x_{2i-1})_{i=1}^\\infty\\sim(x_{2i})_{i=1}^\\infty\\big\\},\n$$\nwe find $\\Gamma\\searrow0$, $M\\nearrow \\infty$ and a reflexive space  $F\\supseteq E$ with an F.D.D. $(F_i)_{i=1}^\\infty$ such that  any element of ${\\rm bb}_{E,\\Gamma, M}(F_i)$ is a  basic sequence all of whose normalised block sequences $(y_i)_{i=1}^\\infty$ satisfy $(y_{2i-1})_{i=1}^\\infty\\sim(y_{2i})_{i=1}^\\infty$.\n\nWe claim that there is a constant $C\\geqslant 1$ such that $(y_{2i-1})_{i=1}^\\infty\\sim_C(y_{2i})_{i=1}^\\infty$ for any such normalised block basis $(y_i)_{i=1}^\\infty$. For if not, then,  by concatenating finite bits of sequences, we would be able to produce  some $(u_i)_{i=1}^\\infty \\in {\\rm bb}_{E,\\Gamma, M}(F_i)$ and a normalised block sequence $(y_i)_{i=1}^\\infty$  of $(u_i)_{i=1}^\\infty$ failing \n$(y_{2i-1})_{i=1}^\\infty\\sim(y_{2i})_{i=1}^\\infty$, which is impossible.\n\n\nSince it suffices to prove the conclusion of the theorem for a cofinite-dimensional subspace of $E$, by considering the cofinite-dimensional subspaces $F_{m_1+1}\\oplus F_{m_1+2}\\oplus F_{m_1+3}\\oplus\\ldots $ and $E\\cap \\big(F_{m_1+1}\\oplus F_{m_1+2}\\oplus F_{m_1+3}\\oplus\\ldots\\big)$ of respectively $F$ and $E$, we can, without loss of generality, assume that $m_1=0$ and thus not worry about the initial offset by $m_1$ in the definition of $M$-separation (cf. Definition \\ref{delta-block}).\n\n\n\nPick $(u_i)_{i=1}^\\infty\\in {\\rm bb}_{E,\\Gamma,M}(F_i)$. Then, for any choice of signs $\\epsilon_i\\in\\{-1,1\\}$, also $(\\epsilon_iu_i)_{i=1}^\\infty\\in {\\rm bb}_{E,\\Gamma,M}(F_i)$ and hence $(\\epsilon_{2i-1}u_{2i-1})_{i=1}^\\infty\\sim(\\epsilon_{2i}u_{2i})_{i=1}^\\infty$. It follows that $(u_{2i-1})_{i=1}^\\infty$ is a basic sequence, equivalent to $(\\epsilon_{2i}u_{2i})_{i=1}^\\infty$ for any choice of signs $\\epsilon_i\\in \\{-1,1\\}$, and thus must be unconditional. \n\nNow $[u_{2i-1}]_{i=1}^\\infty$ can be equivalently renormed so that $(u_{2i-1})_{i=1}^\\infty$ is $1$-unconditional and, by a result of A. Pe\\l czy\\'nski \\cite{pelczynski}, this renorming extends to an equivalent renorming of $F$. So, without loss of generality, we shall assume that $(u_{2i-1})_{i=1}^\\infty$ is $1$-uncondi\\-tio\\-nal and has the shift property with some constant $C$.\nMoreover, as $E$ is reflexive, it follows by a theorem of R. C. James (Theorem 3.2.13 in \\cite{albiac}) that $(u_{2i-1})_{i=1}^\\infty$ is both shrinking and boundedly complete.\n\nWe let $v_i=u_{2i+1}$, whence $(v_i)_{i=1}^\\infty$ is the subsequence of $(u_{2i-1})_{i=1}^\\infty$ omitting the first term.\nChoose also $\\sigma_i<\\gamma_{2i-1}$ such that $\\sum_{i=1}^\\infty\\sigma_i<\\frac 1{24KC^2}$, , where $K$ denotes the constant of the decomposition $(F_i)_{i=1}^\\infty$.\n\n\n\nSince $(u_i)_{i=1}^\\infty$ is an $M$-separated $\\Gamma$-block sequence,  $\\N$ can be partitioned into successive finite intervals\n$$\nL_1<I_1<R_1\\;\\;\\;<\\;\\;\\;L_2<I_2<R_2\\;\\;\\;<\\;\\;\\;L_3<I_3<R_3\\;\\;\\;<\\ldots\n$$\nsuch that\n\\begin{itemize}\n\\item[(a)] $\\norm {I_i(v_i)-v_i}<\\gamma_{2i+1}$,\n\\item[(b)] for every $i>1$ there is a $j$ such that $[m_j,m_{j+1}]\\subseteq L_i$\n\\item[(c)] and for every $i$ there is a $j$ such that $[m_j,m_{j+1}]\\subseteq R_i$.\n\\end{itemize}\nMoreover, for \n$$\nH_i=\\sum_{j\\in L_i\\cup I_i\\cup R_i}F_j,\n$$\nlet $(a_i)_{i=0}^\\infty$ be given as in Proposition \\ref{overlap} and set\n$$\nA_i=H_{a_{i-1}+1}\\oplus\\ldots\\oplus H_{a_i}.\n$$\nWe define a new norm $\\triple\\cdot$ on ${\\rm span}(\\bigcup_{i=1}^\\infty A_i)$ by setting\n$$\n\\triple y=\\Big\\|\\sum_{i=1}^\\infty \\|A_iy\\|v_{a_i}\\Big\\|.\n$$\nSince $(v_i)_{i=1}^\\infty$ is  $1$-unconditional,  $\\triple\\cdot$ is\nindeed a norm and we can therefore consider the completion $V=\\overline{\\rm span}^{\\triple\\cdot}\\big(\\bigcup_{i=1}^\\infty A_i\\big)$.\nMoreover, we claim that the mapping \n$$\nT\\colon x\\in E\\mapsto \\sum_{i=1}^\\infty A_ix\\in V\n$$\nis a well-defined isomorphic embedding of $E$ into $V$.\n\n\n\n\nTo see this, suppose $x\\in S_E$ is fixed and let $(x_i)_{i=1}^\\infty$, $(b_i)_{i=0}^\\infty$ and $D\\subseteq \\N$\nbe given as in Proposition \\ref{overlap}. Let also\n$$\nB_i=H_{b_{i-1}+1}\\oplus\\ldots\\oplus H_{b_i-1}.\n$$\nThen the decomposition $F=F_1\\oplus F_2\\oplus F_3\\oplus\\ldots$ blocks as\n\\[\\begin{split}\nF=&A_1\\oplus A_2\\oplus A_3\\oplus\\ldots\\\\\n=&B_1\\oplus H_{b_1}\\oplus B_2\\oplus H_{b_2}\\oplus B_3\\oplus H_{b_3}\\oplus\\ldots,\n\\end{split}\\]\nwhere, moreover, \n$$\nA_i\\;\\subseteq\\; B_i\\oplus H_{b_i}\\oplus B_{i+1}\n$$\nand, letting $A_0$ be the trivial space $\\{0\\}$,\n$$\nB_{i}\\;\\subseteq\\; A_{i-1}\\oplus A_{i}.\n$$\nIt follows that with respect to the ordering of the original decomposition $(F_i)_{i=1}^\\infty$, we have\n\\begin{equation}\nB_1<L_{b_1}<I_{b_1}<R_{b_1}<B_2<L_{b_2}<I_{b_2}<R_{b_2}<B_3<\\ldots.\n\\end{equation}\nNow, by condition (4) of Proposition \\ref{overlap},\n$$\n\\big | \\|B_ix\\|-\\|x_i\\|\\big|\\leqslant \\|B_ix-x_i\\|<\\sigma_i,\n$$\nand so, using condition (5) of Proposition \\ref{overlap}, we have\n\\begin{displaymath}\\begin{split}\n\\|A_ix\\|\n\\leqslant&2K\\Norm{[B_i\\oplus H_{b_i}\\oplus B_{i+1}]x}\\\\\n\\leqslant& 2K \\big(\\|B_ix\\|+\\norm{H_{b_i}x}+\\|B_{i+1}x\\|\\big)\\\\\n<& 2K \\big(\\|x_i\\|+\\|x_{i+1}\\|+3\\sigma_i\\big).\n\\end{split}\\end{displaymath}\nNote also that\n$$\n\\|x_i\\|\\leqslant \\|B_ix\\|+\\sigma_i\\leqslant 2K\\|A_{i-1}x\\|+2K\\|A_ix\\|+\\sigma_i,\n$$\nand, by condition (3) of Proposition \\ref{overlap}, for any $i\\in D$, we have\n$$\n\\Norm{B_ix_i-x_i}<\\sigma_i\\norm {x_i}<\\gamma_{2i-1}\\norm {x_i}.\n$$\nList now $D$ increasingly as $D=\\{d_1,d_2,d_3,\\ldots\\}$ and note that, as $2i< 2b_{d_i}+1$,\n$$\n\\norm{I_{b_{d_i}}(v_{b_{d_i}})-v_{b_{d_i}}}<\\gamma_{2b_{d_i}+1}\\leqslant  \\gamma_{2i}.\n$$\nTherefore, by the ordering (2) above, we see that\n$$\n\\Big(\\frac{x_{d_1}}{\\norm{x_{d_1}}},v_{b_{d_1}},\\frac{x_{d_2}}{\\norm{x_{d_2}}},v_{b_{d_2}},\\frac{x_{d_3}}{\\norm{x_{d_3}}},v_{b_{d_3}},\\frac{x_{d_4}}{\\norm{x_{d_4}}},\\ldots\\Big)\n$$\nis an $M$-separated $\\Gamma$-block sequence, as witnessed by the sequence of interval projections\n$$\nB_{d_1},I_{b_{d_1}},B_{d_2},I_{b_{d_2}},B_3,I_{b_{d_3}},B_4,\\ldots,\n$$\nand hence $\\big(\\frac{x_{i}}{\\norm{x_{i}}}\\big)_{i\\in D}\\sim_C\\big(v_{b_i}\\big)_{i\\in D}$.\nFurthermore, as $(v_i)_{i=1}^\\infty$ has the shift property with constant $C$ and $b_1\\leqslant a_1<b_2\\leqslant a_2<\\ldots$, we have \n\\begin{equation}\\label{equiv}\n(v_{b_{i+1}})_{i=1}^\\infty\\sim_C(v_{b_i})_{i=1}^\\infty\\sim_C(v_{a_i})_{i=1}^\\infty\\sim_C(v_{a_{i+1}})_{i=1}^\\infty\n\\end{equation}\nand therefore $\\big(\\frac{x_{i}}{\\norm{x_{i}}}\\big)_{i\\in D}\\sim_{C^2}\\big(v_{a_i}\\big)_{i\\in D}$. Since now $\\sum_{i=1}^\\infty x_i$ converges and $\\sum_{i\\notin D}\\norm{x_i}<\\sum_{i\\notin D}\\sigma_i<\\infty$, it follows that also $\\sum_{i=1}^\\infty \\norm{x_i}v_{a_i}$ and  $\\sum_{i=1}^\\infty \\norm{x_{i+1}}v_{a_i}$ converge. Since $\\norm{A_ix}\\leqslant2K \\big(\\|x_i\\|+\\|x_{i+1}\\|+3\\sigma_i\\big)$ and $(v_i)_{i=1}^\\infty$ is unconditional,  we finally see that the sum $\\sum_{i=1}^\\infty \\norm{A_ix}v_{a_i}$ converges and hence that $Tx=\\sum_{i=1}^\\infty A_ix\\in V$ is well-defined.\n\nBy the same mode of reasoning, one verifies the following sequence of inequalities.\n\\begin{displaymath}\\begin{split}\n\\|x\\|\n=&\\Norm{\\sum_{i=1}^\\infty x_i}\\\\\n\\leqslant&\\Big\\|\\sum_{i\\in D}\\|x_i\\|\\frac{x_i}{\\|x_i\\|}\\Big\\|+\\Norm{\\sum_{i\\notin D} x_i}\\\\\n\\leqslant&C\\Big\\|\\sum_{i\\in D}\\|x_i\\|v_{b_i}\\Big\\|+\\sum_{i\\notin D}\\|x_i\\|\\\\\n\\leqslant&C^2\\Big\\|\\sum_{i=1}^\\infty \\|x_i\\|v_{a_i}\\Big\\|+\\sum_{i\\notin D}\\sigma_i\\\\\n\\leqslant&C^2\\Big\\|\\sum_{i=1}^\\infty \\big(2K\\|A_{i-1}x\\|+2K\\|A_ix\\|+\\sigma_i\\big)v_{a_i}\\Big\\|+\\frac14\\\\\n\\leqslant&2KC^2\\big\\|\\sum_{i=1}^\\infty \\|A_{i-1}x\\|v_{a_{i}}\\big\\|+2KC^2\\big\\|\\sum_{i=1}^\\infty \\|A_ix\\|v_{a_i}\\big\\|+C^2\\sum_{i=1}^\\infty \\sigma_i+\\frac14\\\\\n\\leqslant&2KC^2(C+1)\\big\\|\\sum_{i=1}^\\infty \\|A_ix\\|v_{a_i}\\big\\|+\\frac12\\\\\n\\leqslant&4KC^3\\triple{Tx}+\\frac12.\n\\end{split}\\end{displaymath}\nThus, as $\\|x\\|-\\frac 12=\\frac12\\|x\\|$, we have $\\|x\\|\\leqslant 8KC^3\\triple{Tx}$. \nA similar argument shows that $\\triple{Tx}\\leqslant 5KC^3\\norm{x}$, whereby $T$ is an isomorphic embedding of $E$ into $V$.\n\n\n\nWe shall now show how to embed $V$ into a space with a basis having the block property, which will finish the proof of the theorem.\nFirst, to simplify notation, we let $w_i=v_{a_i}$. Fix also  $k_i\\geqslant 1$ such that $A_i$ embeds\nwith constant $2$ into $Z_i=\\ell_\\infty^{k_i}$. Then $V$ clearly embeds with\nconstant $2$ into $Z=\\sum_{i=1}^\\infty Z_i$ equipped with the norm\n$$\n\\triple{y}'=\\Big\\|\\sum_{i=1}^\\infty\\|Z_iy\\|w_i\\Big\\|.\n$$\nMoreover, since $(w_i)_{i=1}^\\infty=(v_{a_i})_{i=1}^\\infty$ is both shrinking and boundedly complete, $Z$ is\nreflexive.\n\nFor each $i$, we let $(e_1^i,e_2^i,\\ldots,e_{k_i}^i)$ be the standard unit\nvector basis for $\\ell_\\infty^{k_i}$. Then\n$$\n(f_i)_{i=1}^\\infty=(e_1^1,e_2^1,\\ldots,e_{k_1}^1,e_1^2,e_2^2,\\ldots,e_{k_2}^2,\\ldots)\n$$\nis a $1$-unconditional basis for $Z$, which we claim has the block property.\nTo see this, suppose $(y_i)_{i=1}^\\infty$ is a normalised\nblock sequence of $(f_i)_{i=1}^\\infty$ and set $r_i=\\min {\\rm supp}\\; y_i$ and\n$$\ni\\in A\\Leftrightarrow \\e j\\;\\; y_i\\in Z_j.\n$$\nNotice that for all $j$ there are at most two distinct $i\\notin A$ such that\n$Z_jy_i\\neq 0$. We can therefore split $\\sim A$ into two sets $B$ and $D$ such that\nfor all $j$ there is at most one $i$ from each of $B$ and $D$ such that\n$Z_jy_i\\neq 0$. By unconditionality, it is enough to show that $(y_i)_{i\\in\nA}\\sim(f_{r_i})_{i\\in A}$, $(y_i)_{i\\in B}\\sim(f_{r_i})_{i\\in B}$ and\n$(y_i)_{i\\in D}\\sim(f_{r_i})_{i\\in D}$. Since the cases $B$ and $D$ are\nsimilar, let us just do $A$ and $B$.\n\n\nFor each $i\\in B$, let $n_i$ and $m_i$ be respectively the minimal and maximal $j$ such that $Z_jy_i\\neq0$, whence\n$y_i=Z_{n_i}y_i+\\ldots+Z_{m_i}y_i$ and $n_i<m_i<n_j<m_j$ for $i<j$ in $B$. In particular, this means that if\n$$\nz_i=\\sum_{j=n_i}^{m_i}\\|Z_jy_i\\|w_{j},\n$$\nthen $(z_i)_{i\\in B}$ is a block sequence of $(w_i)_{i=1}^\\infty$ and\n$$\n\\norm{z_i}=\\Norm{\\sum_{j=n_i}^{m_i}\\|Z_jy_i\\|w_{j}}=\\triple{y_i}'=1.\n$$ \nAs $(w_i)_{i=1}^\\infty$ has the shift property, this means that $(z_i)_{i\\in B}\\sim(w_{n_i})_{i\\in B}\\\n\\sim(f_{r_i})_{i\\in B}$.\nOn the other hand, if $(\\lambda_i)_{i=1}^\\infty\\in c_{00}$, then\n\\begin{displaymath}\\begin{split}\n\\Triple{\\sum_{i\\in B}\\lambda_iy_i}' \n=& \\Big\\|\\sum_{i\\in B}\\sum_{j=n_i}^{m_i}\\|Z_j\\lambda_iy_i\\|w_{j}\\Big\\|\\\\\n=& \\Big\\|\\sum_{i\\in B}|\\lambda_i|\\sum_{j=n_i}^{m_i}\\|Z_jy_i\\|w_{j}\\Big\\|\\\\\n=& \\Big\\|\\sum_{i\\in\nB}|\\lambda_i|z_i\\Big\\|.\n\\end{split}\\end{displaymath}\nSince $(z_i)_{i\\in B}$ is unconditional, it follows that $(y_i)_{i\\in\nB}\\sim(z_i)_{i\\in B}\\sim(f_{r_i})_{i\\in B}$.\n\nWe now partition $A$ into finite sets $a_j$ by setting\n$$\ni\\in a_j\\Leftrightarrow y_i\\in Z_j.\n$$\nThen for all $(\\lambda_i)_{i=1}^\\infty\\in c_{00}$\n\\begin{displaymath}\\begin{split}\n\\Triple{\\sum_{i\\in A}\\lambda_iy_i}' \n=& \\Big\\|\\sum_{j=1}^\\infty\\Norm{\\sum_{i\\in a_j}\\lambda_iy_i}w_{j}\\Big\\|\\\\\n=& \\Big\\|\\sum_{j=1}^\\infty\\;\\big(\\sup_{i\\in a_j}|\\lambda_i|\\big)w_{j}\\Big\\|\\\\\n=& \\Big\\|\\sum_{j=1}^\\infty\\Norm{\\sum_{i\\in a_j}\\lambda_if_{r_i}}w_{j}\\Big\\|\\\\\n=&\\Triple{\\sum_{i\\in A}\\lambda_if_{r_i}}'.\n\\end{split}\\end{displaymath}\nSo $(y_i)_{i\\in A}\\sim(f_{r_i})_{i\\in A}$, which finishes the proof.\n\\end{proof}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLearned image compression (LIC) has outperformed traditional methods like JPEG~\\cite{jpeg} and BPG~\\cite{bpg} in terms of PSNR and MS-SSIM. Its outstanding rate-distortion performance mostly derives from jointly training all parameters instead of handcrafted and independent module design. Hyperprior and context frameworks~\\cite{imagecnn5, imagecnn7, minnen2020channel, elic} dramatically improve compression gain across all ranges of bit rates. However, traditional methods have their own merits for industrial applications including smaller memory footprint, less computational complexity and hardware-friendly characteristics. LIC has an extremely high demand for hardware resources. Large model size and intensive computation hinder real-world deployment of LIC. Furthermore, correct decoding across platforms is not guaranteed due to the non-determinism of model inference and error can propagate catastrophically in entropy decoding~\\cite{balle2018integer}. All these obstacles restrict commercial deployment of LIC even though it obtains a better trade-off between bit rate and PSNR or MS-SSIM. In this paper, we report the performance of our optimized LIC implementation, which outperforms the fastest industrial-level codec implementation, i.e., jpeg-turbo~\\cite{jpegturbo} regarding rate-distortion performance and visual quality while achieving comparable speed. Compared with the most competitive next-generation image codec JPEG XL~\\cite{alakuijala2019jpeg}, our model shows superior compression performance with several times of advantage regarding inference speed.\n\nTo obtain a practical LIC model, we apply neural architecture search (NAS) and quantization to make LIC lighter and faster. NAS has been proved useful in finding efficient models for high-level vision tasks, such as object detection~\\cite{chen2019detnas},  recognition~\\cite{ding2020autospeech} and segmentation~\\cite{cubuk2019autoaugment}, while researches on NAS for low-level tasks such as denoising, super-resolution and image compression are still in early stages. In this paper, we aim to design a more efficient network and hardware-friendly architecture for LIC to enable its industrial deployment. Hardware-aware NAS is dedicated to designing promising networks for various devices.\n\nBesides, quantization has exhibited promising results for decreasing model size as well as accelerating inference. Methods for neural network quantization can be divided into two categories of quantization-aware training (QAT) and post-training quantization (PTQ). We exploit quantization not only to accelerate the inference of LIC but also to ensure the correctness of cross-platform decoding~\\cite{balle2018integer}. \n\nThe paper is organized as follows. Section \\uppercase\\expandafter{\\romannumeral2} introduces related works. Section \\uppercase\\expandafter{\\romannumeral3} gives an overview of our architecture and pipeline. Section \\uppercase\\expandafter{\\romannumeral4} describes the details of our method. Experimental results and discussion are given in section \\uppercase\\expandafter{\\romannumeral5}, and conclusion is drawn in section \\uppercase\\expandafter{\\romannumeral6}.\n\n\\section{Related Work}\n\\subsection{NAS}\nSearch space, optimization methods and search strategy are the major components of NAS. In this paper, we follow the framework of BigNAS~\\cite{yu2020bignas}, which only trains one big model and searches efficient sub-networks for direct deployment without additional training. For optimization methods, Yu \\emph{et al.}~\\cite{yu2018slimmable, yu2019universally} propose a slimmable neural network with parameter sharing which can adaptively prune output channels with little performance degradation. HyperNet~\\cite{ha2016hypernetworks} trains an auxiliary network to generate weight parameters for the trainable model. For search strategy,  an effective evolution method~\\cite{real2019regularized}  is proposed to obtain architecture with higher accuracy, and HyperBand~\\cite{li2017hyperband} can assess candidate neural architectures to avoid redundant training which results in worse performance.  \n\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{imgs\/framework.pdf}\n    \\caption{Framework of our architecture. $x, \\hat{x}$ are the original image and the reconstructed one. $y, \\hat{y}, z, \\hat{z}$ are latent variables. $\\mu, \\sigma$ are mean and variance values of Gaussian distribution. $Q$ represents quantization. \\textit{AE} and \\textit{AD} are the arithmetic encoding and decoding. The blue blocks are (transposed) convolutional layers controlled by \\textit{supernet} in Fig.~\\ref{fig:supernet}, and $k5s2$ means $5 \\times 5$ kernel with stride $2$. \\textit{ReLU, ReLU6} are the activation functions.}\n    \\label{fig:framework}\n\\end{figure}\n\n\\subsection{Quantization}\n\nQuantization-Aware Training (QAT)~\\cite{choi2018pact, gong2019differentiable, esser2019learned} and Post-Training Quantization (PTQ)~\\cite{nagel2020up, li2021brecq, wei2022qdrop} are two major categories of model quantization. QAT has attracted much attention because it compensates for the quantization error in an end-to-end optimization manner and achieves relatively better performance even in extremely low bit-width scenarios. Besides speeding up the model inference, the quantization techniques are also used in LIC for another purpose: eliminating the cross-platform inconsistency. Ball{\\'e} \\emph{et al.}~\\cite{balle2018integer} first introduces a QAT-like integer-only arithmetic technique to determinize the hyperprior model. Sun \\emph{et al.}~\\cite{sun2021learned} instead adopts PTQ-based approaches and He \\emph{et al.}\\cite{he2022post} extends the deterministic inference to more elaborate joint autoregressive models.\n\n\n\\section{Architecture}\nOur framework is shown in Fig.~\\ref{fig:framework}. We follow the pipeline in ~\\cite{imagecnn5} but use less complex activation functions and supernet to search the parameters of convolution layers, whose weight and bias are generated by supernet in Fig.~\\ref{fig:supernet}. \n\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=0.95\\linewidth]{imgs\/supernet.pdf}\n    \\caption{Supernet of our NAS architecture. $w, b$ are the weight and bias parameters. The notation $kmsn$ means $m \\times m$ kernel with stride $n$. \\textit{Search Space} is the combination of parameters for channels. \\textit{Linear} is the linear convolutional layer. \\textit{GeLU, Softmax} are the activation functions where $GeLU(x)=0.5 x(1+\\tanh{[\\sqrt{\\frac{\\pi}{2}} (x+0.004715 x^3)]})$, and $Softmax(x_i)=\\frac{e^{x_i}}{\\sum^l_{i=0} e^{x_i}}$, $l$ is the length of input vector $\\mathrm{x}$. The blue block is the convolution adopted in Fig.~\\ref{fig:framework}. }\n    \\label{fig:supernet}\n\\end{figure}\n\n\\section{Methodology}\n\n\\subsection{Conditional and Hardware-aware NAS}\nFirstly we define the search space. The minimal and maximal numbers of parameters to be searched are given empirically, and the search stride is predefined for each convolutional layer. BigNAS can train several models at the same time, and the sandwich rule with combinations of ${min, random, random, max}$ is embedded to guarantee the effectiveness of sub-models, where ${min, random}$ and ${max}$ denote the sub-models with minimal, random and maximal number of parameters. After training, we can directly deploy sub-models for validation. \n\nFor each convolutional layer in Fig.~\\ref{fig:framework}, weight and bias parameters are generated by supernet in Fig.~\\ref{fig:supernet}. A group of weights from the initial transposed convolution or convolution layers will be combined by \\textit{concat} operation. Then the dot product of concatenated tensor and latent variables extracted from search space parameters by multiple layer perception (MLP), which consists of a single \\textit{Linear} layer and a \\textit{Softmax} layer, will be calculated to generate weight. Meanwhile, bias will be directly generated by latent variables from MLP, which consists of two \\textit{Linear} layers and a \\textit{GeLU} activation layer. Then the original parameters are replaced with generated weight and bias for training and testing. \n\nDuring the searching process, we balance the rate-distortion performance and computational complexity by taking FLOPS and inference latency into consideration. FLOPS is calculated to restrict the model size and a latency lookup table obtained from the statistical characteristics of various operations on specific hardware is used to constrain the inference latency. \n\n\\subsection{Other Optimization}\\label{otheropt}\nEfforts have been made in engineering, resulting in a highly optimized software implementation. We leverage multi-threading and instruction-level parallelism to accelerate our implementation, reaching throughput higher than the reciprocal of latency. For entropy coding, multi-threading and AVX-512 are also used for acceleration.\n\n\\section{Experiments}\n\\subsection{Setup}\\label{setup}\nWe train our model with 8000 images picked from ImageNet~\\cite{deng2009imagenet} dataset and randomly crop each image to $256\\times256$. For evaluation, we use 24 Kodak~\\cite{1993kodak} images.\n\nFor training settings, we set learning rate to $1e-4$ and train the network for 2000 epochs. We follow~\\cite{minnen2020channel} and use the mixed quantization trick to train our network. Channel numbers in main and hyper networks are obtained with NAS. In terms of quantization experiments, we apply LSQ~\\cite{esser2019learned} to quantize all layers in our network following the general quantization rules of NVIDIA's TensorRT Framework. We use the full precision model to initialize weights. Then we train the network for 400 epochs with learning rate set to $1e-4$ and another 100 epochs with learning rate decreased to its $\\frac{1}{10}$. \n\nFor traditional codecs, we measure the latency for executing the whole pipeline including disk i\/o and png format conversion for a fair comparison.\nFor JPEG XL~\\cite{alakuijala2019jpeg}, we tune the distance parameter for rate-distortion trade-off and leave other parameters as default. For AVIF, we test with AOM codec by adjusting its quantization step tuned by PSNR unless otherwise specified. \n\nTo test the average latency of our LIC model and traditional codecs, we use 100 1080p images captured from a video and measure its average processing time. For a fair comparison, when comparing latency between LIC model and jpeg-turbo~\\cite{jpegturbo}, we test these images saved in \\emph{bmp} file format while we use images saved in \\emph{png} format when comparing LIC model, JPEG XL and AVIF. Unless otherwise mentioned, \nwe test on a Intel(R) Core(TM) i9-10900X CPU @ 3.70GHz and a Tesla T4 GPU for 1080p dataset with png format.   \n\\subsection{Rate Distortion Performance}\n\\begin{figure}[tb]\n\\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/bpp_psnr_avif_ssim.pdf}%\n        \\caption{BPP-PSNR curves of codecs}\n        \\label{fig:bpp-psnr}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/bpp_msssim_avif_ssim.pdf}%\n        \\caption{BPP-MS-SSIM curves of codecs}\n        \\label{fig:bpp-msssim}\n    \\end{subfigure}\n    \\caption{Performance of our LIC model compared with traditional codecs with respect to bpp v.s. psnr and bpp v.s. ms-ssim. ``avif-0.10.0 (ssim)\" denotes AVIF tuned by ssim.}\n\\end{figure}\nAs shown in Fig.~\\ref{fig:bpp-psnr}, our LIC model achieves superior performance with a large margin regarding BPP-PSNR compared with jpeg-turbo, especially in low bit rates. It also defeats JPEG XL at low bit rates. Its performance is still comparable to JPEG XL when bpp exceeds 1.0. For AVIF, our LIC model is comparable at low bit rates but lags as bit rate increases.\n\nOur model surpasses traditional codecs in terms of BPP-MS-SSIM, as shown in Fig.~\\ref{fig:bpp-msssim}. MS-SSIM has a closer relationship with perceptual quality~\\cite{cheng2019perceptual} and for deep networks, they tend to obtain a larger advantage over traditional codecs in terms of BPP-MS-SSIM, which can be verified from the fact that both LIC (Learned Image Compression) and DVC (Deep Video Compression) surpass traditional codecs in terms of BPP-MS-SSIM before BPP-PSNR~\\cite{imagecnn7, lu2019dvc}. We follow the same rationale by using a hybrid loss of PSNR and MS-SSIM~\\cite{sun2021hlic} to optimize perceptual quality, which leads to our model performing better than traditional codecs including AVIF tuned by ssim by a large margin in terms of BPP-MS-SSIM but worse than AVIF in terms of BPP-PSNR.\n\\subsection{Speed Analysis}\nWe compare the inference latency of our LIC model with traditional codecs and the results are shown in Fig.~\\ref{fig:latency}. Our model achieves \\textbf{18\/12ms} for encoding\/decoding latency, which is at most 20\\% higher in encoding latency and at least 35\\% lower in decoding latency than jpeg-turbo with \\emph{bmp} format input. Compared with JPEG XL and AVIF, our LIC model is several times faster, reaching \\textbf{50\/19ms} for encoding and decoding processes respectively. To further demonstrate the practicality of our implementation of the LIC model, we also test its latency on a less expensive GPU (NVIDIA GeForce GTX 1660 SUPER) and a less powerful CPU (Intel(R) Core(TM) i7-8700 CPU @ 3.20GHz). Encoding\/decoding latency is \\textbf{58\/25ms}. In addition, we have also tested the latency of our LIC model on CPU, reaching encoding\/decoding latency of \\textbf{199\/284ms, comparable to JPEG XL} as shown in Fig.\\ref{fig:latency-xl-avif}. We leave the optimization of CPU deployment as future work.\n\nIn terms of throughput, with techniques mentioned in Section \\ref{otheropt}, when processing images in parallel, our implementation reaches stunning \\textbf{145\/208fps} for \\emph{bmp} format input and \\textbf{126\/199fps} when input is saved in \\emph{png} format for encoding and decoding processes. On a less expensive GPU (NVIDIA GeForce GTX 1660 SUPER), our model still reaches throughput of \\textbf{92\/82fps}.\n\n\n\\begin{figure}[tb]\n\\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \\captionsetup{width=80mm}\n        \\includegraphics[width=\\linewidth]{imgs\/comparison_latency_jpeg_turbo.pdf}%\n        \\caption{Comparison of latency between our LIC model and jpeg-turbo for 1080p \\emph{bmp} format images.}\n        \\label{fig:latency-turbo}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \\captionsetup{width=80mm}\n        \\includegraphics[width=\\linewidth]{imgs\/comparison_latency_jpeg_xl_avif.pdf}%\n        \\caption{Comparison of latency between our LIC model, AVIF and JPEG XL for 1080p \\emph{png} format images.}\n        \\label{fig:latency-xl-avif}\n    \\end{subfigure}\n\\caption{Latency comparison of our LIC model and traditional codecs. \n``Quality\" stands for jpeg qp while ``Reference Bpp\" stands for bit rates on 1080p dataset for JPEG XL and AVIF.\n}\n\\label{fig:latency}\n\\end{figure}\n\nWe also report latency of each subnetwork in our model in Table.~\\ref{table:latency} which is obtained using TensorRT's trtexec benchmarking tool with an input of resolution $1088\\times1920$.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ c |c c c c}\n\\hline \\hline\n \\textbf{GPU Type}& $g_a$ & $g_s$ &$h_a$&$h_s$ \\\\\n \\hline\n \\textbf{Tesla T4} & 7.25&\t5.48&\t1.15&\t1.17 \\\\  \n \\textbf{GeForce GTX 1660 SUPER} & 10.08 & 10.75 & 1.68 & 2.80   \\\\\n \\hline\n\\end{tabular}\n\\caption{Latency of each subnetwork in our model. $g_a$, $g_s$, $h_a$ and $h_s$ above refer to analysis and synthesis transforms of main and hyperprior autoencoders.}\n\\label{table:latency}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Memory Analysis}\nWe show the memory footprint of our model during encoding and decoding processes of 100 1080p images in Fig.~\\ref{fig:mem}, peaking at 1.8GB and 1.5GB for CPU and GPU respectively. \n\\begin{figure}[tb]\n\\centering\n    \\begin{subfigure}[t]{0.23\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/cpu_mem.pdf}%\n        \\label{fig:cpu_mem}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.23\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/gpu_mem.pdf}%\n        \\label{fig:gpu_mem}\n    \\end{subfigure}\n    \\caption{CPU and GPU memory usage during encoding and decoding.}\n    \\label{fig:mem}\n\\end{figure}\n\n\\subsection{Visual Quality}\nIn this subsection, we compare the visual quality of our LIC model with that of traditional codecs on Kodak~\\cite{1993kodak} dataset. As can be seen in Fig.~\\ref{fig:kodim24}, the image compressed by JPEG~\\cite{jpeg} displays blocking artifacts while the image compressed by AVIF is oversmoothed, losing details. Image compressed by JPEG XL~\\cite{alakuijala2019jpeg} suffers from blocking artifacts, similar to JPEG, but to a lesser extent. The image compressed by our LIC model, on the other hand, shows better perceptual quality.\n\n\n\\begin{figure}[tb]\n\\centering\n    \\begin{subfigure}[t]{0.15\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/display\/kodim24_box.png}%\n        \\caption{Original image}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_30_512x768_box.png}%\n        \\caption{JPEG \\\\44.4KB}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_35_512x768.png}%\n        \\caption{AVIF \\\\43.75KB}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_3.35_512x768.png}%\n        \\caption{JPEG XL \\\\44.35KB}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/kodim24.png}%\n        \\caption{our LIC \\\\44.44KB}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_30_512x768_region.png}%\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_35_512x768_region.png}%\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/output_decoded_kodim24_3.35_512x768_region.png}%\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.11\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{imgs\/kodim24_region.png}%\n    \\end{subfigure}\n    \\caption{Visual quality comparison of kodim24.png~\\cite{1993kodak}. Images on the third row are obtained by zooming in the blue square box region of images above.}\n    \\label{fig:kodim24}\n\\end{figure}\n\\subsection{NAS}\n For NAS, we reduce about $10\\%$ inference time while improving rate-distortion performance.\n We get lower bpp ($0.7281 \\rightarrow 0.7278$), with higher PSNR ($33.5550 \\rightarrow 34.0997$) and higher MS-SSIM ($0.9876 \\rightarrow 0.9880$), but with lower latency ($15.2059 \\rightarrow 13.9574$) as shown in Table.~\\ref{tab:Bignas}.\n\n\\begin{table}[]\n\\setlength{\\tabcolsep}{3pt}\n\\begin{tabular}{ccccc}\n\\hline \\hline\nStructure   & $g_a$          & $h_a$       & $h_s$       & $g_s$     \\\\ \\hline\nOrigin      &48,96,112,176& 176,246,176 & 246,176,176 & 112,96,3  \\\\ \nNAS      & 32,120,104,220 & 248,224,256 & 236,200,220 & 112,112,3 \\\\ \\hline \\hline\nPerformance & BPP            & PSNR        & MS-SSIM     & Lantency(ms)  \\\\ \\hline\nOrigin      & 0.7281         & 33.5550     & 0.9876      & 15.2059   \\\\ \nNAS      & 0.7278         & 34.0997     & 0.9880      & 13.9547   \\\\ \\hline\n\\end{tabular}\n\\caption{Output channels of each (de)convolution block in $g_a$, $g_s$, $h_a$ and $h_s$  and rate-distortion performance with latency are listed above.}\\label{tab:Bignas}\n\\end{table}\n\n\\subsection{Quantization}\n\\begin{table}[]\n\\centering\n\\setlength{\\tabcolsep}{6pt}\n\\begin{tabular}{c|ccc}\n\\hline \\hline\nmodel   & bpp         & psnr     &ms-ssim       \\\\ \\hline\nFP32      &0.728 & 33.55 & 0.9876 \\\\ \nQuantized      & 0.704     & 32.91    & 0.9839 \\\\\n$g_a$-$h_a$-Quantized      &0.716\t&33.49&\t0.9873 \\\\\\hline\n\n\\end{tabular}\n\\caption{BPP-PSNR-MS-SSIM of quantized models. ``Quantized\" stands for the model with all layers quantized, evaluated on a NVIDIA GeForce GTX 1660 SUPER GPU. ``$g_a$-$h_a$-Quantized\" denotes the model with its main and hyper encoders quantized, evaluated in pytorch's fake quantize mode.} \n\\label{table:quant}\n\\end{table}\nWe show the rate-distortion performance of the quantized models in Table.~\\ref{table:quant}. We notice that quantizing main and hyper encoders barely hurt performance while quantizing all layers leads to a minor loss in rate-distortion performance. As for speedup on a real device, quantization brings about approximately 20\\% to 35\\% increase in throughput (images processed in parallel) on a NVIDIA GeForce GTX 1660 SUPER GPU, from 92\/82 fps to 114\/112 fps.\n\n\n\\section{Conclusion}\nWe evaluate the practicality of learned image compression by optimizing and assessing rate-distortion performance, latency, throughput, memory footprint and visual quality. We compare these results with those of traditional codecs. LIC has comparable or even faster inference speed compared with the fastest industrial codec, i.e., jpeg-turbo, while being superior in terms of rate-distortion performance and perceptual quality. Besides, LIC achieves astonishing throughput of 145\/208 fps for 1080p images, making it applicable for industrial scenarios. In addition, NAS and quantization show promising results for obtaining more efficient and more effective LIC models. All these results show that learned image compression is becoming ready for industrial applications. Future works will be devoted to optimizing LIC for mobile devices.\n\n\n\\balance\n{\\small\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}