---
abstract: 'We study the multi-channel Gel’fand-Calderón inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation $-\Delta \psi + v(x) \psi = 0$, $x\in D$, where $v$ is a smooth matrix-valued potential defined on a bounded planar domain $D$. We give an exact global reconstruction method for finding $v$ from the associated Dirichlet-to-Neumann operator.  This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.' 
address: 'CNRS (UMR 7641), Centre de Mathématiques Appliquées, École Polytechnique, 91128, Palaiseau, France'
author:
- 'Roman G. Novikov'
- Matteo Santacesaria
title: 'Global uniqueness and reconstruction for the multi-channel Gel’fand-Calderón inverse problem in two dimensions'
---

Introduction
============

Let $D$ be an open bounded domain in $\R^2$ with with $C^2$ boundary and let $v \in C^1(\bar D, M_{n}(\C))$, where $\matn$ is the set of the $n \times n$ complex-valued matrices.  The Dirichlet-to-Neumann map associated to $v$ is the operator $\Phi : C^1(\partial D, \matn) \to L^p(\partial D, \matn), \; p < \infty$ defined by: $$\Phi(f) = \left.\frac{\partial \psi}{\partial \nu}\right|_{\partial D}$$ where $f \in C^1(\partial D, \matn)$, $\nu$ is the outer normal of $\partial D$ and $\psi$ is the $H^1 (\bar D, \matn)$-solution of the Dirichlet problem $$\label{equa}
-\Delta \psi + v(x) \psi =0 \; \; \textrm{on } D, \; \; \psi|_{\partial D} = f;$$ here we assume that $$\label{direig}
0 \textrm{ is not a Dirichlet eigenvalue for the operator } - \Delta + v \textrm{ in } D.$$ Equation arises, in particular, in quantum mechanics, acoustics, electrodynamics; formally, it looks like the Schrödinger equation with potential $v$ at zero energy. 

In addition, comes up as a 2D-approximation for the 3D equation (see section \[sec3d\]). 

The following inverse boundary value problem arises from this construction. 

[**Problem 1.**]{}  Given $\Phi$, find $v$.

This problem can be considered as the Gel’fand inverse boundary value problem for the multi-channel 2D Schrödinger equation at zero energy (see [@G], [@N1]) and can also be seen as a generalization of the 2D Calderón problem for the electrical impedance tomography (see [@C], [@N1]).  In addition, the history of inverse problems for the two-dimensional Schrödinger equation at fixed energy goes back to [@DKN] (see also [@N3], [@Gr] and references therein).  Note also that Problem 1 can be considered as a model problem for the monochromatic ocean tomography (e.g. see [@Bur] for similar problems arising in this tomography). 

In the case of complex-valued potentials the global injectivity of the map $v \to \Phi$ was firstly proved in [@N1] for $D \subset \R^d$ with $d \geq 3$ and in [@B] for $d = 2$ with $v \in L^p$: in particular, these results were obtained by the use of global reconstructions developed in the same papers. 

This is the first paper which gives global (uniqueness and reconstruction) results for Problem 1 with $\matn$-valued potentials with $n \ge 2$. Results in this direction were only known for potentials with many restrictions (e.g. see [@X]). 

We emphasize that Problem 1 is not overdetermined, in the sense that we consider the reconstruction of a $\matn$-valued function $v(x)$ of two variables, $x \in D \subset \R^2$, from a $\matn$-valued function $\Phi(\theta,\theta')$ of two variables, $(\theta, \theta') \in \partial D \times \partial D$, where $\Phi(\theta,\theta')$ is the Schwartz kernel of the Dirichlet-to-Neumann operator $\Phi$: this is one of the principal differences between Problem 1 and its analogue for $D \subset \R^d$ with $d \ge 3$. At present, very few global results are proved for non-overdetermined inverse problems for the Schrödinger equation on $D \subset \R^d$ with $d \geq 2$. Concerning these results, our paper develops the two-dimensional works [@B], [@NS] and indicates 3D applications of the method.  The non-overdetermined inverse problems, including multi-channel ones, are much more developed for the Schrödinger equation in dimension $d =1$ (e.g. see [@AM], [@ZS]). 

We recall that in global results one does not assume that the potential $v$ is small in some sense or is (piecewise) real analytic or is subject to some other serious restrictions. 

Our global reconstruction procedure for Problem 1 follows the same scheme as in the scalar case given in [@N1], with some fundamental modifications inspired by [@B]. 

Let us identify $\R^2$ with $\C$ and use the coordinates $z= x_1 + i x_2, \; \bar z = x_1 - i x_2$, where $(x_1, x_2) \in \R^2$. We define a special family of solutions of equation , which we call the Buckhgeim analogues of the Faddeev solutions: $\psi_{z_0}(z,\lambda)$, for $z, z_0 \in \bar D, \lambda \in \C$, such that $-\Delta \psi + v (x) \psi =0$ over $D$, where in particular $\psi_{z_0}(z,\lambda) \to e^{\lambda (z-z_0)^2}I$ for $\lambda \to \infty$ (i.e. for $|\lambda| \to + \infty$) and $I$ is the identity matrix. 

More precisely, for a matrix valued potential $v$ of size $n$, we define $\psi_{z_0}(z,\lambda)$ as $$\psi_{z_0}(z,\lambda) = e^{\lambda (z-z_0)^2}\mu_{z_0}(z,\lambda),$$ where $\mu_{z_0}(\cdot,\lambda)$ solves the integral equation $$\label{defmu}
\mu_{z_0} (z,\lambda) = I+ \int_D g_{z_0}(z,\zeta,\lambda) v(\zeta) \mu_{z_0}(\zeta,\lambda) d \Ree \zeta \, d \Imm \zeta,$$ $I$ is the identity matrix of size $n \in \N$, $z,z_0 \in \bar D, \; \lambda \in \C$ and $$\label{defg}
g_{z_0}(z,\zeta, \lambda) = \frac{e^{\lambda(\zeta-z_0)^2-\bar \lambda(\bar \zeta - \bar z_0)^2}}{4 \pi^2} \int_D \frac{e^{-\lambda(\eta -z_0)^2+\bar \lambda(\bar \eta -\bar z_0)^2}}{(z- \eta)(\bar \eta -\bar \zeta)} d\Ree \eta \, d \Imm \eta$$ is a Green function of the operator $4\left( \frac{\partial}{\partial z} +2\lambda (z-z_0) \right) \frac{\partial}{\partial \bar z}$ in $D$, for $z_0 \in D$. We consider equation , at fixed $z_0$ and $\lambda$, as a linear integral equation for $\mu_{z_0}( \cdot, \lambda) \in C^1_{\bar z} (\bar D)$: we will see that it is uniquely solvable for $|\lambda| > \rho_1(D,N_1,n)$, where $\|v\|_{C^1_{\bar z}(\bar D, \matn)} < N_1$ (see Proposition \[prop3\]). 

In order to state the reconstruction method we also define the Bukhgeim analogue of the Faddeev generalized scattering amplitude $$\label{defh}
h_{z_0} (\lambda) = \int_D e^{\lambda (z-z_0)^2 - \bar \lambda (\bar z - \bar z_0)^2} v(z) \mu_{z_0} (z,\lambda) d \mathrm{Re}z \, d \mathrm{Im} z,$$ for $z_0 \in \bar D, \; \lambda \in \C$.

\[theo2\] Let $D \subset \R^2$ be an open bounded domain with $C^2$ boundary and let $v \in C^1 (\bar D, \matn)$ be a matrix-valued potential which satisfies and $v|_{\partial D} = 0$. Consider, for $z_0 \in D$, the functions $h_{z_0}$, $\psi_{z_0}$, $g_{z_0}$ defined above and $\Phi, \Phi_0$ the Dirichlet-to-Neumann maps associated to the potentials $v$ and $0$, respectively.  Then the following reconstruction formulas and equation hold: $$\begin{aligned}
 \label{recv}
v(z_0) &= \lim_{\lambda \to \infty} \frac{2 }{\pi}|\lambda| h_{z_0}(\lambda), \\ \label{rech}
h_{z_0} (\lambda) &= \int_{\partial D} e^{-\bar \lambda (\bar z - \bar z_0)} (\Phi - \Phi_0) \psi_{z_0} (z,\lambda) |dz|,\\ \label{bordpsi}
\psi_{z_0}(z,\lambda)|_{\partial D}&= e^{\lambda(z-z_0)^2}I + \int_{\partial D}G_{z_0}(z,\zeta,\lambda)(\Phi-\Phi_0) \psi_{z_0}(\zeta,\lambda) |d \zeta|,\end{aligned}$$ where $$\label{defG}
G_{z_0}(z,\zeta, \lambda) = e^{\lambda (z-z_0)^2}g_{z_0} (z,\zeta,\lambda)e^{-\lambda (\zeta-z_0)^2},$$ $z_0 \in D$, $z, \zeta \in \partial D$, $\lambda \in \C$, $|\lambda| > \rho_1(D,N_1,n)$, where $\|v\|_{C^1_{\bar z}(\bar D, \matn)} < N_1$.

In addition, if $v \in C^2(\bar D, \matn)$ with $\|v\|_{C^2(\bar D, \matn)} < N_2$ and $\frac{\partial v}{\partial \nu}|_{\partial D} = v|_{\partial D}= 0$ then the following estimates hold:

\[estimv\] $$\begin{aligned}
 \label{esttv}
\left|v(z_0) - \frac{2}{\pi}|\lambda| h_{z_0}(\lambda)\right| &\leq a(D,n) \frac{\log(3|\lambda|)}{|\lambda|^{1/2}}N_2(N_2+1), \\
\label{esttv2}
\left|v(z_0) - \frac{2}{\pi}|\lambda| h_{z_0}(\lambda)\right| &\leq b(D,n) \frac{(\log(3|\lambda|))^2}{|\lambda|^{3/4}}N_2(N_2^2+1),\end{aligned}$$

for $|\lambda| > \rho_2(D,N_1,n)$, $z_0 \in D$.

Note that in Theorem \[theo2\], $\rho_j = \rho_j(D,N_1,n)$, $j=1,2$ (where $\|v\|_{C^1_{\bar z}(\bar D, \matn)} < N_1$), are arbitrary fixed positive constants such that $$\begin{split}
2n\frac{c_2(D)}{ |\lambda|^{\frac 1 2}}\|v\|_{C^1_ {\overline z}(\bar D)} < 1,\ |\lambda|\ge 1, \textrm{ if} \; |\lambda| > \rho_1, \\
2n\frac{c_2(D)}{ |\lambda|^{\frac 1 2}}\|v\|_{C^1_ {\overline z}(\bar D)} \le \frac 1 2 ,\ |\lambda|\ge 1, \textrm{ if} \; |\lambda| > \rho_2,
\end{split}$$ where $c_2$ is the constant in Lemma \[lem1\]. 

Note that estimate is not strictly stronger than because of the presence of the $N_2^3$ factor. 

In order to make use of the reconstruction given by Theorem \[theo2\], the following two propositions are necessary:

\[prop2\] Under the assumptions of Theorem \[theo2\] (without the additional assumptions used for ), equation is a Fredholm linear integral equation of the second kind for $\psi_{z_0} \in C(\partial D)$.

\[prop3\] Under the assumptions of Theorem \[theo2\] (without the additional assumptions used for ), for $|\lambda| > \rho_1(D,N_1,n)$, where $\|v\|_{C^1_{\bar z}(\bar D, \matn)} < N_1$, equations and are uniquely solvable in the spaces of continuous functions on $\bar D$ and $\partial D$, respectively. 

\[rem3\] Note that the assumption that $v|_{\partial D} = 0$ is unnecessary for formula , equation and Propositions \[prop2\], \[prop3\].  In addition, formula also holds without this assumption if $$\label{condbord}
\int_{\partial D} e^{\lambda (z-z_0)^2 - \bar \lambda (\bar z - \bar z_0)^2} w(z)|dz| \to 0 \qquad \text{as } |\lambda| \to \infty,$$ for fixed $z_0 \in D$ and each $w \in C^1(\partial D)$. The class of domains $D$ for which holds for each $z_0 \in D$ is large and includes, for example, all ellipses. 

Note also that if $v|_{\partial D} \neq 0$ but $v \equiv \Lambda \in \matn$ on some open neighborhood of $\partial D$ in $\bar D$, then estimates hold with $h_{z_0} (\lambda)$ replaced by $$\label{hplus}
h^+_{z_0}(\lambda) = h_{z_0}(\lambda) + \int_{\R^2 \setminus D} e^{\lambda(z-z_0)^2 - \bar \lambda (\bar z - \bar z_0)^2} \Lambda \chi(z) d \Ree z \, d \Imm z,$$ where $\chi \in C^2(\R^2 , \R)$, $\chi \equiv 1$ on $D$, $\mathrm{supp} \chi$ is compact, and with the constants $a,b$ depending also on $\chi$. The aforementioned matrix $\Lambda$, for example, can be related with a diagonal matrix composed by the eigenvalues $\{ \lambda_i \}_{1 \leq i \leq n}$ arising in section \[sec3d\]. 

Theorem \[theo2\] and Propositions \[prop2\], \[prop3\] yield the following corollary:

\[theo1\] Let $D \subset \R^2$ be an open bounded domain with $C^2$ boundary, let $v_1, v_2 \in C^1(\bar D, \matn)$ be two matrix-valued potentials which satisfy and $\Phi_1, \Phi_2$ the corresponding Dirichlet-to-Neumann operators.  If $\Phi_1 = \Phi_2$ then $v_1 = v_2$.

Theorem \[theo2\], Propositions \[prop2\], \[prop3\] and Corollary \[theo1\] are proved in section \[secpf\]. 

The global reconstruction of Theorem \[theo2\] is fine in the sense that is consists in solving Fredholm linear integral equations of the second type and using explicit formulas; nevertheless this reconstruction is not optimal with respect to its stability properties: see [@BRS], [@N4], [@BKM] for discussions and numerical implementations of the aforementioned similar (but overdetermined) reconstruction of [@N1] for $d=3$ and $n=1$. An approximate but more stable reconstruction method for Problem 1 will be published in another paper. 

The present paper is focused on global uniqueness and reconstruction for Problem 1 for $n \geq 2$. In addition, using the techniques developed in the present work and following the scheme of [@NS] it is also possible to obtain a global logarithmic stability estimate for Problem 1 in the multi-channel case.  Following inverse problem traditions (e.g. see [@A], [@N4], [@NS]) this result will be published in another paper. 

[**Acknowledgements.**]{}  We thank V. A. Burov, O. D. Rumyantseva, S. N. Sergeev for very useful discussions. 

Approximation of the 3D equation {#sec3d}
================================

In this section we recall how the multi-channel two-dimensional Schrödinger equation can be seen as an approximation of the scalar 3D equation in a cylindrical domain; in this framework, three-dimensional inverse problems can be approximated by two-dimensional ones. 

Let $L= [a,b]$ for some $a,b \in \R$ and consider the complex-valued potential $v(x,z)$ defined on the set $D \times L$, where $x = (x_1, x_2) \in D \subset \R^2, \; z \in L$. We consider the equation $$\label{3d}
-\Delta \psi(x,z) + v(x,z) \psi(x,z) = 0 \quad \textrm{in } D \times L.$$ Now, for every $x \in D$ we can write $\psi(x,z) = \sum_{j=1}^{\infty} \psi_j(x) \phi_j(z)$, where $\{\phi_j\}$ is the orthonormal basis of $L^2 (L)$ given by the eigenfunctions of $- \frac{d^2}{dz^2}$: more precisely $$\begin{aligned}
 \label{cond1}
-\frac{d^2}{dz^2} \phi_j(z) &= \lambda_j \phi_j(z) \textrm{ for } z \in L, \\ \label{cond2}
\phi_j|_{\partial L} &= 0 \quad \textrm{(for example)} \\ \nonumber
\int_{L}\bar \phi_i(z) \phi_j(z) dz &= \delta_{ij} \end{aligned}$$ and $\psi_j(x) = \int_L \psi(x,z) \bar \phi_j(z) dz$. Now equation reads $$\begin{aligned}
 \label{cond3}
\sum_{j=1}^{\infty} \left( -\Delta_x \psi_j(x) \phi_j(z) - \psi_j(x) \Delta_z \phi_j(z) \right) + v(x,z)\sum_{j=1}^{\infty} \psi_j(x) \phi_j(z) = 0.\end{aligned}$$ Using - and the properties of $\{ \phi_j(z) \}$, we obtain that equation is equivalent to the following infinite-dimensional system $$\begin{aligned}
-\Delta_x \psi_i(x) + \lambda_i \psi_i(x) + \sum_{j=1}^{\infty} V_{ij}(x) \psi_j(x) = 0, \textrm{ for } i = 1,\ldots,\end{aligned}$$ where $$\begin{aligned}
V_{ij}(x) = \int_L \bar \phi_i(z) v(x,z) \phi_j(z) dz.\end{aligned}$$ Notice that if $\bar v =v$ then $V^{\ast} = V$. Now, if we impose $1 \leq i,j \leq n$ for some $n \in \N$, we find equation . 

We also give here the relation between the Dirichlet-to-Neumann (D-t-N) operators of the 3D equation and that of the 2D multi-channel equation.  If $\Phi(\theta,z, \theta',z')$ is the Schwartz kernel of the D-t-N operator of the 3D problem, and $(\Phi_{ij}(\theta,\theta'))_{i,j \geq 1}$ that of the 2D infinity-channel problem, we have $$\label{diri3}
\Phi_{ij}(\theta,\theta') = \int_{L \times L} \Phi(\theta,z, \theta',z')\bar \phi_i(z) \phi_j(z')dz \, dz',$$ where $\theta, \theta' \in \partial D$, $z,z' \in L$. This follows from $$\begin{aligned}
\int_{\partial D \times L} \Phi(\theta,z,\theta',z')f(\theta',z')d\theta' \, dz' = \sum_{i=1}^{\infty} \left( \sum_{j=1}^{\infty} \int_{\partial D} \Phi_{ij}(\theta, \theta')f_j(\theta') d\theta' \right) \phi_i(z),\end{aligned}$$ for every $f \in C^{1}(\partial (D \times L))$ such that $f|_{D \times \partial L} = 0$ and $f(\theta,z) = \sum_{j=1}^{\infty} f_j(\theta) \phi_j(z)$.

Let us remark that reductions of 3D direct and inverse problems to multi-channel 2D problems are well known in the physical literature for a long time (e.g. see [@Bur]).  Nevertheless, we do not know a reference containing formula in its precise form. 

Preliminaries
=============

In this section we introduce and give details about the above-mentioned family of solutions of equation , which will be used throughout all the paper. 

Let us define the function spaces $C^1_{\bar z}(\bar D) = \{ u : u, \frac{\partial u}{ \partial \bar z} \in C(\bar D, \matn) \}$ with the norm $\|u \|_{C^1_{\bar z}(\bar D)} = \max ( \|u\|_{C(\bar D)}, \| \frac{\partial u}{\partial \bar z} \|_{C(\bar D)} )$, $\|u\|_{C(\bar D)} = \sup_{z \in \bar D} |u|$ and $|u| = \max_{1 \leq i,j \leq n} |u_{i,j}|$; we define also $C^1_{z}(\bar D) = \{ u : u, \frac{\partial u}{ \partial z} \in C(\bar D, \matn) \}$ with an analogous norm. 

The functions $G_{z_0}(z,\zeta, \lambda), \; g_{z_0}(z,\zeta, \lambda), \; \psi_{z_0}(z,\lambda), \; \mu_{z_0}(z,\lambda)$ defined in Section 1, satisfy

$$\begin{aligned}
 \label{eq21}
4 \frac{\partial^2}{\partial z \partial \bar z} &G_{z_0} (z, \zeta, \lambda) = \delta(z-\zeta), \\ \label{zetaaa}
4\frac{\partial^2}{\partial \zeta \partial \bar \zeta} &G_{z_0}(z,\zeta,\lambda) = \delta(\zeta-z),\\ \label{eq22}
4\left(\frac{\partial}{\partial z} + 2\lambda (z-z_0) \right) \frac{\partial}{\partial \bar z} &g_{z_0}(z,\zeta, \lambda) = \delta(z-\zeta), \\
4\frac{\partial}{\partial \bar \zeta} \left(\frac{\partial}{\partial \zeta} - 2\lambda (\zeta-z_0) \right) &g_{z_0}(z,\zeta, \lambda) = \delta(\zeta-z),\\ \label{eq23}
-4 \frac{\partial^2}{\partial z \partial \bar z} &\psi_{z_0} (z,\lambda) + v(z) \psi_{z_0}(z,\lambda) = 0,\\ \label{eq24}
-4 \left(\frac{\partial}{\partial z} + 2\lambda (z-z_0) \right) \frac{\partial}{\partial \bar z} &\mu_{z_0}(z, \lambda) + v(z) \mu_{z_0} (z,\lambda) = 0,\end{aligned}$$

where $z, z_0 , \zeta \in D$, $\lambda \in \C$, $\delta$ is the Dirac’s delta. ( In addition, it is assumed that is uniquely solvable for $\mu_{z_0}( \cdot, \lambda) \in C^1_{\bar z} (\bar D)$ at fixed $z_0$ and $\lambda$.) Formulas - follow from , , and from $$\begin{aligned}
\frac{\partial}{\partial \bar z} &\frac{1}{\pi (z-\zeta)} = \delta(z-\zeta), \\
\left(\frac{\partial}{\partial z} + 2\lambda (z-z_0) \right) &\frac{e^{-\lambda(z-z_0)^2+\bar \lambda (\bar z- \bar z_0)^2}}{\pi (\bar z -\bar \zeta) } e^{\lambda(\zeta - z_0)^2- \bar \lambda(\bar \zeta -\bar z_0)^2}  = \delta(z-\zeta),\end{aligned}$$ where $z, \zeta, z_0 , \lambda \in \C$.

We say that the functions $G_{z_0}$, $g_{z_0}$, $\psi_{z_0}$, $\mu_{z_0}$, $h_{z_0}$ are the Bukhgeim-type analogues of the Faddeev functions (see [@NS]).  We recall that the history of these functions goes back to [@F] and [@BC]. 

Now we state some fundamental lemmata.  Let $$\label{green}
g_{z_0, \lambda} u(z) = \int_D g_{z_0} (z, \zeta, \lambda) u(\zeta) d \mathrm{Re}\zeta \, d \mathrm{Im}\zeta, \; z \in \bar D, \; z_0, \lambda \in \C,$$ where $g_{z_0}(z, \zeta, \lambda)$ is defined by and $u$ is a test function. 

\[lem1\] Let $g_{z_0, \lambda} u$ be defined by .  Then, for $z_0, \lambda \in \C$, the following estimates hold: $$\begin{aligned}
 \label{contg}
&g_{z_0, \lambda} u \in C^1_{\bar z}(\bar D), \quad \textrm{for} \; u \in C(\bar D), \\ \label{estcontg}
\| &g_{z_0, \lambda} u \|_{C^1(\bar D)} \leq c_1(D,\lambda) \|u\|_{C(\bar D)}, \quad \textrm{for} \; u \in C(\bar D), \\ \label{est1}
\| &g_{z_0, \lambda} u \|_{C^1_{\bar z}(\bar D)} \leq \frac{c_2(D)}{|\lambda|^{\frac 1 2}} \|u \|_{C^1_{\bar z}(\bar D)}, \quad \textrm{for} \; u \in C^1_{\bar z}(\bar D), \; |\lambda| \geq 1. 
$$

Given a potential $v \in C^1_{\bar z}(\bar D)$ we define the operator $g_{z_0,\lambda} v$ simply as $(g_{z_0,\lambda} v) u(z) = g_{z_0,\lambda} w(z), \; w=vu$, for a test function $u$. If $u \in C^1_{\bar z}(\bar D)$, by Lemma \[lem1\] we have that $g_{z_0,\lambda} v : C^1_{\bar z}(\bar D)  \to C^1_{\bar z}(\bar D)$, $$\label{estsolmu}
\| g_{z_0,\lambda} v \|^{op}_{C^1_{\bar z}(\bar D)} \leq 2n \| g_{z_0,\lambda} \|^{op}_{C^1_{\bar z}(\bar D)} \|v\|_{C^1_{\bar z}(\bar D)},$$ where $\| \cdot \|^{op}_{C^1_{\bar z}(\bar D)}$ denotes the operator norm in $C^1_{\bar z}(\bar D)$, $z_0, \lambda \in \C$. In addition, $\| g_{z_0,\lambda} \|^{op}_{C^1_{\bar z}(\bar D)}$ is estimated in Lemma \[lem1\].  Inequality and Lemma \[lem1\] implies existence and uniqueness of $\mu_{z_0}(z, \lambda)$ (and thus also $\psi_{z_0}(z,\lambda)$) for $|\lambda| > \rho_1(D,N_1,n)$.

Let $$\begin{aligned}
\mu^{(k)}_{z_0}(z, \lambda) &= \sum_{j=0}^k (g_{z_0,\lambda} v)^j I, \\
h^{(k)}_{z_0}(\lambda) &= \int_D e^{\lambda (z-z_0)^2 -\bar \lambda (\bar z- \bar z_0)^2}v(z) \mu^{(k)}_{z_0}(z,\lambda) d\Ree z \, d \Imm z,\end{aligned}$$ where $z,z_0 \in D$, $\lambda \in \C$, $k \in \N \cup \{ 0 \}$.

\[lem2\] For $v \in C^1_{\bar z}(\bar D)$ such that $v|_{\partial D} = 0$ the following formula holds: $$\label{estp}
v(z_0) = \frac{2}{\pi} \lim_{\lambda \to \infty} |\lambda| h^{(0)}_{z_0}(\lambda), \qquad z_0 \in D.$$ In addition, if $v \in C^2(\bar D)$, $v|_{\partial D}= 0$ and $\frac{\partial v}{\partial \nu}|_{\partial D} = 0$ then $$\label{estm}
\left|v(z_0) - \frac{2}{\pi}|\lambda| h^{(0)}_{z_0}(\lambda) \right| \leq c_3(D,n) \frac{\log(3|\lambda|)}{|\lambda|}\|v\|_{C^2(\bar D)},$$ for $z_0 \in D$, $\lambda \in \C$, $|\lambda| \geq 1$.

Following the proof of [@NS Lemma 6.2] and assuming , we have that limit is valid without the assumption that $v|_{\partial D} =0$. In addition, if $v|_{\partial D} \neq 0$ but $v \equiv \Lambda \in \matn$ on some open neighborhood of $\partial D$ in $\bar D$, then estimate holds with $h^{(0)}_{z_0} (\lambda)$ replaced by $$h^{(0),+}_{z_0}(\lambda) = h^{(0)}_{z_0}(\lambda) + \int_{\R^2 \setminus D} e^{\lambda(z-z_0)^2 - \bar \lambda (\bar z - \bar z_0)^2} \Lambda \chi(z) d \Ree z \, d \Imm z,$$ where $\chi \in C^2(\R^2 , \R)$, $\chi \equiv 1$ on $D$, $\mathrm{supp} \chi$ is compact, and the constant $c_3$ depending also on $\chi$.

Let $$W_{z_0}(\lambda)=\int_D e^{\lambda(z-z_0)^2-\bar\lambda(\bar z-\bar z_0)^2}
w(z)d\Ree\,z d\Imm\,z,$$ where $z_0\in\bar D$, $\lambda\in\C$ and $w$ is some $\matn$-valued function on $\bar D$. (One can see that $W_{z_0}=h_{z_0}^{(0)}$ for $w=v$.)

\[lem3\] For $w\in C_{\bar z}^1(\bar D)$ the following estimate holds: $$\label{estw}
|W_{z_0}(\lambda)|\le c_4(D) \frac{\log\,(3|\lambda|)}{ |\lambda|}  \|w\|_{C_{\bar z}^1(\bar D)},\ z_0\in\bar D,\ |\lambda|\ge 1.$$

\[lem4\] For $v \in C^1_{\bar z}(\bar D)$ and for $\|g_{z_0,\lambda} v \|^{op}_{C^1_{\bar z}(\bar D)} \leq \delta < 1$ we have that $$\begin{aligned}
 \label{estmuk}
&\|\mu_{z_0}(\cdot, \lambda) - \mu_{z_0}^{(k)}(\cdot, \lambda)\|_{C^1_{\bar z}(\bar D)} \leq \frac{\delta^{k+1}}{1-\delta}, \\ \label{esth}
&|h_{z_0}(\lambda)-h^{(k)}_{z_0}(\lambda)| \leq c_5(D,n)\frac{\log(3|\lambda|)}{|\lambda|} \frac{\delta^{k+1}}{1-\delta} \|v\|_{C^1_{\bar z}(\bar D)},\end{aligned}$$ where $z_0 \in D, \; \lambda \in \C,\; |\lambda |\ge 1, \; k \in \N \cup \{ 0 \}$.

The proof of Lemma \[lem4\] in the scalar case can be found in [@NS]: the generalization to the matrix-valued case is straightforward. 

\[lem5\] The function $g_{z_0}(z,\zeta,\lambda)$ satisfies the following properties: $$\begin{aligned}
&g_{z_0}(z,\zeta,\lambda) \quad \textrm{is continuous for } z,\zeta \in \bar D, \;z \neq \zeta, \; z_0 \in D, \\ \label{estgg}
|&g_{z_0}(z,\zeta,\lambda)| \leq c_6(D) |\log |z- \zeta| |, \qquad z, \zeta \in \bar D, \; z_0 \in D,\end{aligned}$$ where $\lambda \in \C$.

These properties follow from the definition and from classical estimates (see [@V]). 

\[lem6\] Under the assumptions of Proposition \[prop2\], the Schwartz kernel $(\Phi-\Phi_0)(z,\zeta)$ of the operator $\Phi-\Phi_0$ satisfies the following properties: $$\begin{aligned}
&(\Phi-\Phi_0)(z,\zeta) \quad \textrm{is continuous for } z,\zeta \in \partial D, \;z \neq \zeta, \\
|&(\Phi-\Phi_0)(z,\zeta)| \leq c_7(D,v,n) |\log |z- \zeta| |, \qquad z, \zeta \in \partial D.\end{aligned}$$

For a proof of this Lemma in the scalar case we refer to [@N1; @N2]: the generalization to the matrix-valued case is straightforward. 

Proofs of Theorem \[theo2\], Propositions \[prop2\], \[prop3\]\
and Corollary \[theo1\] {#secpf}
===============================================================

We begin with a matrix version of Alessandrini’s identity (see [@A] for the scalar case): $$\label{aless1}
\int_{\partial D} u_0(z) (\Phi - \Phi_0)u(z) |dz| = \int_D u_0(z) v(z) u(z) d \Ree z \, d \Imm z$$ for any sufficiently regular $\matn$-valued function $u$ (resp.  $u_0$) such that $\Delta u_0 = 0$ (resp.  $(-\Delta +v) u = 0$) in $D$. This follows from Stokes’s theorem, exactly as in the scalar case. 

The general matrix version of Alessandrini’s identity (that will not be used) $$\label{aless2}
\int_{\partial D} u_1(z) (\Phi_2 - \Phi_1)u_2(z) |dz| = \int_D u_1(z) (v_2(z) -v_1(z)) u_2(z) d \Ree z \, d \Imm z$$ for $u_1,u_2 \in C^2(\bar D, \matn)$ such that $(-\Delta + v_j) u_j =0$ in $D$, works if $u_1$ and $v_1$ commute each other (but does not work in general). 

Let us begin with the proof of formulas and : we have indeed $$\begin{aligned}
 \label{convg}
\left|v(z_0) - \frac{2}{\pi} |\lambda| h_{z_0}(\lambda)\right| \leq \left|v(z_0) - \frac{2}{\pi}|\lambda| h^{(0)}_{z_0}(\lambda)\right|+\frac{2}{\pi}|\lambda||h_{z_0}(\lambda) - h^{(0)}_{z_0}(\lambda)|.\end{aligned}$$ The first term in the right side goes to zero as $|\lambda| \to \infty$ by Lemma \[lem2\], while the other by Lemmata \[lem1\] and \[lem4\].  In addition, for $v \in C^2(\bar D,\matn)$ with $\|v\|_{C^2(\bar D)} <N_2$ and $\frac{\partial v}{\partial \nu}|_{\partial D} =0$, using , , and we obtain, from : $$\begin{aligned}
\left|v(z_0) - \frac{2}{\pi} |\lambda| h_{z_0}(\lambda)\right| &\leq c_3(D,n)\frac{\log(3|\lambda|)}{|\lambda|}\|v\|_{C^2(\bar D)} \\
&\qquad + c_5(D,n)\frac{\log(3|\lambda|)}{|\lambda|^{1/2}}\|v\|^2_{C^1_{\bar z}(\bar D)} \\
&\leq c_8(D,n)\frac{\log(3|\lambda|)}{|\lambda|^{1/2}}(\|v\|_{C^2(\bar D)}+ \|v\|^2_{C^1_{\bar z}(\bar D)}),\end{aligned}$$ for $\lambda$ such that $$\begin{aligned}
2n\frac{c_2(D)}{ |\lambda|^{\frac 1 2}}\|v\|_{C^1_ {\overline z}(\bar D)} \le \frac 1 2,\ |\lambda|\ge 1,\end{aligned}$$ which implies .  In order to prove we will need the following lemma:

\[lem11\] Let $g_{z_0, \lambda} u$ be defined by , where $u \in C^1_{\bar z}(\bar D)$, $z_0, \lambda \in \C$. Then the following estimate holds: $$\begin{aligned}
\| &g_{z_0, \lambda} u \|_{C(\bar D)} \leq \eta(D) \frac{\log(3|\lambda|)}{|\lambda|^{\frac 3 4}} \|u \|_{C^1_{\bar z}(\bar D)}, \; |\lambda| \geq 1.\end{aligned}$$

[*Proof of Lemma \[lem11\].*]{}  As in the proof of [@NS Lemma 3.1], we can write $g_{z_0, \lambda} = \frac 1 4 T \bar T_{z_0, \lambda}$, for $z_0, \lambda \in \C$, where $$\begin{aligned}
&T u(z) = - \frac{1}{\pi} \int_D \frac{u(\zeta)}{\zeta - z} d \Ree\zeta \, d \Imm \zeta, \\
&\bar T_{z_0, \lambda} u(z) = -\frac{e^{-\lambda(z- z_0)^2 + \bar \lambda(\bar z - \bar z_0)^2}}{\pi} \int_D \frac{e^{\lambda(\zeta-z_0)^2-\bar \lambda(\bar \zeta - \bar z_0)^2}}{\bar \zeta- \bar z} u(\zeta)  d \Ree\zeta \, d \Imm \zeta,\end{aligned}$$ for $z \in \bar D$ and $u$ a test function.  We have that (see [@NS]): $$\begin{aligned}
&Tw \in C^1_{\bar z}(\bar D), \\ \label{est32}
&\| Tw \|_{C^1_{\bar z}(\bar D)} \leq \eta_1(D) \| w \|_{C(\bar D)}, \; \textrm{where } w \in C(D), \\ \label{est33}
&\bar T_{z_0,\lambda} u \in C(\bar D), \\ \label{est34}
&\|\bar T_{z_0, \lambda} u\|_{C(\bar D)} \leq \frac{\eta_2(D)}{|\lambda|^{\frac 1 2}} \|u \|_{C^1_{\bar z}(\bar D)}, \; |\lambda| \geq 1, \\ \label{est35}
&\|\bar T_{z_0, \lambda} u\|_{C(\bar D)} \leq \frac{\log (3 |\lambda|)(1+|z-z_0|)\eta_3(D)}{|\lambda| |z- z_0|^2}\|u\|_{C^1_{\bar z}(\bar D)}, \; |\lambda| \geq 1,\end{aligned}$$ where $u \in C^1_{\bar z}(\bar D)$, $z_0, \lambda \in \C$.

Let $z_0 \in D$, $0 < \delta < \frac 1 2$ and $B_{z_0,\delta} = \{ z \in \C : |z- z_0| < \delta \}$. We have $$\begin{aligned}
 \label{estg}
|4\pi g_{z_0, \lambda} u(z)| &= \left| \int_D \frac{\bar T_{z_0, \lambda} u(\zeta)}{\zeta - z} d \Ree\zeta \, d \Imm \zeta \right|  \\ \nonumber
&\leq \int_{B_{z_0,\delta} \cap D} \frac{|\bar T_{z_0, \lambda} u(\zeta)|}{|\zeta - z|} d \Ree\zeta \, d \Imm \zeta + \int_{D \setminus B_{z_0,\delta}} \frac{|\bar T_{z_0, \lambda} u(\zeta)|}{|\zeta - z|} d \Ree\zeta \, d \Imm \zeta \\ \nonumber
&\leq 2 \pi \delta \frac{\eta_2(D)}{|\lambda|^{\frac 1 2}} \|u \|_{C^1_{\bar z}(\bar D)} + \frac{\log (3 |\lambda|)\eta_4(D)}{|\lambda| \delta}\|u\|_{C^1_{\bar z}(\bar D)},\end{aligned}$$ where we used the following estimate: $$\begin{aligned}
&\int_{D \setminus B_{z_0,\delta}} \frac{1}{|\zeta - z||\zeta - z_0|^2} d \Ree\zeta \, d \Imm \zeta \\
&\qquad = \int_{B_{z,\delta} \cap (D \setminus B_{z_0,\delta})} \frac{1}{|\zeta - z||\zeta - z_0|^2} d \Ree\zeta \, d \Imm \zeta \\ \nonumber
&\qquad  \qquad + \int_{D \setminus ( B_{z,\delta} \cup B_{z_0,\delta})} \frac{1}{|\zeta - z||\zeta - z_0|^2} d \Ree\zeta \, d \Imm \zeta \\
&\qquad \leq \frac{2 \pi}{\delta} + \int_{D \setminus ( B_{z,\delta} \cup B_{z_0,\delta})} \frac{1}{|\zeta - z|^3}+ \frac{1}{|\zeta - z_0|^3} d \Ree\zeta \, d \Imm \zeta \\
&\qquad \leq \frac{\eta_5(D)}{\delta}.\end{aligned}$$ Putting $\delta =\frac{1}{2} |\lambda|^{-\frac{1}{4}}$ in we obtain the result.  Thus Lemma \[lem11\] is proved. 

We now come back to the proof of .  Proceeding from and Lemma \[lem2\] we obtain: $$\begin{aligned}
 \label{convg2}
\left|v(z_0) - \frac{2}{\pi} |\lambda| h_{z_0}(\lambda)\right| \leq c_3(D,n)\frac{\log(3|\lambda|)}{|\lambda|}\|v\|_{C^2(\bar D)}+\frac{2}{\pi}|\lambda||h_{z_0}(\lambda) - h^{(0)}_{z_0}(\lambda)|,\end{aligned}$$ for $|\lambda| \geq 1$. In addition, from the definitions of $h^{(k)}\!, \;  \mu^{(k)} \!$, Lemmata \[lem1\] and \[lem4\], we have $$\begin{aligned}
&|h_{z_0}(\lambda) - h^{(0)}_{z_0}(\lambda)| \\
&\quad \leq \left|\int_D e^{\lambda (z-z_0)^2 -\bar \lambda (\bar z- \bar z_0)^2}v(z) g_{z_0,\lambda}v(z) d\Ree z \, d \Imm z \right| + O\left(\frac{\log(3|\lambda|)}{|\lambda|^2} \right)n^2 \|v\|^3_{C^1_{\overline z}(\bar D)},\end{aligned}$$ for $\lambda$ such that $2n\frac{c_2(D)}{ |\lambda|^{1/2}}\|v\|_{C^1_ {\overline z}(\bar D)} \le \frac 1 2,\ |\lambda|\ge 1$.

Repeating the proof of [@NS Lemma 3.3] and using also Lemma \[lem11\], we have, for $0<\varepsilon \leq 1$, $$\begin{aligned}
 \label{ultima}
&\left|\int_D e^{\lambda (z-z_0)^2 -\bar \lambda (\bar z- \bar z_0)^2}v(z) g_{z_0,\lambda}v(z) d\Ree z \, d \Imm z \right| \\ \nonumber
&\leq \int_{D \cap B_{z_0,\varepsilon}}\! \! \! \! \! \! \! \!   \|v(z) g_{z_0,\lambda}v(z)\|_{C(\bar D)} d \Ree z \, d \Imm z +\frac{1}{4 |\lambda|} \int_{\partial (D \setminus B_{z_0,\varepsilon})} \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!  \frac{\|v(z) g_{z_0,\lambda}v(z)\|_{C(\bar D)}}{|\bar z - \bar z_0|}|dz| \\ \nonumber
&\qquad + \frac{1}{2|\lambda|}\int_{D \setminus B_{z_0,\varepsilon}}\left| \frac{\partial}{\partial \bar z}\left( \frac{v(z) g_{z_0,\lambda}v(z)}{\bar z-\bar z_0}\right) \right| d \Ree z \, d \Imm z 
 \\ \nonumber
&\leq  \sigma_1(D,n) \|v\|_{C(\bar D)} \|v\|_{C^1_{\overline z}(\bar D)} \frac{\varepsilon^2  \log(3 |\lambda|)}{|\lambda|^{3/4}} \\ \nonumber
&\qquad+ \sigma_2(D,n) \|v\|_{C(\bar D)} \|v\|_{C^1_{\overline z}(\bar D)} \frac{\log(3 \varepsilon^{-1}) \log(3 |\lambda|) }{|\lambda|^{1+3/4}} \\ \nonumber
&\qquad + \frac{1}{8|\lambda|} \left|\int_{D \setminus B_{z_0,\varepsilon}} e^{\lambda (z-z_0)^2 -\bar \lambda (\bar z- \bar z_0)^2} v(z) \frac{\bar T_{z_0, \lambda}v(z)}{\bar z- \bar z_0} d\Ree z \, d \Imm z \right|, \quad |\lambda| \geq 1,\end{aligned}$$ where we also used integration by parts and the fact that $\frac{\partial}{\partial \bar z}g_{\lambda,z_0}u(z)= \frac{1}{4}\bar T_{z_0,\lambda}u(z)$. The last term in can be estimated independently on $\varepsilon$ by $$\begin{aligned}
\sigma_3(D,n) \frac{\log(3|\lambda|)}{|\lambda|^{1+3/4}} \|v\|_{C(\bar D)} \|v\|_{C^1_{\overline z}(\bar D)} \end{aligned}$$ using the same argument as in the proof of Lemma \[lem11\] (see estimate ).  Now putting $\varepsilon = |\lambda|^{-1/2}$ in we obtain $$\begin{aligned}
|\lambda||h_{z_0}(\lambda) - h^{(0)}_{z_0}(\lambda)| \leq \sigma_4(D,n)\frac{(\log(3|\lambda|))^2}{|\lambda|^{3/4}}\|v\|^2_{C^1_{\overline z}(\bar D)}( \|v\|_{C^1_{\overline z}(\bar D)} +1),\end{aligned}$$ for $|\lambda| > \rho_2(D,N_1,n),$ which, together with , gives us . 

The proofs of the other formulas of Theorem \[theo2\] are based on identity .  As $\mu_{z_0}(z,\lambda) = e^{-\lambda (z-z_0)^2} \psi_{z_0}(z,\lambda)$, we can write the generalized scattering amplitude as $$\nonumber
h_{z_0}(\lambda) = \int_D e^{-\bar \lambda (\bar z-\bar z_0)^2} v(z) \psi_{z_0}(z,\lambda) d \Ree z \, d \Imm z.$$ Now identity with $u_0(z) = e^{-\bar \lambda (\bar z-\bar z_0)^2}I$ and $u(z) = \psi_{z_0}(z,\lambda)$ reads $$\nonumber
\int_{\partial D} e^{-\bar \lambda (\bar z-\bar z_0)^2}(\Phi - \Phi_0)\psi_{z_0}(z,\lambda) |dz| = \int_D e^{-\bar \lambda (\bar z-\bar z_0)^2} v(z) \psi_{z_0}(z,\lambda) d \Ree z \, d \Imm z$$ which gives formula . 

Since $\mu_{z_0}$ is a solution of equation , $\psi_{z_0}(z,\lambda)$ satisfies the equation $$\label{defpsi}
\psi_{z_0}(z,\lambda) = e^{\lambda(z-z_0)^2}I + \int_D G_{z_0}(z,\zeta,\lambda)v(\zeta) \psi_{z_0}(\zeta,\lambda) d \Ree \zeta \, d \Imm \zeta,$$ for $z_0, z \in \bar D$, $\lambda \in \C$, $|\lambda| > \rho_1(D,N_1,n)$. Thus again by identity , with $u_0 = G_{z_0}(z,\zeta,\lambda)I$ and $u(z) = \psi_{z_0}(\zeta , \lambda)$, by and we obtain, for $z \in \partial D$, $$\begin{aligned}
\int_{\partial D}\! \! \!  G_{z_0}(z,\zeta, \lambda) (\Phi-\Phi_0) \psi_{z_0}(\zeta,\lambda) |d \zeta| &= \int_D G_{z_0}(z,\zeta,\lambda) v(\zeta) \psi_{z_0}(\zeta,\lambda) d \Ree \zeta \, d\Imm \zeta \\ \nonumber
&= \psi_{z_0}(z,\lambda) - e^{\lambda(z-z_0)^2}I.\end{aligned}$$ This finish the proof of Theorem \[theo2\]. 

By we have that $G_{z_0}(z,\zeta, \lambda)$ satisfies the same properties as $g_{z_0}(z,\zeta, \lambda)$ in Lemma \[lem5\], with the difference that the constant in depends also on $\lambda$. This observation, along with Lemma \[lem6\], implies that the operator $A(\lambda)$ defined as $$\nonumber
A(\lambda)u(z) = \int_{\partial D} G_{z_0} (z, \zeta, \lambda) (\Phi-\Phi_0)u(\zeta) |d \zeta|, \qquad z \in \partial D,$$ for a test function $u$, is compact on the space of continuous functions on $\partial D$. Thus equation is a Fredholm linear integral equation of the second kind in the space of continuous functions on $\partial D$.

First we have that equations and are well defined (i.e. Fredholm linear integral equations of the second type) on the spaces of continuous functions on $\bar D$ and $\partial D$ respectively.  This follows from for the first equation and from Proposition \[prop2\] for the second one. 

Now if admits a solution $\mu_{z_0}(z,\lambda) \in C(\bar D)$, then by and one readily obtains $\mu_{z_0}(z,\lambda) \in C^1_{\bar z} (\bar D)$. This solution is unique by Lemma \[lem1\] for $|\lambda| > \rho_1(D,N_1,n)$ and by the same arguments as in the proof of Theorem \[theo2\] one has that $\psi_{z_0}(z,\lambda)|_{z \in \partial D}$ satisfies equation . 

Conversely, suppose that $\psi_{z_0}(z,\lambda) \in C(\partial D)$ satisfies equation : we have to show that $\psi_{z_0}(z,\lambda)$, defined on $\bar D$ as the solution of the Dirichlet problem $(-\Delta +v)\psi_{z_0}(z,\lambda) = 0$ with boundary values given by a solution of equation , satisfies . 

By identity , $\psi_{z_0}(z,\lambda)$ satisfies already equation with $z \in \partial D$. Now, the function $$\varphi(z) = \psi_{z_0}(z,\lambda)-e^{\lambda(z-z_0)^2}I -\int_D G_{z_0}(z,\zeta,\lambda)v(\zeta) \psi_{z_0}(\zeta,\lambda) d \Ree \zeta \, d \Imm \zeta$$ satisfies $\Delta \varphi =0$ in $D$ and $\varphi|_{\partial D} = 0$, so $\varphi \equiv 0$ in $D$. Proposition \[prop3\] is proved. 

If $v_j|_{\partial D} = 0$, for $j=1,2$, then we can apply Theorem \[theo2\] and Propositions \[prop2\], \[prop3\].  As $\Phi_1 = \Phi_2$, then $\psi^1_{z_0}(\cdot,\lambda)|_{\partial D} = \psi^2_{z_0}(\cdot,\lambda)|_{\partial D}$ for $|\lambda| > \rho_1(D,N_1,n)$ (where we called $\psi^j_{z_0}(z,\lambda)$ the Bukhgeim analogues of the Faddeev solutions corresponding to $v_j$, for $j=1,2$).  Thus we also have equality between the corresponding generalized scattering amplitudes, $h^1_{z_0}(\lambda) = h^2_{z_0}(\lambda)$ for $|\lambda| > \rho_1(D,N_1,n)$, which yields $v_1(z_0) = v_2(z_0)$ for $z_0 \in D$.

If $v_j|_{\partial D} \neq 0$, for $j=1,2$, and $D$ is such that holds, then by Remark \[rem3\] we can apply Theorem \[theo2\] and argue as above. 

The general case follows from stability estimates which will be published in another paper, following the scheme of [@NS]. 

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