Monochromatic reconstruction algorithms for two-dimensional multi-channel inverse problems

TL;DR

This paper introduces two efficient algorithms for reconstructing matrix-valued potentials in two-dimensional multi-channel Schrödinger inverse problems, achieving Lipschitz stability and improved accuracy at high energies.

Contribution

The paper presents novel algorithms for multi-channel inverse Schrödinger problems that ensure Lipschitz stability and enhanced reconstruction accuracy as energy increases.

Findings

Potential reconstructed with Lipschitz stability

Error decreases as energy increases

Applicable to matrix-valued potentials with smoothness and decay

Abstract

We consider two inverse problems for the multi-channel two-dimensional Schr\"odinger equation at fixed positive energy, i.e. the equation $-\Delta \psi + V(x)\psi = E \psi$ at fixed positive $E$, where $V$ is a matrix-valued potential. The first is the Gel'fand inverse problem on a bounded domain $D$ at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane $\R^2$. We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases we show that the potential $V$ is reconstructed with Lipschitz stability by these algorithms up to $O(E^{-(m-2)/2})$ in the uniform norm as $E \to +\infty$, under the assumptions that $V$ is $m$-times differentiable in $L^1$, for $m \geq 3$, and has sufficient boundary decay.