Inverse Problem for the Schr\"odinger Equation with Non-self-adjoint Matrix Potential

TL;DR

This paper develops a method to recover non-self-adjoint matrix potentials in the Schr"odinger equation from boundary control data, advancing inverse problem techniques for complex quantum systems.

Contribution

It introduces a novel approach combining spectral data recovery with boundary control methods for inverse problems involving non-self-adjoint matrix potentials.

Findings

Spectral data can be reconstructed from dynamic boundary data.

The inverse problem for the Schr"odinger equation is solvable using the proposed method.

The approach extends inverse problem techniques to non-self-adjoint matrix potentials.

Abstract

We consider the dynamical system with boundary control for the vector Schr\"odinger equation on the interval with a non-self-adjoint matrix potential. For this system, we study the inverse problem of recovering the matrix potential from the dynamical Dirichlet--to--Neumann operator. We first provide a method to recover spectral data for an abstract system from dynamic data and apply it to the Schr\"odinger equation. We then develop a strategy for solving the inverse problem for the Schr\"odinger equation using this method with other techniques of the Boundary control method.