Inverse problem with transmission eigenvalues for the discrete Schr\"odinger equation

TL;DR

This paper investigates the inverse problem of reconstructing potentials in a discrete Schrödinger equation from transmission eigenvalues, demonstrating unique solutions and special cases with multiple or infinite solutions, supported by explicit examples.

Contribution

It applies the Marchenko and Gelfand-Levitan methods to the discrete Schrödinger inverse problem, identifying conditions for uniqueness and multiplicity of solutions.

Findings

Unique solution in most cases using transmission eigenvalues.

Existence of multiple or infinite solutions in an 'unusual' case.

Explicit examples illustrating the theoretical results.

Abstract

The discrete Schr\"odinger equation with the Dirichlet boundary condition is considered on a half-line lattice when the potential is real valued and compactly supported. The inverse problem of recovery of the potential from the so-called transmission eigenvalues is analyzed. The Marchenko method and the Gel'fand-Levitan method are used to solve the inverse problem uniquely, except in one "unusual" case where the sum of the transmission eigenvalues is equal to a certain integer related to the support of the potential. It is shown that in the unusual case there may be a unique solution corresponding to certain sets of transmission eigenvalues, there may be a finite number of distinct potentials for some sets of transmission eigenvalues, or there may be infinitely many potentials for some sets of transmission eigenvalues. The theory presented is illustrated with several explicit examples.