Ambarzumian-type problems for discrete Schr\"{o}dinger operators

TL;DR

This paper investigates the unique determination of finite discrete Schrödinger operators from spectral data, extending classical inverse spectral problems to various boundary conditions and establishing new uniqueness results.

Contribution

It introduces novel Ambarzumian-type inverse spectral results for discrete Schrödinger operators with diverse boundary conditions, including mixed inverse problems.

Findings

Unique determination of operators from spectral data under various boundary conditions

Proof of a mixed inverse spectral problem with partial diagonal entries and eigenvalues

Extension of classical inverse spectral theory to discrete Schrödinger operators

Abstract

We discuss the problem of unique determination of the finite free discrete Schr\"{o}dinger operator from its spectrum, also known as Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: Diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schr\"{o}dinger operator.