Ambarzumian-type mixed inverse spectral problems for Jacobi matrices

TL;DR

This paper explores whether a Jacobi matrix can be uniquely reconstructed when most of its diagonal entries are unknown, but a set of eigenvalues and some diagonal entries are known, extending inverse spectral problem theory.

Contribution

It introduces a new inverse spectral problem for Jacobi matrices involving partial diagonal data and eigenvalues, providing conditions for unique reconstruction.

Findings

Unique determination of Jacobi matrices under specified partial data

Extension of inverse spectral theory to mixed known/unknown diagonal entries

Potential applications in spectral analysis and matrix reconstruction

Abstract

We investigate Ambarzumian-type mixed inverse spectral problems for Jacobi matrices. Specifically, we examine whether the Jacobi matrix can be uniquely determined by knowing all but the first $m$ diagonal entries and a set of $m$ ordered eigenvalues.