On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices

TL;DR

This paper proves the unique reconstruction of semi-infinite Jacobi matrices from two spectra and provides conditions for spectral sequences to correspond to such operators, extending classical inverse spectral results.

Contribution

It establishes a discrete analogue of the Borg-Marchenko theorem for Jacobi matrices and characterizes spectral sequences for different boundary conditions.

Findings

Unique reconstruction from two spectra proved

Necessary and sufficient conditions for spectral sequences provided

Extension of classical inverse spectral theory to discrete case

Abstract

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg-Marchenko theorem for Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.