The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case

TL;DR

This paper introduces a method to reconstruct semi-infinite Jacobi operators in the limit circle case using spectra from two self-adjoint extensions, providing conditions for spectral sequences to correspond to such operators.

Contribution

It offers a novel reconstruction technique and necessary and sufficient spectral conditions for Jacobi operators in the limit circle case.

Findings

Reconstruction method from two spectra

Necessary and sufficient spectral conditions

Application to semi-infinite Jacobi matrices

Abstract

We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case.