Stability of determinacy and inverse spectral problems for Jacobi operators

TL;DR

This paper explores the stability of spectral measure determinacy and inverse spectral problems for Jacobi operators, analyzing how perturbations affect these properties and providing new characterizations and inverse problem solutions.

Contribution

It introduces new results on the stability of the index of determinacy under perturbations and characterizes this index via cyclic vectors, advancing inverse spectral theory for Jacobi operators.

Findings

The index of determinacy remains constant under certain perturbations.

Characterization of the index in terms of cyclic vectors.

A new inverse problem relating perturbation location to the index.

Abstract

This work studies the interplay between Green functions, the index of determinacy of spectral measures and interior finite rank perturbations of Jacobi operators. The index of determinacy quantifies the stability of uniqueness of solutions of the moment problem. We give results on the constancy of this index in terms of perturbations of the corresponding Jacobi operators. The permanence of the $N$-extremality of a measure is also studied. A measure $\mu$ is $N$-extremal when the polynomials are dense in $L_2(\mathbb{R},\mu)$. As a by-product, we give a characterization of the index in terms of cyclic vectors. We consider a new inverse problem for Jacobi operators in which information on the place where the interior perturbation occurs is obtained from the index of determinacy.