Spectral and resonance problem for perturbations of periodic Jacobi operators

TL;DR

This paper establishes conditions for spectral measures of perturbed periodic Jacobi operators, solving the inverse resonance problem and demonstrating stability under small perturbations.

Contribution

It provides a complete solution to the inverse resonance problem for Jacobi operators, including existence, uniqueness, and stability results.

Findings

Necessary and sufficient conditions for spectral measures of perturbed Jacobi operators.

Complete solution to the inverse resonance problem.

Proof of stability of the inverse resonance problem under small perturbations.

Abstract

Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range or exponentially decaying perturbation of a periodic Jacobi operator. As a corollary we can fully solve the inverse resonance problem: given resonances and eigenvalues we can recover the spectral measure of the Jacobi operator; we provide necessary and sufficient conditions under which such an operator exists and is unique; and we show that the inverse resonance problem is stable under small perturbations.