Inverse problems for Jacobi operators with finitely supported perturbations

TL;DR

This paper addresses the inverse problem for Jacobi operators with finitely supported perturbations, utilizing resonance data, and establishes conditions for resonances and their multiplicities.

Contribution

It introduces a novel approach to solve inverse problems for Jacobi operators using resonance analysis and polynomial theory, including forbidden resonance domains.

Findings

Resolved inverse problems for Jacobi operators with finite perturbations

Identified forbidden resonance domains and resonance multiplicities

Connected inverse eigenvalue problems with polynomial theory

Abstract

We solve the inverse problem for Jacobi operators on the half lattice with finitely supported perturbations, in particular, in terms of resonances. Our proof is based on the results for the inverse eigenvalue problem for specific finite Jacobi matrices and theory of polynomials. We determine forbidden domains for resonances and maximal possible multiplicities of real and complex resonances.