Finite range perturbations of finite gap Jacobi and CMV operators

TL;DR

This paper characterizes spectral measures of finite range perturbations of finite gap Jacobi and CMV operators, solving an open problem and establishing that operators are uniquely determined by their eigenvalues and resonances.

Contribution

It provides necessary and sufficient conditions for spectral measures and solves the inverse resonance problem for these operators, including the special case of eventually periodic operators.

Findings

Characterization of spectral measures for finite range perturbations.

Solution to the inverse resonance problem.

Resolution of an open problem for eventually periodic operators.

Abstract

Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range perturbation of a Jacobi or CMV operator from a finite gap isospectral torus. The special case of eventually periodic operators solves an open problem of Simon [25, D.2.7]. We also solve the inverse resonance problem: it is shown that an operator is completely determined by the set of its eigenvalues and resonances, and we provide necessary and sufficient conditions on their configuration for such an operator to exist.