Inverse resonance scattering for Jacobi operators

TL;DR

This paper solves inverse problems for Jacobi operators with compactly supported coefficients, characterizing the relationship between the operator parameters and their spectral data, including zeros of the reflection coefficient and bound states.

Contribution

It provides explicit solutions and characterizations for inverse problems involving Jacobi operators, including the set of operators sharing the same spectral data.

Findings

Solution to inverse problems for Jacobi operators

Characterization of operators with identical resonances and bound states

Description of the set of iso-resonance operators

Abstract

We consider the Jacobi operator $(Jf)_n= a_{n-1}f_{n-1}+a_nf_{n+1}+b_nf_n$ on $\Z$ with a real compactly supported sequences $(a_n-1)_{n\in\Z}$ and $(b_n)_{n\in\Z}$. We give the solution of two inverse problems (including characterization): $ (a,b)\to \{$zeros of the reflection coefficient$\}$ and $(a,b)\to \{$bound states and resonances$\}$. We describe the set of "iso-resonance operators $J$", i.e., all operators $J$ with the same resonances and bound states.