Periodic Jacobi operator with finitely supported perturbations: the inverse resonance problem

TL;DR

This paper addresses the inverse resonance problem for a periodic Jacobi operator with finitely supported perturbations, establishing a unique correspondence between perturbations and scattering data, and exploring reconstruction methods from spectral information.

Contribution

It proves the bijective relationship between finitely supported perturbations and scattering data for the operator, enabling reconstruction from spectral information.

Findings

The mapping from perturbations to scattering data is one-to-one and onto.

Reconstruction of perturbations is possible from eigenvalues, resonances, and zeros of a specific function.

The inverse problem is well-posed with explicit spectral data.

Abstract

We consider a periodic Jacobi operator $H$ with finitely supported perturbations on ${\Bbb Z}.$ We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the scattering data: the inverse of the transmission coefficient and the Jost function on the right half-axis, is one-to-one and onto. We consider the problem of reconstruction of the scattering data from all eigenvalues, resonances and the set of zeros of $R_-(\lambda)+1,$ where $R_-$ is the reflection coefficient.