Periodic Jacobi operator with finitely supported perturbation on the half-lattice

TL;DR

This paper analyzes the spectral properties of a periodic Jacobi operator with finitely supported perturbations on the half-lattice, including eigenvalues, resonances, and inverse problems, providing explicit reconstruction methods.

Contribution

It characterizes all eigenvalues and resonances of the perturbed operator and establishes a unique reconstruction of perturbations from spectral data.

Findings

Complete description of eigenvalues and resonances.

Proof of one-to-one correspondence between perturbations and Jost functions.

Explicit reconstruction method for perturbations from spectral data.

Abstract

We consider the periodic Jacobi operator $J$ with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of $J$ and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from the eigenvalues, resonances and the set of zeros of $S(\l)-1,$ where $S(\l)$ is the scattering matrix.