Scattering theory for difference equations with operator coefficients

TL;DR

This paper develops a scattering theory framework for second order difference equations with operator-valued coefficients, analyzing spectral properties and the scattering matrix for perturbations of the discrete Laplacian.

Contribution

It introduces a novel approach to scattering theory for difference equations with operator coefficients, extending spectral analysis techniques.

Findings

Characterization of the spectrum for operator-valued perturbations

Analysis of the scattering matrix properties

Results on the spectral stability under perturbations

Abstract

We consider a second order difference equation with operator-valued coefficients. More precisely, we study either compact or trace class perturbations of the discrete Laplacian in the Hilbert space of bi-infinite square-summable sequence with entries in a fixed Hilbert space. We discuss its continuous and discrete spectrum, as well as properties of the associated scattering matrix.