Levinson theorem for discrete Schr\"odinger operators on the line with matrix potentials having a first moment

TL;DR

This paper establishes a Levinson theorem for discrete matrix Schr"odinger operators with perturbations having a finite first moment, linking scattering data to spectral properties.

Contribution

It introduces a Levinson theorem for matrix-valued discrete Schr"odinger operators with non-compact support and first moment conditions, expanding spectral and scattering theory.

Findings

Proves a Levinson theorem relating scattering data and spectral properties.

Develops stationary scattering theory using Jost solutions at complex energies.

Analyzes operators with perturbations having a finite first moment.

Abstract

This paper proves new results on spectral and scattering theory for matrix-valued Schr\"odinger operators on the discrete line with non-compactly supported perturbations whose first moments are assumed to exist. In particular, a Levinson theorem is proved, in which a relation between scattering data and spectral properties (bound and half bound states) of the corresponding Hamiltonians is derived. The proof is based on stationary scattering theory with prominent use of Jost solutions at complex energies that are controlled by Volterra-type integral equations.