Scattering theory for lattice operators in dimension $d\geq 3$

TL;DR

This paper develops a detailed scattering theory framework for lattice operators in dimensions three and higher, providing explicit formulas for wave operators and proving a Levinson theorem that accounts for embedded eigenvalues and threshold effects.

Contribution

It introduces an explicit formula for wave operators using a dilation operator, extending scattering theory to higher-dimensional lattice Hamiltonians with impurities.

Findings

Explicit wave operator formula in dimension d≥3

Levinson theorem proved with embedded eigenvalues

Analysis of threshold singularities in scattering

Abstract

This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension $d\geq 3$ the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in presence of embedded eigenvalues and threshold singularities.