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

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 $d\ge 3$ 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.

Minor errors and misprints corrected; new result on absense of embedded eigenvalues for potential scattering; to appear in RMP

References in corpus (3)

• On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators
• Levinson's theorem for Schroedinger operators with point interaction: a topological approach
• The topological meaning of Levinson's theorem, half-bound states included

Cited by in corpus (7)

• One-dimensional Dirac operators with zero-range interactions: Spectral, scattering, and topological results
• On the wave operators for the Friedrichs-Faddeev model
• New expressions for the wave operators of Schroedinger operators in R^3
• The density of surface states as the total time delay
• Levinson's theorem as an index pairing
• Dimensional reduction and scattering formulation for even topological invariants
• Resonances under Rank One Perturbations