Limit absorption and Green function estimates for matrix-valued periodic operators

TL;DR

This paper establishes limit absorption principles and Green function estimates for matrix-valued periodic operators, analyzing resolvent boundary behavior in different dimensions and near critical energy points using advanced mathematical techniques.

Contribution

It introduces new methods to analyze resolvent limits and Green functions for matrix-valued periodic Hamiltonians, especially near van Hove singularities and Weyl points.

Findings

Limit absorption principle holds in 3D and away from critical points in 2D.

Green function estimates are derived near critical points using Morse and stationary phase methods.

New oscillatory integral techniques are developed for Weyl point analysis.

Abstract

The boundary value of the resolvent of a generic periodic tight-binding Hamiltonian with matrix symbols is shown to satisfy a limit absorption principle which is continuous in energy in dimensions $d=3$, and in dimension $d=2$ away from critical points of the energy bands corresponding to van Hove singularities. The analysis away from critical points of the energy bands is based on the coarea formula, while at the critical points it involves a parametric Morse lemma and stationary phase arguments. In particular, at Weyl points a new type of oscillatory integrals is dealt with.