Asymptotics of resolvent integrals: The suppression of crossings for analytic lattice dispersion relations

TL;DR

This paper investigates the conditions under which analytic dispersion relations in lattice systems suppress crossings, showing that non-constant Morse functions do suppress crossings, with examples of exceptions.

Contribution

It establishes a precise criterion for crossing suppression in lattice dispersion relations and connects it to the Morse property, providing new theoretical insights.

Findings

Dispersion relations suppress crossings iff not constant on any affine hyperplane

Morse functions always suppress crossings under the given conditions

Examples of lattice systems with non-suppressing dispersion relations are provided

Abstract

We study the so called crossing estimate for analytic dispersion relations of periodic lattice systems in dimensions three and higher. Under a certain regularity assumption on the behavior of the dispersion relation near its critical values, we prove that an analytic dispersion relation suppresses crossings if and only if it is not a constant on any affine hyperplane. In particular, this will then be true for any dispersion relation which is a Morse function. We also provide two examples of simple lattice systems whose dispersion relations do not suppress crossings in the present sense.