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

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 behaviour 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 applies to any dispersion relation which is an analytic Morse function. We also provide two examples of simple lattice systems whose dispersion relations do not suppress crossings in the present sense.