Band spectrum singularities for Schrödinger operators

Abstract

In this paper, we develop a systematic framework to study the dispersion surfaces of Schr{ö}dinger operators $-\hat{I}”+V$, where the potential $V\in C^{\infty }({\mathbb{R}}^n,\mathbb{R})$ is periodic with respect to a lattice $\hat{I}›\subset {\mathbb{R}}^n$ and respects the symmetries of $\hat{I}›$. Our analysis combines the theory of holomorphic families of operators of type (A) with the seminal work of Fefferman--Weinstein \cite{feffer12}. It allows us to extend results on the existence of spectral degeneracies past a perturbative regime. As an application, we describe the generic structure of some singularities in the band spectrum of Schrödinger operators invariant under the three-dimensional simple, body-centered and face-centered cubic lattices.

37 pages, 3 figures