Band spectrum singularities for Schrödinger operators

Abstract: In this paper, we develop a systematic framework to study the dispersion surfaces of Schr\"odinger operators $ -\Delta + V$, where the potential $V \in C^\infty(\mathbb{R}^n,\mathbb{R})$ is periodic with respect to a lattice $\Lambda \subset \mathbb{R}^n$ and respects the symmetries of $\Lambda$. Our analysis relies on an abstract result, that expands on a seminal work of Fefferman--Weinstein \cite{feffer12}: if an operator depends analytically on a parameter, then so do its eigenvalues and eigenprojectors away from a discrete set. As an application, we describe the generic structure of some singularities in the band spectrum of Schr\"odinger operators invariant under the three-dimensional simple, body-centered and face-centered cubic lattices.