Schrödinger Operators with Lattice Invariant Potentials

Abstract

We develop a systematic framework to study the dispersion surfaces of Schrödinger operators H = −∆+V, where the potential V is both periodic with respect to a lattice Λ and respects its symmetries. Our analysis relies on an abstract result, previously proven by Franz Rellich [Rel40] and which we prove using an alternative approach inspired by methods developed by Tosio Kato [Kat95]: if a self-adjoint operator depends analytically on a parameter, then so do its eigenvalues and eigenprojectors in a neighborhood of the real line. Using this and techniques from Floquet-Bloch theory and representation theory, we prove a series of results that can be used to analyze the operator H where the lattice Λ is arbitrary. As an application of this framework, we describe the generic structure of some singularities in the band spectrum of Schrödinger operators invariant under various two- and three-dimensional lattices. Specifically, we study the square, hexagonal, rectangular, simple cubic, body-centered cubic, face-centered cubic, and stacked hexagonal lattices, in the process reproducing results due to [Kel+18] and [FW12], and also proving a conjecture of [GZZ22].