Asymptotic expansion of the spectrum for periodic Schr\"{o}dinger operators

TL;DR

This paper derives an asymptotic expansion for the eigenvalues and eigenfunctions of periodic Schr"{o}dinger operators, comparing the spectrum to a homogenized operator and detailing high-order corrections.

Contribution

It introduces a method to obtain high-order asymptotic expansions for spectra of periodic Schr"{o}dinger operators with confining potentials.

Findings

Asymptotic expansion for eigenvalues and eigenfunctions derived

Spectrum compared to homogenized operator with detailed corrections

High-order correction terms characterized

Abstract

We prove an asymptotic expansion for the eigenvalues and eigenfunctions of Schr\"{o}dinger-type operator with a confining potential and with principle part a periodic elliptic operator in divergence form. We compare the spectrum to the homogenized operator and characterize the corrections up to arbitrarily high order.