Spectral asymptotics of periodic elliptic operators

TL;DR

This paper analyzes the spectral properties of periodic elliptic operators using decomposition into boundary condition variants, demonstrating convergence of rescaled semigroups to homogenized operators and establishing their analytic dependence.

Contribution

It introduces a method to analyze spectral asymptotics of periodic elliptic operators via decomposition and homogenization, with results on kernel regularity and analytic dependence.

Findings

Semigroups have kernels with Gaussian bounds and Hölder continuity.

Rescaled semigroups converge in trace norm to homogenized semigroups.

Analytic dependence of semigroups on complex parameters is established.

Abstract

We demonstrate that the structure of complex second-order strongly elliptic operators $H$ on ${\bf R}^d$ with coefficients invariant under translation by ${\bf Z}^d$ can be analyzed through decomposition in terms of versions $H_z$, $z\in{\bf T}^d$, of $H$ with $z$-periodic boundary conditions acting on $L_2({\bf I}^d)$ where ${\bf I}=[0,1>$. If the semigroup $S$ generated by $H$ has a H\"older continuous integral kernel satisfying Gaussian bounds then the semigroups $S^z$ generated by the $H_z$ have kernels with similar properties and $z\mapsto S^z$ extends to a function on ${\bf C}^d\setminus\{0\}$ which is analytic with respect to the trace norm. The sequence of semigroups $S^{(m),z}$ obtained by rescaling the coefficients of $H_z$ by $c(x)\to c(mx)$ converges in trace norm to the semigroup $\hat{S}^z$ generated by the homogenization $\hat{H}_z$ of $H_z$. These convergence properties allow asymptotic analysis of the spectrum of $H$.