Generic Simplicity of Spectral Edges and Applications to Homogenization

TL;DR

This paper demonstrates that small perturbations can simplify spectral edges of elliptic operators with periodic coefficients, enabling homogenization at internal edges even with eigenvalue multiplicities.

Contribution

It introduces spectral tools to achieve simplicity of internal spectral edges through perturbations, facilitating Bloch wave homogenization in complex spectral scenarios.

Findings

Small perturbations can make multiple Bloch eigenvalues simple.

Homogenization at internal spectral edges is possible despite eigenvalue multiplicity.

All crossing Bloch modes contribute to homogenization, while others do not.

Abstract

We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the presence of multiplicity by employing the perturbed Bloch eigenvalues. We show that all the crossing Bloch modes contribute to the homogenization at the internal edge and that higher and lower modes do not contribute to the homogenization process.