Periodic Schr\"odinger operators with local defects and spectral pollution

TL;DR

This paper investigates spectral pollution in numerical methods for perturbed periodic Schr"odinger operators, demonstrating conditions under which spectral pollution can be eliminated, especially using approximate spectral projectors.

Contribution

It provides a detailed analysis of spectral pollution, proves the supercell model avoids it, and extends criteria for pollution-free spectral calculations using approximate projectors.

Findings

Supercell model does not produce spectral pollution.

Spectral pollution can be eliminated using approximate spectral projectors.

Extended criteria for no-pollution in spectral gaps.

Abstract

This article deals with the numerical calculation of eigenvalues of perturbed periodic Schr\"odinger operators located in spectral gaps. Such operators are encountered in the modeling of the electronic structure of crystals with local defects, and of photonic crystals. The usual finite element Galerkin approximation is known to give rise to spectral pollution. In this article, we give a precise description of the corresponding spurious states. We then prove that the supercell model does not produce spectral pollution. Lastly, we extend results by Lewin and S\'er\'e on some no-pollution criteria. In particular, we prove that using approximate spectral projectors enables one to eliminate spectral pollution in a given spectral gap of the reference periodic Sch\"odinger operator.