Localization effect for a spectral problem in a perforated domain with Fourier boundary cconditions

TL;DR

This paper studies how eigenfunctions localize in a perforated domain with Fourier boundary conditions, revealing the asymptotic behavior of the spectrum and eigenfunctions through homogenization and auxiliary operators.

Contribution

It introduces a novel analysis of spectral localization effects in perforated domains with Fourier boundary conditions, including asymptotic descriptions and convergence estimates.

Findings

Eigenfunctions exhibit localization due to periodic coefficients.

The lower spectrum converges to an auxiliary harmonic oscillator.

Asymptotic eigenpair behavior is characterized with convergence rates.

Abstract

We consider a homogenization of elliptic spectral problem stated in a perforated domain, Fourier boundary conditions being imposed on the boundary of perforation. The presence of a locally periodic coefficient in the boundary operator gives rise to the effect of a localization of the eigenfunctions. Moreover, the limit behaviour of the lower part of the spectrum can be described in terms of an auxiliary harmonic oscillator operator. We describe the asymptotics of the eigenpairs and derive the estimates for the rate of convergence.