Subcritical perturbation of a locally periodic elliptic operator

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues and eigenfunctions for a locally periodic elliptic operator with oscillating coefficients, as the oscillation scale tends to zero, under specific potential assumptions.

Contribution

It provides the leading asymptotic terms for eigenvalues and eigenfunctions of a singularly perturbed elliptic operator with locally periodic coefficients and a potential with a unique interior minimum.

Findings

Asymptotic formulas for eigenvalues and eigenfunctions derived

Leading terms depend on the local minimum of the potential

Results applicable to operators with non-degenerate Hessian at the minimum

Abstract

We consider a singularly perturbed Dirichlet spectral problem for an elliptic operator of second order. The coefficients of the operator are assumed to be locally periodic and oscillating in the scale $\varepsilon$. We describe the leading terms of the asymptotics of the eigenvalues and the eigenfunctions to the problem, as the parameter $\varepsilon$ tends to zero, under structural assumptions on the potential. More precisely, we assume that the local average of the potential has a unique global minimum point in the interior of the domain and its Hessian is non-degenerate at this point.