Subcritical perturbation of a locally periodic elliptic operator

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 ϵ. We describe the leading terms of the asymptotics of the eigenvalues and the eigenfunctions to the problem, as the parameter ϵ 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. Copyright © 2016 John Wiley & Sons, Ltd.

Spectral Asymptotics for the Schrödinger Operator on the Line with Spreading and Oscillating Potentials

“…This transpires for instance in [2,3,1] where, in order to deal with large potentials, the authors introduce a factorization principle which is similar to our normal form transformation (see below), although without the change of variable. We would also like to mention the recent works [25,7,26], where similar multiscale problem as ours are studied.…”

Spectral asymptotics for the Schrödinger operator on the line with spreading and oscillating potentials

“…This transpires for instance in [2,3,1] where, in order to deal with large potentials, the authors introduce a factorization principle which is similar to our normal form transformation (see below), although without the change of variable. We would also like to mention the recent works [22,6,23], where similar multiscale problem as ours are studied. 1.4.2.…”