Eigenfunctions Localised on a Defect in High-Contrast Random Media
Matteo Capoferri, Mikhail Cherdantsev, Igor Velčić · SIAM Journal on Mathematical Analysis · 2023
Abstract. We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrum of a high-contrast random operator. We consider a family of elliptic operators [Formula: see text] in divergence form whose coefficients are random, possess double porosity type scaling, and are perturbed on a fixed-size compact domain (a defect). Working in the gaps of the limiting spectrum of the unperturbed operator [Formula: see text], we show that the point spectrum of [Formula: see text] converges in the sense of Hausdorff to the point spectrum of the limiting two-scale operator [Formula: see text] as [Formula: see text]. Furthermore, we prove that the eigenfunctions of [Formula: see text] decay exponentially at infinity uniformly for sufficiently small [Formula: see text]. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of [Formula: see text].