Wildly perturbed manifolds: norm resolvent and spectral convergence

TL;DR

This paper investigates the spectral and resolvent convergence of Laplace operators on manifolds with numerous small removed regions, revealing how different configurations of obstacles influence the limiting operator.

Contribution

It provides new results on norm resolvent convergence for Laplacians on manifolds with complex, wildly perturbed domains, extending previous work to non-compact and infinite obstacle arrangements.

Findings

Neumann Laplacian converges to the unperturbed Laplacian when obstacles vanish.

In the Dirichlet case, sparse obstacles lead to the unperturbed limit, while concentrated obstacles create a solid region affecting the limit.

The work extends operator convergence results to more general, non-compact manifold settings.

Abstract

Since the publication of the important work of Rauch and Taylor (Potential and scattering theory on wildly perturbed domains, JFA, 1975) a lot has been done to analyse wild perturbations of the Laplace operator. Here we present results concerning the norm convergence of the resolvent. We consider a (not necessarily compact) manifold with many small balls removed, the number of balls can increase as the radius is shrinking, the number of balls can also be infinite. If the distance of the balls shrinks less fast than the radius, then we show that the Neumann Laplacian converges to the unperturbed Laplacian, i.e., the obstacles vanish. In the Dirichlet case, we have two cases: if the balls are too sparse, the limit operator is again the unperturbed one, while if the balls concentrate at a certain region (they become "solid" in a region), the limit operator is the Dirichlet Laplacian on the complement outside the solid region. Our work is based on a norm convergence result for operators acting in varying Hilbert spaces developed by the second author.