Manifolds with many small wormholes: norm resolvent and spectral convergence

TL;DR

This paper investigates how adding many small wormholes to manifolds affects the spectral properties of the Laplace-Beltrami operator, showing different limit behaviors depending on the distribution of the handles.

Contribution

It provides new results on the norm resolvent convergence for Laplace-Beltrami operators on manifolds with numerous small handles, extending previous analysis to non-compact cases.

Findings

Sparse handles lead to the unperturbed operator as the limit.

Dense handles cause the limit operator to enforce function equality across connected parts.

Results apply to both compact and non-compact manifolds.

Abstract

We present results concerning the norm convergence of resolvents for wildperturbations of the Laplace-Beltrami operator. This article is a continuation of ouranalysis on wildly perturbed manifolds presented in [AP21]. We study here manifoldswith an increasing number of small (i.e., short and thin) handles added. The handlescan also be seen as wormholes, as they connect different parts being originally far away.We consider two situations: if the small handles are distributed too sparse the limitoperator is the unperturbed one on the initial manifold, the handles are fading. Onthe other hand, if the small handles are dense in certain regions the limit operator isthe Laplace-Beltrami operator acting on functions which are identical on the two partsjoined by the handles, the handles hence produce adhesion. Our results also apply tonon-compact manifolds. Our work is based on a norm convergence result for operatorsacting in varying Hilbert spaces described in the book [P12] by the second author.