Pré-Publication, Document De Travail Année : 2025

Manifolds with many small wormholes: norm resolvent and spectral convergence

Résumé

We present results concerning the norm convergence of resolvents for wild perturbations of the Laplace-Beltrami operator. This article is a continuation of our analysis on wildly perturbed manifolds presented in [AP21]. We study here manifolds with an increasing number of small (i.e., short and thin) handles added. The handles can 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 limit operator is the unperturbed one on the initial manifold, the handles are fading. On the other hand, if the small handles are dense in certain regions the limit operator is the Laplace-Beltrami operator acting on functions which are identical on the two parts joined by the handles, the handles hence produce adhesion. Our results also apply to non-compact manifolds. Our work is based on a norm convergence result for operators acting in varying Hilbert spaces described in the book [P12] by the second author.
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Dates et versions

hal-04909154 , version 1 (24-01-2025)

Identifiants

  • HAL Id : hal-04909154 , version 1

Citer

Colette Anné, Olaf Post. Manifolds with many small wormholes: norm resolvent and spectral convergence. 2025. ⟨hal-04909154⟩
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