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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