Manifolds with cylindrical ends having a finite and positive number of embedded eigenvalues

We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a hyperbolic torus with a hole. To our knowledge, this is the first example of a manifold with a cylindrical e...

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Christiansen, T J, Datchev, K
Format: Article
Language:English
Subjects:
Asymptotic series
Eigenvalues
Manifolds
Toruses
Wave equations
Online Access:Get full text
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Summary:We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a hyperbolic torus with a hole. To our knowledge, this is the first example of a manifold with a cylindrical end whose number of eigenvalues is known to be finite and nonzero. The construction can be varied to give examples with arbitrary genus and with an arbitrarily large finite number of eigenvalues. The constructed surfaces also have resonance-free regions near the continuous spectrum and long-time asymptotic expansions of solutions to the wave equation.
ISSN:2331-8422
DOI:10.48550/arxiv.2007.04092