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

Abstract

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.

7 pages, 3 figures

References in corpus (6)

• Spectral gaps without the pressure condition
• Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian
• Resolvent estimates on asymptotically cylindrical manifolds and on the half line
• Wave asymptotics for waveguides and manifolds with infinite cylindrical ends
• Inverse resonance scattering for on rotationally symmetric manifolds
• Absolute continuity of the spectrum in a twisted Dirichlet-Neumann waveguide