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

TL;DR

This paper constructs a surface with a cylindrical end that uniquely has a finite, positive number of embedded Laplace eigenvalues, providing new examples in spectral geometry.

Contribution

It introduces the first known example of a manifold with a cylindrical end having a finite, nonzero number of embedded eigenvalues, with adaptable constructions for various topologies.

Findings

Finite number of embedded eigenvalues constructed

Resonance-free regions near the continuous spectrum

Long-time asymptotic expansions for wave solutions

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.