Exponential Decay of Eigenfunctions and Accumulation of Eigenvalues on Manifolds with Axial Analytic Asymptotically Cylindrical Ends

TL;DR

This paper investigates the decay properties of eigenfunctions and the accumulation behavior of eigenvalues for the Laplacian on manifolds with specific cylindrical ends, revealing that eigenvalues only accumulate at thresholds and eigenfunctions decay exponentially.

Contribution

It extends previous work by proving exponential decay of eigenfunctions and characterizing eigenvalue accumulation on manifolds with axial analytic asymptotically cylindrical ends.

Findings

Eigenfunctions decay exponentially at a rate determined by the distance to the next threshold.

Eigenvalues can only accumulate at thresholds and from below.

Eigenvalues are of finite multiplicity.

Abstract

In this paper we continue our study of the Laplacian on manifolds with axial analytic asymptotically cylindrical ends initiated in~arXiv:1003.2538. By using the complex scaling method and the Phragm\'{e}n-Lindel\"{o}f principle we prove exponential decay of the eigenfunctions corresponding to the non-threshold eigenvalues of the Laplacian on functions. In the case of a manifold with (non-compact) boundary it is either the Dirichlet Laplacian or the Neumann Laplacian. We show that the rate of exponential decay of an eigenfunction is prescribed by the distance from the corresponding eigenvalue to the next threshold. Under our assumptions on the behaviour of the metric at infinity accumulation of isolated and embedded eigenvalues occur. The results on decay of eigenfunctions combined with the compactness argument due to Perry imply that the eigenvalues can accumulate only at thresholds and only from below. The eigenvalues are of finite multiplicity.