A scattering approach to a surface with hyperbolic cusp

TL;DR

This paper develops a scattering theory for a family of surfaces transitioning to a hyperbolic cusp, analyzing spectral properties of the Laplacian and approximating eigenfunctions in the limit.

Contribution

It introduces a novel approach to spectral and scattering analysis on surfaces with hyperbolic cusps using a parameter-dependent metric.

Findings

Spectral and scattering theory described for surfaces with hyperbolic cusp

Zero Fourier coefficient of eigenfunctions approximates hyperbolic cusp case

Results hold for large spectral parameter values

Abstract

Let $X$ be a two-dimensional smooth manifold with boundary $S^{1}$ and $Y=[1,\infty)\times S^{1}$. We consider a family of complete surfaces arising by endowing $X\cup_{S^{1}}Y$ with a parameter dependent Riemannian metric, such that the restriction of the metric to $Y$ converges to the hyperbolic metric as a limit with respect to the parameter. We describe the associated spectral and scattering theory of the Laplacian for such a surface. We further show that on $Y$ the zero $S^{1}$-Fourier coefficient of the generalized eigenfunction of this Laplacian, as a family with respect to the parameter, approximates in a certain sense, for large values of the spectral parameter, the zero $S^{1}$-Fourier coefficient of the generalized eigenfunction of the Laplacian for the case of a surface with hyperbolic cusp.