Resonance-free regions for negatively curved manifolds with cusps

TL;DR

This paper establishes the existence of resonance-free regions at high frequencies for negatively curved cusp manifolds, using a semi-classical parametrix construction, and explores resonance behavior in specific examples.

Contribution

It introduces a semi-classical parametrix for the scattering determinant on negatively curved cusp manifolds, proving the existence of resonance-free zones at high frequencies.

Findings

Resonance-free regions exist at high frequency for manifolds with one cusp.

Such regions also exist for manifolds with multiple cusps and generic metrics.

Explicit examples show sequences of resonances away from the spectrum.

Abstract

The Laplace-Beltrami operator on cusp manifolds has continuous spectrum. The resonances are complex numbers that replace the discrete spectrum of the compact case. They are the poles of a meromorphic function $\varphi(s)$, $s\in \mathbb{C}$, the \emph{scattering determinant}. We construct a semi-classical parametrix for this function in a half plane of $\mathbb{C}$ when the curvature of the manifold is negative. We deduce that for manifolds with one cusp, there is a zone without resonances at high frequency. This is true more generally for manifolds with several cusps and generic metrics. We also study some exceptional examples with almost explicit sequences of resonances away from the spectrum.