Resonance-free regions for negatively curved manifolds with cusps

DOI10.1353/AJM.2018.0020zbMATH Open1396.53061arXiv1505.02600OpenAlexW2234731580MaRDI QIDQ4582823

Yannick Bonthonneau

Publication date: 24 August 2018

Published in: American Journal of Mathematics (Search for Journal in Brave)

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 $varphi(s)$, $sinmathbbC$, the emph{scattering determinant}. We construct a semi-classical parametrix for this function in a half plane of $mathbbC$ 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.

Full work available at URL: https://arxiv.org/abs/1505.02600

zbMATH Keywords

Dirichlet seriesparametrixresonancescalar Laplaciannegative curvaturecusps

Mathematics Subject Classification ID

Global Riemannian geometry, including pinching (53C20) Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) (58J60) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21)

Related Items (1)

Sur les fonctions propres positives des variétés de Cartan-Hadamard. (On positive eigenfunctions of Cartan-Hadamard manifolds.)

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