Resonance free regions for nontrapping manifolds with cusps

TL;DR

This paper proves the existence of large resonance free regions for nonpositively curved manifolds with cusps using complex scaling and escape functions, marking a first in this area.

Contribution

It introduces a novel method combining escape functions and complex scaling to establish resonance free regions in manifolds with cusps.

Findings

Resonance free regions are logarithmically large for certain manifolds.

First proof of large resonance free regions in manifolds with cusps.

Method applicable to nonpositively curved perturbations of parabolic cylinders.

Abstract

This paper has been withdrawn because of a mistake in Lemma 7.4. For a corrected version of the main theorem, as well as various further developments and applications, see arXiv:1210.7736. For nonpositively curved perturbations of parabolic cylinders we establish the existence of a logarithmically large resonance free region. We use an escape function construction in a compact part of the manifold and near infinity we use the method of complex scaling. To the author's knowledge this is the first proof of a large resonance free region for manifolds with cusps.