Resonance free regions for nontrapping manifolds with cusps

TL;DR

This paper establishes resolvent estimates for nontrapping manifolds with cusps, demonstrating the existence of wide resonance free regions and implications for wave and Schrödinger equations.

Contribution

It introduces new resolvent estimates for manifolds with cusps, showing optimality and implications for resonance free regions and wave expansions.

Findings

Existence of arbitrarily wide resonance free strips

Lossless local smoothing estimates for Schrödinger equation

Optimality of resolvent estimates in certain contexts

Abstract

We prove resolvent estimates for nontrapping manifolds with cusps which imply the existence of arbitrarily wide resonance free strips, local smoothing for the Schrodinger equation, and resonant wave expansions. We obtain lossless limiting absorption and local smoothing estimates, but the estimates on the holomorphically continued resolvent exhibit losses. We prove that these estimates are optimal in certain respects.