A Quantitative Vainberg Method for Black Box Scattering

TL;DR

This paper develops a quantitative version of Vainberg's method linking resonance-free regions to wave propagation properties, providing sharp bounds on resonances for nontrapping geometries with conic points.

Contribution

It introduces a quantitative approach connecting resonance-free regions to wave singularity propagation, with applications to nontrapping manifolds and polygonal domains.

Findings

Logarithmic resonance free regions near the real axis with polynomial resolvent bounds.

Resonances in logarithmic strips related to wave trace singularities at large times.

Sharp bounds on resonance free regions for nontrapping geometries with conic points.

Abstract

We give a quantitative version of Vainberg's method relating pole free regions to propagation of singularities for black box scatterers. In particular, we show that there is a logarithmic resonance free region near the real axis of size $\tau$ with polynomial bounds on the resolvent if and only if the wave propagator gains derivatives at rate $\tau$. Next we show that if there exist singularities in the wave trace at times tending to infinity which smooth at rate $\tau$, then there are resonances in logarithmic strips whose width is given by $\tau$. As our main application of these results, we give sharp bounds on the size of resonance free regions in scattering on geometrically nontrapping manifolds with conic points. Moreover, these bounds are generically optimal on exteriors of nontrapping polygonal domains.