Resolvent estimates, wave decay, and resonance-free regions for star-shaped waveguides

TL;DR

This paper establishes resolvent estimates, wave decay, and resonance-free regions for star-shaped waveguides and related domains, advancing understanding of spectral properties and scattering phenomena in unbounded geometries.

Contribution

It introduces a new notion of star-shapedness for unbounded domains and derives resolvent estimates that impact wave decay and resonance analysis.

Findings

Resolvent estimates near the continuous spectrum for star-shaped domains.

Wave decay results for domains with infinite cylindrical ends.

Resonance-free regions established for certain waveguide geometries.

Abstract

Using coordinates $(x,y)\in \mathbb R\times \mathbb R^{d-1}$, we introduce the notion that an unbounded domain in $\mathbb R^d$ is star shaped with respect to $x=\pm \infty$. For such domains, we prove estimates on the resolvent of the Dirichlet Laplacian near the continuous spectrum. When the domain has infinite cylindrical ends, this has consequences for wave decay and resonance-free regions. Our results also cover examples beyond the star-shaped case, including scattering by a strictly convex obstacle inside a straight planar waveguide.