Mathematical Research Letters

Volume 29 (2022)

Number 1

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

Pages: 101 – 130



T. J. Christiansen (Department of Mathematics, University of Missouri, Columbia, Mo., U.S.A.)

K. Datchev (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


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.

The authors gratefully acknowledge the partial support of the Simons Foundation (TC, collaboration grant for mathematicians), an MU Research Leave (TC), and the National Science Foundation (KD, Grant DMS-1708511). The authors thank Peter Hislop for helpful conversations.

Received 4 November 2019

Accepted 18 August 2020

Published 6 September 2022