Generic simplicity of resonances in obstacle scattering

TL;DR

This paper proves that in obstacle scattering, all resonances are generically simple for a broad class of obstacles with smooth boundaries, in both odd and even dimensions.

Contribution

It establishes the generic simplicity of resonances in Dirichlet obstacle scattering across various dimensions and boundary smoothness classes.

Findings

Resonances are generically simple in odd dimensions.

Resonances are generically simple in even dimensions.

The result applies to obstacles with $C^k$ and $C^\infty$ boundaries.

Abstract

We show that all resonances in Dirichlet obstacle scattering (in $\mathbb{C}$ in odd dimensions and in the logarithmic cover of $\mathbb{C}\setminus\{0\}$ in even dimensions) are generically simple in the class of obstacles with $C^k$ (and $C^\infty$) boundaries, $k \geq 2$.