Counting the Resonances in High and Even Dimensional Obstacle Scattering

TL;DR

This paper establishes a polynomial lower bound on the number of resonances for obstacle scattering in high even-dimensional Euclidean spaces using a Poisson summation formula and wave trace analysis.

Contribution

It introduces a novel approach combining Poisson summation and wave trace singularity analysis to bound resonances in even-dimensional obstacle scattering.

Findings

Polynomial lower bound for resonances in even dimensions

Application of Poisson summation formula in resonance counting

Use of wave trace singularity to estimate resonance distribution

Abstract

In this paper, we give a polynomial lower bound for the resonances of $-\Delta$ perturbed by an obstacle in even-dimensional Euclidean spaces, $n\geq4$. The proof is based on a Poisson Summation Formula which comes from the Hadamard factorization theorem in the open upper complex plane. We take advantage of the singularity of regularized wave trace to give the pole/resonance counting function over the principal branch of logarithmic plane a lower bound.