Resonances and balls in obstacle scattering with Neumann boundary conditions

TL;DR

This paper proves that in odd-dimensional spaces, if an obstacle's scattering resonances match those of a ball, then the obstacle must be a ball, establishing a uniqueness result in obstacle scattering with Neumann boundary conditions.

Contribution

It demonstrates a resonance-based uniqueness theorem for obstacles in odd dimensions, extending the understanding of inverse scattering problems with Neumann boundary conditions.

Findings

Resonance data uniquely determines spherical obstacles.

Results extend to unions of multiple identical balls.

Provides conditions under which obstacles are characterized by their resonances.

Abstract

We consider scattering by an obstacle in $\Real^d$, $d\geq 3 $ odd. We show that for the Neumann Laplacian if an obstacle has the same resonances as the ball of radius $\rho$ does, then the obstacle is a ball of radius $\rho$. We give related results for obstacles which are disjoint unions of several balls of the same radius.