A Super-Exponential Decaying Property of Odd-Dimensional Wave Scattered by an Obstacle

TL;DR

This paper investigates how super-exponential decay of wave propagators along back-scattered geodesics implies the obstacle must be trivial, leading to the conclusion that the wave vanishes entirely if such decay occurs.

Contribution

It establishes a super-exponential decay property for wave scattering in odd dimensions and links this decay to the triviality of the obstacle, using finite speed of propagation.

Findings

Super-exponential decay implies obstacle is trivial.

Fundamental solution decay leads to wave vanishing for all time.

Finite speed of propagation is key to the analysis.

Abstract

We examine an inverse backscattering property of wave motion imposed by an obstacle. We show that if the wave propagator decays super-exponentially along the back-scattered geodesics, then the involved scatterer must be trivial. In particular, if the fundamental solution decays super-exponentially some time after t=0, it vanishes for all time. We use finite speed of propagation in this article.