On uniqueness in the inverse obstacle problem via the positive supersolutions of the Helmholtz equation

Abstract

This paper is concerned with an inverse obstacle scattering problem of an acoustic wave for a single incident plane wave and a wave number. The Colton-Sleeman theorem states the unique recovery of sound-soft obstacles with a smooth boundary from the far-field pattern of the scattered wave for a single incident plane wave at a fixed wave number. The wave number has a bound given by the first Dirichlet eigenvalue of the negative Laplacian in an open ball that contains the obstacles. In this paper, another proof of the Colton-Sleeman theorem that works also for the case when we have a known {\it unbounded} set that contains obstacles is given. Unlike original one, the proof given here is not based on the monotonicity of the first Dirichlet eigenvalue of the negative Laplacian. Instead, it relies on a {\it positive supersolution} of the Helmholtz equation in a known domain that contains obstacles. Some corollaries which are new and not covered by the Colton-Sleeman Theorem are also given.