Generalized Rellich's lemmas, uniqueness theorem and inside-out duality for scattering poles

TL;DR

This paper extends Rellich's lemmas to complex wavenumbers, establishing uniqueness in inverse scattering and exploring inside-out duality, with numerical validation of identifying scattering poles without prior obstacle knowledge.

Contribution

It introduces generalized Rellich's lemmas for complex wavenumbers and applies them to prove uniqueness and identify scattering poles without prior obstacle information.

Findings

Generalized Rellich's lemmas for complex wavenumbers

Uniqueness results for inverse scattering problems

Numerical validation of pole identification methods

Abstract

Scattering poles correspond to non-trivial scattered fields in the absence of incident waves and play a crucial role in the study of wave phenomena. These poles are complex wavenumbers with negative imaginary parts. In this paper, we prove two generalized Rellich's lemmas for scattered fields associated with complex wavenumbers. These lemmas are then used to establish uniqueness results for inverse scattering problems. We further explore the inside-out duality, which characterizes scattering poles through the linear sampling method applied to interior scattering problems. Notably, we demonstrate that exterior Dirichlet/Neumann poles can be identified without prior knowledge of the actual sound-soft or sound-hard obstacles. Numerical examples are provided to validate the theoretical results.