A Proof of Schiffer's Conjecture in Starlike Domain by Far-Field Patterns

TL;DR

This paper proves Schiffer's conjecture in spectral geometry for starlike domains by demonstrating the uniqueness of inverse scattering solutions using far-field pattern analysis across all incident angles.

Contribution

It introduces a novel approach linking spectral geometry conjectures to scattering theory and proves the conjecture for starlike domains through inverse scattering techniques.

Findings

Schiffer's conjecture holds for starlike domains

Uniqueness of inverse scattering solutions is established

Far-field patterns determine the domain uniquely

Abstract

We formulate the Schiffer's conjecture in spectral geometry in the context of scattering theory. The problem is equivalent to finding a non-trivial solution in an interior transmission problem. We compare the back-scattering data of the perturbation along all incident angles. The uniqueness of the inverse scattering problem along each incident direction proves the Schiffer's conjecture.