Some Inverse Spectral Results in Exterior Transmission Problem

TL;DR

This paper develops an inverse spectral theory for a domain with an inhomogeneous cavity, linking eigenvalue distributions to inhomogeneity characteristics through far-field measurements, advancing understanding of inverse problems in wave scattering.

Contribution

It introduces a novel inverse spectral approach for penetrable inhomogeneous media, connecting ODE and PDE systems via analytic continuation to achieve uniqueness results.

Findings

Eigenvalue density described in the complex plane for each scattering angle

Proved inverse uniqueness of inhomogeneity from far-field measurements

Constructed an ODE system linking inside and outside solutions

Abstract

We consider an inverse spectral theory in a domain with the cavity that is bounded by a penetrable inhomogeneous medium. An ODE system is constructed piecewise through the solutions inside and outside the cavity. The ODE system is connected to the PDE system via the analytic continuation. For each scattered angle, we describe its eigenvalue density in the complex plane, and prove an inverse uniqueness on the inhomogeneity by the measurements in the far-fields.