Approximation of the zero-index transmission eigenvalues with a conductive boundary and parameter estimation

TL;DR

This paper introduces a spectral-Galerkin method to approximate zero-index transmission eigenvalues with conductive boundaries, providing convergence analysis, numerical examples, and a parameter estimation technique for the refractive index.

Contribution

The paper develops a new spectral-Galerkin approach for eigenvalues with conductive boundary conditions, including convergence proof and a method for refractive index estimation.

Findings

Convergence rate established using Weyl's law.

Numerical eigenvalues computed for disk and square domains.

Method for refractive index estimation based on conductivity parameter.

Abstract

In this paper, we present a Spectral-Galerkin Method to approximate the zero-index transmission eigenvalues with a conductive boundary condition. This is a new eigenvalue problem derived from the scalar inverse scattering problem for an isotropic media with a conductive boundary condition. In our analysis, we will consider the equivalent fourth-order eigenvalue problem where we establish the convergence when the approximation space is the span of finitely many Dirichlet eigenfunctions for the Laplacian. We establish the convergence rate of the spectral approximation by appealing to Weyl's law. Numerical examples for computing the eigenvalues and eigenfunctions for the unit disk and unit square are presented. Lastly, we provide a method for estimating the refractive index assuming the conductivity parameter is either sufficiently large or small but otherwise unknown.