Existence and stability of electromagnetic Stekloff eigenvalues with a trace class modification

TL;DR

This paper investigates a generalized electromagnetic Stekloff eigenvalue problem with a trace class modification, proving the existence of infinitely many eigenvalues under certain conditions and establishing their stability and convergence properties.

Contribution

It introduces a new trace class approach to electromagnetic Stekloff eigenvalues, demonstrating their existence, stability, and convergence as the smoothing parameter varies.

Findings

Infinitely many eigenvalues exist for high smoothing parameters.

Eigenvalues are stable with respect to material coefficient variations.

Eigenvalues converge to the standard class as smoothing diminishes.

Abstract

A recent area of interest is the development and study of eigenvalue problems arising in scattering theory that may provide potential target signatures for use in nondestructive testing of materials. We consider a generalization of the electromagnetic Stekloff eigenvalue problem that depends upon a smoothing parameter, for which we establish two main results that were previously unavailable for this type of eigenvalue problem. First, we use the theory of trace class operators to prove that infinitely many eigenvalues exist for a sufficiently high degree of smoothing, even for an absorbing medium. Second, we leverage regularity results for Maxwell's equations in order to establish stability results for the eigenvalues with respect to the material coefficients, and we show that this generalized class of Stekloff eigenvalues converges to the standard class as the smoothing parameter approaches zero.