Electromagnetic Stekloff eigenvalues: existence and behavior in the selfadjoint case

TL;DR

This paper analyzes the spectrum of electromagnetic Stekloff eigenvalues in the selfadjoint case, revealing a structured spectrum with essential and discrete parts, and provides a mathematical framework for their existence and behavior.

Contribution

It offers a detailed spectral analysis of the original Stekloff eigenvalue problem in the selfadjoint case, including the existence of infinite eigenvalue sequences and their accumulation points.

Findings

Spectrum has three parts: zero, positive eigenvalues, negative eigenvalues.

Infinite sequences of eigenvalues exist, accumulating at infinity and zero.

The analysis uses block operator representation and fixed point techniques.

Abstract

In [Camano, Lackner, Monk, SIAM J. Math. Anal., Vol. 49, No. 6, pp. 4376-4401 (2017)] it was suggested to use Stekloff eigenvalues for Maxwell equations as target signature for nondestructive testing via inverse scattering. The authors recognized that in general the eigenvalues due not correspond to the spectrum of a compact operator and hence proposed a modified eigenvalue problem with the desired properties. The Fredholmness and the approximation of both problems were analyzed in [Halla, arXiv:1909.00689 (2019)]. The present work considers the original eigenvalue problem in the selfadjoint case. We report that apart for a countable set of particular frequencies, the spectrum consists of three disjoint parts: The essential spectrum consisting of the point zero, an infinite sequence of positive eigenvalues which accumulate only at infinity and an infinite sequence of negative eigenvalues which accumulate only at zero. The analysis is based on a representation of the operator as block operator. For small/big enough eigenvalue parameter the Schur-complements with respect to different components can be build. For each Schur-complement the existence of an infinite sequence of eigenvalues is proved via a fixed point technique similar to [Cakoni, Haddar, Applicable Analysis, 88:4, 475-493 (2009)]. The modified eigenvalue problem considered in the above references arises as limit of one of the Schur-complements.