Contents Online
Communications in Mathematical Sciences
Volume 20 (2022)
Number 5
Spectral structure of electromagnetic scattering on arbitrarily shaped dielectrics
Pages: 1363 – 1395
DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n5.a7
Author
Abstract
Spectral analysis is performed on the Born equation, a strongly singular integral equation modeling the interactions between electromagnetic waves and arbitrarily shaped dielectric scatterers. Compact and Hilbert–Schmidt operator polynomials are constructed from the Green operator of electromagnetic scattering on scatterers with smooth boundaries. As a consequence, it is shown that the strongly singular Born equation has a discrete spectrum, and that the spectral series $\sum_\lambda {\lvert \lambda \rvert}^2 {\lvert 1+2 \lambda \rvert$^4$ is convergent, counting multiplicities of the eigenvalues \lambda . This reveals a shape-independent optical resonance mode corresponding to a critical dielectric permittivity $\epsilon_r = -1$.
Keywords
electromagnetic scattering, optical resonance, Green operator, compact operator, Hilbert–Schmidt operator
2010 Mathematics Subject Classification
Primary 47G20. Secondary 35P25, 35Q60, 78A45.
Received 25 April 2021
Received revised 1 November 2021
Accepted 2 December 2021
Published 26 May 2022