Communications in Mathematical Sciences

Volume 20 (2022)

Number 5

Spectral structure of electromagnetic scattering on arbitrarily shaped dielectrics

Pages: 1363 – 1395



Yajun Zhou (Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey, U.S.A.; and Academy of Advanced Interdisciplinary Studies (AAIS), Peking University, Beijing, China)


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 $\epsilon_r = -1$.


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