The Maxwell system (1865) in time-harmonic formulation has always an infinite dimensional kernel (given by gradient fields), even in bounded domains; therefore, the Maxwell essential spectrum is always not-empty, and many standard spectral theory techniques fail. Even more dramatically, dissipative Maxwell systems in bounded domains might have segments of essential spectrum along the imaginary axis.
I will discuss a few results regarding the spectrum of the time-harmonic Maxwell system in domains with interesting geometry. For product domains, I will show that the classical TE-TM modes decomposition of the eigenvalues generalises to the "curved case", where the domain is in the form $\Sigma \times I$, $\Sigma$ being a two-dimensional manifold. In particular, for thin domains $\Sigma_\epsilon = \Sigma \times (0, \epsilon)$, the eigenvalues of the Maxwell system converge to the eigenvalues of the Dirichlet Laplacian on $\Sigma$, as $\epsilon \to 0^+$. In unbounded domains, I will show a few examples showing that, depending on the geometry at infinity, the essential spectrum of the Maxwell system might assume very different shapes. Time-permitting, I will generalise this picture to the case of dissipative Maxwell systems, featuring non-constant, discontinuous, complex-valued coefficients.
Based on joint work with S. B\"ogli (Durham), M. Marletta (Cardiff), L. Provenzano (Sapienza Rome) and C. Tretter (Bern).
