| Abstract: |
| 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 non-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.
If the $L^\infty$ coefficients $\epsilon$, $\mu$, and $\sigma$ are asymptotically constant, I will show that the essential spectrum of the Maxwell system in anisotropic conductive media with perfectly conducting boundary conditions can be characterised as the union of the essential spectra of a bounded operator and of an unbounded selfadjoint $\mathrm{\curl} \mathrm{\curl}$ operator. I will further discuss how we can concretely compute the spectrum of the original Maxwell system avoiding spectral pollution phenomena. I will conclude with some results aimed at relating the geometry of the domain $\Omega$ with the spectrum of the $\mathrm{\curl} \mathrm{\curl}$ operator.
Based on joint work with S. B\ogli (Durham), M. Marletta (Cardiff), L. Provenzano (Sapienza Rome) and C. Tretter (Bern). |
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