On the limiting amplitude principle for a transmission problem between a dielectric and a metamaterial
In this talk, we are interested in a transmission problem between a dielectric (the vacuum) and a metamaterial. The question we consider is the following: does the limiting amplitude principle hold in such a medium? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation. An answer is proposed here in the case of a two-layered medium composed of the vacuum and a particular metamaterial (Drude model). In this context, we reformulate the time-dependent Maxwell’s equations as a Schrödinger equation and perform its complete spectral analysis . This permits a quasi-explicit representation of the solution via the “generalized diagonalization” of the associated unbounded self-adjoint operator. As an application of this study, we show finally that the limiting amplitude principle holds except for a particular frequency, called the critical frequency, characterized by a ratio of permittivities and permeabilities equal to −1 across the interface. This frequency is a resonance of the system and the response to this excitation blows up linearly in time.
 M. Cassier, C. Hazard and P. Joly, Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part I: Generalized Fourier transform, Communications in Partial Differential Equations (2017), 42 (11), 1707-1748.