A pr 2 00 7 A Rigorous Time - Domain Analysis of Full – Wave Electromagnetic Cloaking ( Invisibility ) ∗ †

There is currently a great deal of interest in the theoretical and practical possibility of cloaking objects from the observation by electromagnetic waves. The basic idea of these invisibility devices [8, 9, 13], [18] is to use anisotropic transformation media whose permittivity and permeability ε , μ, are obtained from the ones, ε 0 , μ λν 0 , of isotropic media, by singular transformations of coordinates. In this paper we study electromagnetic cloaking in the time-domain using the formalism of time-dependent scattering theory [23]. This formalism provides us with a rigorous method to analyze the propagation of electromagnetic wave packets with finite energy in transformation media. In particular, it allows us to settle in an unambiguous way the mathematical problems posed by the singularities of the inverse of the permittivity and the permeability of the transformation media on the boundary of the cloaked objects. Von Neumann’s theory of self-adjoint extensions of symmetric operators plays an important role on this issue. We write Maxwell’s equations in Schrödinger form with the electromagnetic propagator playing the role of the Hamiltonian. We prove that the electromagnetic propagator outside of the cloaked objects is essentially self-adjoint. This means that it has only one self-adjoint extension, AΩ, and that this self-adjoint extension generates the only possible unitary time evolution, with constant energy, for finite energy electromagnetic waves, propagating outside of the cloaked objects. PACS classification scheme 2006: 41.20.Jb, 02.30.Tb,02.30.Zz, 02.60.Lj. Research partially supported by CONACYT under Project P42553F. Fellow Sistema Nacional de Investigadores.