Reviews in Mathematical Physics

Using adequate strategies, we localize the spectrum of a class of differential operators. We consider the conditioning of the pseudo-spectrum for a family of nonselfadjoint convection-diffusion operators defined on an unbounded open set of ${ℝ}^n$.

This article explores a number of issues associated with the problem of calculating and detecting electromagnetic quantum induced energy and stress in a stationary dielectric material with a smooth inhomogeneous polarizability. By concentrating on a particular system composed of an ENZ-type (epsilon-near-zero) meta-material, chosen to have a particular anisotropic and smooth inhomogeneous permittivity, confined in an infinitely long perfectly conducting open rectangular waveguide, we are able to deduce analytically from the source-free Maxwell’s equations and their boundary conditions a complete set of bounded harmonic electromagnetic evanescent eigen-modes and their associated eigen-frequencies. Since these solutions prohibit the existence of asymptotic scattering states in the guide, the application of the conventional Lifshitz approach to the Casimir stress problem becomes uncertain. An alternative approach is adopted based upon the spectral properties of the system and a regularization scheme constructed with direct applicability to more general systems composed of dielectrics with smooth inhomogeneous permittivities and open systems that may only admit evanescent modes. This more general scheme enables one, for the first time, to prescribe precise criteria for the extraction of finite quantum expectation values from regularized mode sums together with error bounds on these values, and is used to derive analytic or numeric results for regularized electromagnetic ground state expectation values in the guide.

This paper deals with the construction of a suitable topological $K$-theory capable of classifying topological phases of dynamically stable systems described by gapped $\eta$-self-adjoint operators on a Krein space with indefinite metric $\eta$.