Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

We study the relative zeta function for the couple of operators A0 and Aα, where A0 is the free unconstrained Laplacian in L2(Rd) (d ≥ 2) and Aα is the singular perturbation of A0 associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter α, which is related to the strength of the perturbation, is of the kind α = α (-Δ||), where -Δ|| is the free Laplacian in L2(Rd-1). Thus α may depend on the components of the wave vector parallel to the hyperplane; in this sense Aα describes a semitransparent hyperplane selecting transverse modes. As an application we give an expression for the associated thermal Casimir energy. Whenever α = χI(-Δ||), where χI is the characteristic function of an interval I, the thermal Casimir energy can be explicitly computed.