Relative Spectral Functions for Two Point Interactions in Three Dimensions

The regularized partition function at finite temperature for a massless scalar field interacting with two delta-like external potentials in R is evaluated in an explicit form making use of a rigorous approach based on the introduction of relative determinants associated with the presence of non compact manifold, as defined by Müller in [14], and on results of Albeverio et al. [1], dealing with the selfadjoint extensions of Laplace operators in R. Recently, there has been a growing interest in the Casimir effect, namely the manifestation of vacuum energy at experimental level as well as at theoretical one (see, for example [13, 12] and references therein). The aim of these notes is to study the Casimir energy related to a massless scalar field in a flat space-time perturbed by the presence of two pointlike (uncharged) “impurities” at a relative distance a in R, modelled by delta-like potentials, and associated with the so called semi-transparent boundary conditions (see [6, 9, 10] and references therein). The case concerning one delta potential has been already treated (see for example [17, 16, 18, 7, 11]). In spite of the increasing interest in the Casimir effect and several explicit results obtained, the model we are going to present deserves interest because it is solvable and due to the issues arising in the mathematics involved in the description of the model itself. In fact, from one side, we need a rigorous mathematical description of the Schröedinger-like operator with delta-like potentials, and from the other side, a technique to regularize the functional determinant of self-adjoint elliptic operators associated with continuous spectrum. These two main difficulties can be faced and overtaken using results of Albeverio et al. [1], and Müller [14], respectively, as we will show. In order to start formulating the problem, we use the approach of Finite Temperature Quantum Field Theory, based on the imaginary time formalism (see for example [15] and [4, 5]). We consider a massless scalar field in four dimensional Minkowski space-time interacting with an external field represented by a potential q. Thus, one is dealing with the manifold X(T ) = S β/2π × M , where S r is the circle of radius r, β = 1 T , period of imaginary compactified time is the inverse of the temperature, and M is a three dimensional manifold. The relevant operator reads H = −∆X(T ) + q = −∂2 u−∆M +q, where ∆Y is the Laplace operator on a manifold Y defined by some Riemannian structure, and q : M → R is a suitable potential. The canonical partition function at temperature T of this model may be formally written as Z = det− 1 2 (lH) , 2000 MATHEMATICS SUBJECT CLASSIFICATION: 58G26 (81T16, 11M06).