No abstract

Spectral functions relevant in the context of quantum field theory under the influence of spherically symmetric external conditions are analysed. Examples comprise heatkernels, determinants and spectral sums needed for the analysis of Casimir energies. First, we summarize that a convenient way of handling them is to use the associated zeta function. A way to determine all its needed properties is derived. Using the connection with the mentioned spectral functions, we provide: i.) a method for the calculation of heat-kernel coefficients of Laplace-like operators on Riemannian manifolds with smooth boundaries and ii.) an analysis of vacuum energies in the presence of spherically symmetric boundaries and external background potentials.

We present a very quick and powerful method for the calculation of heat-kernel coefficients. It makes use of rather common ideas, as integral representations of the spectral sum, Mellin transforms, non-trivial commutation of series and integrals and skilful analytic continuation of zeta functions on the complex plane. We apply our method to the case of the heat-kernel expansion of the Laplace operator on a D-dimensional ball with either Dirichlet, Neumann or, in general, Robin boundary conditions. The final formulas are quite simple. Using this case as an example, we illustrate in detail our scheme -which serves for the calculation of an (in principle) arbitrary number of heat-kernel coefficients in any situation when the basis functions are known. We provide a complete list of new results for the coefficients B 3 , ..., B 10 , corresponding to the D-dimensional ball with all the mentioned boundary conditions and D = 3, 4, 5.

We consider zeta functions and heat-kernel expansions on the bounded, generalized cone in arbitrary dimensions using an improved calculational technique. The specific case of a global monopole is analysed in detail and some restrictions thereby placed on the A 5/2 coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced.PACS number(s): 02.30.-f

We analyze the ultraviolet divergences in the ground state energy for a penetrable sphere and a dielectric ball. We argue that for massless fields subtraction of the ''empty space'' or the ''unbounded medium'' contribution is not enough to make the ground state energy finite whenever the heat kernel coefficient a 2 is not zero. It turns out that a 2 0 for a penetrable sphere, a general dielectric background, and the dielectric ball. To our surprise, for more singular configurations, as in the presence of sharp boundaries, the heat kernel coefficients behave to some extent better than in the corresponding smooth cases, making, for instance, the dilute dielectric ball a well-defined problem. ͓S0556-2821͑99͒03508-0͔