Klaus Kirsten

Heat kernel coefficients of the Laplace operator on the D-dimensional ball

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 skillful 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 B3,..., B10, corresponding to the D‐dimensional ball with all the mentioned boundary conditions and D=3,4,5.

Heat-kernels and functional determinants on the generalized cone

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

Functional determinants by contour integration methods

A_{5/2}

Casimir energies for massive scalar fields in a spherical geometry

coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced.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 theA5/2 coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced.

Functional determinants for general Sturm-Liouville problems

Abstract We present a simple and accessible method which uses contour integration methods to derive formulae for functional determinants. To make the presentation as clear as possible, the general idea is first illustrated on the simplest case: a second order differential operator with Dirichlet boundary conditions. The method is applicable to more general situations, and we discuss the way in which the formalism has to be developed to cover these cases. In particular, we also show that simple and elegant formulae exist for the physically important case of determinants where zero modes exist, but have been excluded.

Casimir energy for a massive fermionic quantum field with a spherical boundary

The Casimir energy corresponding to a massive scalar field with Dirichlet boundary conditions on a spherical bag is obtained. The field is considered, separately, inside and outside the bag. The renormalization procedure that is necessary to apply in each situation is studied in detail, in particular the differences occurring with respect to the case when the field occupies the whole space. The final result contains several constants that experience renormalization and can be determined only experimentally. The non-trivial finite parts that appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy for the first time. PACS: 02.30.+g, 02.40.+m hep-th/9608071 E-mail address: bordag@qft.physik.uni-leipzig.d400.de E-mail address: eli@zeta.ecm.ub.es E-mail address: kirsten@tph100.physik.uni-leipzig.de E-mail address: lese@zeta.ecm.ub.es 1

Zeta function determinant of the Laplace operator on the D-dimensional ball

Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of self-adjoint operators of the Sturm– Liouville type with arbitrary linear boundary conditions. The results hold whether or not the operators have negative eigenvalues. The physically important case of functional determinants of operators with a zero mode, but where that mode has been extracted, is studied in detail for the same range of situations as when no zero mode exists. The method of proof uses the properties of generalized zeta-functions. The general form of the final results is the same for the entire range of problems considered.

BOSE-EINSTEIN CONDENSATION OF ATOMIC GASES IN A GENERAL HARMONIC-OSCILLATOR CONFINING POTENTIAL TRAP

The vacuum energies corresponding to massive Dirac fields with the boundary conditions of the MIT bag model are obtained. The calculations are carried out with the fields occupying the regions inside and outside the bag, separately. The renormalization procedure for each of the situations is studied in detail, in particular the differences occurring with respect to the case when the field extends over the whole space. The final result contains several constants that undergo renormalization and can be determined experimentally only. The non-trivial finite parts which appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy in the fermionic case. The vacuum energy behaves as an inverse power of the mass, for large mass of the field.The vacuum energies corresponding to massive Dirac fields with the boundary conditions of the MIT bag model are obtained. The calculations are done with the fields occupying the regions inside and outside the bag, separately. The renormalization procedure for each of the situations is studied in detail, in particular the differences occurring with respect to the case when the field extends over the whole space. The final result contains several constants undergoing renormalization, which can be determined only experimentally. The non-trivial finite parts which appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy for the first time, in the fermionic case. The vacuum energy behaves like inverse powers of the mass for large masses. PACS: 11.10.Gh, 02.30.-f Running title: Casimir energy in the MIT bag E-mail address: eli@zeta.ecm.ub.es, elizalde@io.ieec.fcr.es E-mail address: michael.bordag@itp.uni-leipzig.de E-mail address: klaus.kirsten@itp.uni-leipzig.de 2 1

Heat kernel asymptotics with mixed boundary conditions

We present a direct approach for the calculation of functional determinants of the Laplace operator on balls. Dirichlet and Robin boundary conditions are considered. Using this approach, formulas for any value of the dimension,D, of the ball, can be obtained quite easily. Explicit results are presented here for dimensionsD=2,3,4,5 and 6.

Kaluza-Klein models as pistons

We present an analysis of Bose-Einstein condensation for a system of non- interacting spin-0 particles in a harmonic oscillator confining potential trap. We discuss why a confined system of particles differs both qualitatively and quanti- tatively from an identical system which is not confined. One crucial difference is that a confined system is not characterized by a critical temperature in the same way as an unconfined system such as the free boson gas. We present the results of both a numerical and analytic analysis of the problem of Bose-Einstein condensation in a general anisotropic harmonic oscillator confining potential.