Nonsmoothness of the boundary and the relevant heat kernel coefficients

Abstract

The contributions to the heat kernel coefficients generated by the corners of the boundary are studied. For this purpose the internal and external sectors of a wedge and a cone are considered. These sectors are obtained by introducing, inside the wedge, a cylindrical boundary. Transition to a cone is accomplished by identification of the wedge sides. The basic result of the paper is the calculation of the individual contributions to the heat kernel coefficients generated by the boundary singularities. In the course of this analysis certain patterns, that are followed by these contributions, are revealed. The implications of the obtained results in calculations of the vacuum energy for regions with nonsmooth boundary are discussed. The rules for obtaining all the heat kernel coefficients for the minus Laplace operator defined on a polygon or in its cylindrical generalization are formulated.

30 pages, LaTeX2e using iopart.cls, 1 figure in eps format, 3 tables; substantial changes are introduced in calculations concerning external sector

References in corpus (7)

• New Developments in the Casimir Effect
• Casimir effect for a perfectly conducting wedge in terms of local zeta function
• The heat kernel coefficients for the dielectric cylinder
• High temperature asymptotics of thermodynamic functions of electromagnetic field subjected to boundary conditions on a sphere and cylinder
• Local Casimir Energy for a Wedge with circular outer boundary
• Casimir Effect for a Dielectric Wedge
• Casimir energy of a dilute dispersive dielectric ball: realistic microscopic model

Cited by in corpus (27)

• Heat kernel expansion: user's manual
• Thermal Casimir effect in ideal metal rectangular boxes
• The number of eigenstates: counting function and heat kernel
• Casimir energy in the Fulling--Rindler vacuum
• Casimir Effect for a Semitransparent Wedge and an Annular Piston
• Repulsive Casimir-Polder forces from cosmic strings
• Spectral analysis of a flat plasma sheet model
• Surface vacuum energy and stresses on a plate uniformly accelerated through the Fulling-Rindler vacuum
• Electromagnetic $\delta$-function sphere
• Electromagnetic Casimir densities for a wedge with a coaxial cylindrical shell
• Electrodynamic Casimir Effect in a Medium-Filled Wedge II
• Electrodynamic Casimir Effect in a Medium-Filled Wedge
• Vacuum energy in conical space with additional boundary conditions
• Casimir effect at nonzero temperature for wedges and cylinders
• Whightman function and scalar Casimir densities for a wedge with a cylindrical boundary
• An approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions
• Investigations of the torque anomaly in an annular sector. I. Global calculations, scalar case
• Casimir effect due to a single boundary as a manifestation of the Weyl problem
• The heat kernel on curvilinear polygonal domains in surfaces
• Wightman function and scalar Casimir densities for a wedge with two cylindrical boundaries
• Semiclassical $p$-branes in hyperbolic space
• Origin of quantum shape effect
• Investigations of the torque anomaly in an annular sector. II. Global calculations, electromagnetic case
• Casimir energy of a cylindrical shell of elliptical cross section
• Electromagnetic Casimir Effect in Wedge Geometry and the Energy-Momentum Tensor in Media
• Thermal Casimir Effect for Rectangular Cavities inside D+1-dimensional Minkowski Spacetime Revisited
• Some Developments of the Casimir Effect in $p$-Cavity of $(D+1)$-Dimensional Spacetime