Nonsmoothness of the boundary and the relevant heat kernel coefficients

TL;DR

This paper analyzes how corners and boundary singularities affect heat kernel coefficients, providing formulas and patterns for calculating these contributions, with implications for vacuum energy in regions with nonsmooth boundaries.

Contribution

It introduces a method to compute heat kernel coefficients contributed by boundary corners and singularities, revealing patterns and formulating rules for polygons and cylindrical regions.

Findings

Calculated individual contributions of boundary corners to heat kernel coefficients

Revealed patterns in contributions from boundary singularities

Formulated rules for heat kernel coefficients in polygonal and cylindrical regions

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.