Multiple reflection expansion and heat kernel coefficients

TL;DR

This paper introduces the multiple reflection expansion method for calculating heat kernel coefficients, demonstrating its application to spheres with various boundary conditions and background potentials, including delta functions.

Contribution

It develops the multiple reflection expansion technique for heat kernel coefficients and applies it to complex boundary and matching conditions on spheres, revealing new relations and limitations.

Findings

Derived heat kernel coefficients for spheres with Dirichlet and Neumann conditions.

Established relations between different boundary condition coefficients.

Showed non-convergence of the expansion with delta potentials.

Abstract

We propose the multiple reflection expansion as a tool for the calculation of heat kernel coefficients. As an example, we give the coefficients for a sphere as a finite sum over reflections, obtaining as a byproduct a relation between the coefficients for Dirichlet and Neumann boundary conditions. Further, we calculate the heat kernel coefficients for the most general matching conditions on the surface of a sphere, including those cases corresponding to the presence of delta and delta prime background potentials. In the latter case, the multiple reflection expansion is shown to be non-convergent.