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

TL;DR

The paper introduces a rapid, versatile method for calculating heat-kernel coefficients of the Laplace operator on D-dimensional balls with various boundary conditions, providing explicit formulas and new results for coefficients B_3 to B_{10}.

Contribution

It develops a general analytic scheme for computing heat-kernel coefficients applicable to any basis functions and boundary conditions, with explicit new results for specific cases.

Findings

Derived simple formulas for heat-kernel coefficients on D-dimensional balls.

Provided new explicit results for coefficients B_3 to B_{10} in 3, 4, 5 dimensions.

Demonstrated the method's effectiveness for arbitrary coefficients and boundary conditions.

Abstract

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 $D=3,4,5$.