Special Functions for Heat Kernel Expansion

TL;DR

This paper develops special functions related to the heat kernel expansion on Riemannian manifolds, providing new tools for analyzing Green's functions and applications in theoretical physics.

Contribution

It introduces two families of functions derived from Seeley-DeWitt coefficients, enhancing the analysis of heat kernels and Green's functions in geometric and physical contexts.

Findings

Constructed new function families with useful properties

Demonstrated the shift property of the Laplace operator on these functions

Applied the functions to decompose Green's functions near the diagonal

Abstract

In this paper, we study an asymptotic expansion of the heat kernel for a Laplace operator on a smooth Riemannian manifold without a boundary at enough small values of the proper time. The Seeley-DeWitt coefficients of this decomposition satisfy a set of recurrence relations, which we use to construct two function families of a special kind. Using these functions, we study the expansion of a local heat kernel for the inverse Laplace operator. We show that the new functions have some important properties. For example, we can consider the Laplace operator on the function set as a shift one. Also we describe various applications useful in theoretical physics and, in particular, we find a decomposition of Green's functions near the diagonal in terms of new functions.