Asymptotic and exact expansions of heat traces

TL;DR

This paper investigates the conditions under which heat traces of certain operators have short-time asymptotic expansions, using spectral zeta-functions and inverse Mellin transforms, with applications to various operators.

Contribution

It establishes necessary conditions for the existence of heat trace asymptotic expansions based on spectral zeta-functions and demonstrates their validity for a broad class of operators.

Findings

Derived conditions for short-time asymptotic expansions.

Proved these conditions hold for many operators.

Provided explicit examples illustrating the results.

Abstract

We study heat traces associated with positive unbounded operators with compact inverses. With the help of the inverse Mellin transform we derive necessary conditions for the existence of a short time asymptotic expansion. The conditions are formulated in terms of the meromorphic extension of the associated spectral zeta-functions and proven to be verified for a large class of operators. We also address the problem of convergence of the obtained asymptotic expansions. General results are illustrated with a number of explicit examples.