Asymptotic Expansion of the Heat Kernel Trace of Laplacians with Polynomial Potentials

TL;DR

This paper demonstrates how to derive the asymptotic expansion of the heat kernel trace for Laplacians with polynomial potentials on unbounded domains using a resummed on-diagonal heat kernel expansion.

Contribution

It extends the method of termwise integration of heat kernel expansions to unbounded domains with polynomial potentials, employing a resummed expansion approach.

Findings

Successfully obtained asymptotic expansion for unbounded domains

Extended classical methods to include polynomial potentials

Provided a new technique for heat kernel trace analysis

Abstract

It is well-known that the asymptotic expansion of the trace of the heat kernel for Laplace operators on smooth compact Riemmanian manifolds can be obtained through termwise integration of the asymptotic expansion of the on-diagonal heat kernel. It is the purpose of this work to show that, in certain circumstances, termwise integration can be used to obtain the asymptotic expansion of the heat kernel trace for Laplace operators endowed with a suitable polynomial potential on unbounded domains. This is achieved by utilizing a resummed form of the asymptotic expansion of the on-diagonal heat kernel.