Asymptotic expansion of the heat kernel trace of Laplacians with polynomial potentials

It is well known that the asymptotic expansion of the trace of the heat kernel for Laplace operators on smooth compact Riemannian manifolds can be obtained through termwise integration of the asymptotic expansion of the on-diagonal heat kernel. The purpose of this work is 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.