Estimates of global bounds for some Schrödinger heat kernels on manifolds

Abstract

We establish global bounds for the heat kernel of Schrödinger operators $-\Delta + V$ where $V$ is a certain long range potential. As a consequence we find some conditions for the heat kernel to have global Gaussian lower and upper bound. Some of the conditions are sharp if the potential does not change sign. We also provide a generalized Liouville theorem for Schrödinger operators and a refined version of the trace formula of Sa Barreto and Zworski [SZ].