Sharp Gaussian estimates for Schr\"odinger heat kernels: $L^p$ integrability conditions

TL;DR

This paper establishes new conditions under which the Schr"odinger heat kernel is comparable to the Gaussian kernel, showing that local $L^p$ integrability of the potential is not necessary for this comparability.

Contribution

The authors provide novel sufficient conditions for the Schr"odinger heat kernel's comparability to the Gaussian kernel, relaxing the need for local $L^p$ integrability of the potential.

Findings

New sufficient conditions for heat kernel comparability

Local $L^p$ integrability not necessary for comparability

Broader applicability of Gaussian estimates for Schr"odinger operators

Abstract

We give new sufficient conditions for comparability of the fundamental solution of the Schr\"odinger equation $\partial_t=\Delta+V$ with the Gauss-Weierstrass kernel and show that local $L^p$ integrability of $V$ for $p> 1$ is not necessary for the comparability.