# Sharp Gaussian estimates for heat kernels of Schr\"odinger operators

## TL;DR

This paper characterizes when the heat kernel of Schrödinger operators with non-positive potentials is comparable to the Gaussian kernel, resolving a question from 1998 and revealing that local integrability conditions are not always necessary.

## Contribution

It provides a complete characterization of potentials for which the heat kernel is comparable to the Gaussian kernel, especially in higher dimensions, answering an open problem.

## Key findings

- Characterization of potentials $V o 0$ for heat kernel comparability.
- In dimensions 4 and higher, conditions are more restrictive than boundedness of the Newtonian potential.
- Local $L^p$ integrability for $p>1$ is not necessary for comparability.

## Abstract

We characterize functions $V\le 0$ for which the heat kernel of the Schr\"o\-dinger operator $\Delta+V$ is comparable with the Gauss-Weierstrass kernel uniformly in space and time. In dimension $4$ and higher the condition turns out to be more restrictive than the condition of the boundedness of the Newtonian potential of $V$. This resolves the question of V.~Liskevich and Y.~Semenov posed in 1998. We also give specialized sufficient conditions for the comparability, showing that local $L^p$ integrability of $V$ for $p>1$ is not necessary for the comparability.