Sharp Gaussian estimates for heat kernels of Schrödinger operators

Abstract

We characterize functions $V\le 0$ for which the heat kernel of the Schrö\-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.

arXiv comment19 pages. Editorial changes. Theorem 1.4 has a shorter proof. The paper merges results of arXiv:1606.03745 and arXiv:1511.07167