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

We characterize functions $V\le 0$ for which the heat kernel of the Schr"o-dinger operator $\mathrm{\Delta }+V$ 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.