Sharp Gaussian Estimates for Heat Kernels of Schrödinger Operators

We characterize functions \(V\le 0\) for which the heat kernel of the Schrodinger 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.