Characterization of sharp global Gaussian estimates for Schr\"odinger   heat kernels
  We investigate when the fundamental solution of the Schr\"odinger equation $\partial_t=\Delta+V$ posseses sharp Gaussian bounds global in space and time. We give a characterization for $V\leq 0$ and a sufficient condition for general $V$. 
Krzysztof Bogdan, Jacek Dziuba\'nski, Karol Szczypkowski
math.FA math.AP