Sharp Gaussian upper bounds for Schr\"odinger heat kernel on gradient shrinking Ricci solitons

On gradient shrinking Ricci solitons, we observe that the study of Schr"odinger heat kernel seems to be more natural than the classical heat kernel. In this paper we derive sharp Gaussian upper bounds for the Schr"odinger heat kernel on complete gradient shrinking Ricci solitons. As applications, we prove sharp upper bounds for the Green’s function of the Schr"odinger operator. We also prove sharp lower bounds for eigenvalues of the Schr"odinger operator. These sharp cases are all achieved at Euclidean Gaussian shrinking Ricci solitons.