  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.
