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

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