In this short note, smoothness of the fundamental solution of Schr\"odinger equations on a complete manifold is studied. It is shown that (1) the fundamental solution is smooth under "mild" trapping conditions; (2) there is a Riemannian manifold which is equal to Euclidean space outside a compact set such that the fundamental solution is not smooth.
On the fundamental solution of an elliptic equation in nondivergence form
The Fundamental Solution of an Elliptic Equation with Singular Drift
Short time asymptotics of the fundamental solutions for Schrödinger equations with non-smooth potentials
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