Smoothness of the fundamental solution of Schr\"odinger equations with mild trapping

TL;DR

This paper investigates the smoothness of the fundamental solution to Schr"odinger equations on manifolds, showing smoothness under mild trapping conditions and providing a counterexample where smoothness fails outside a compact set.

Contribution

It establishes conditions for smoothness of the fundamental solution and constructs a manifold where smoothness does not hold, advancing understanding of Schr"odinger equations on manifolds.

Findings

Fundamental solution is smooth under mild trapping conditions

Existence of a manifold where the fundamental solution is not smooth

Smoothness depends on geometric trapping conditions

Abstract

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.