Construction of fundamental solutions to Schröidinger equations on compact manifolds by Feynman path integral methods (Spectral and Scattering Theory and Related Topics)

Abstract

We construct fundamental solutions to Schröidinger equations on compact Riemannian manifolds. We employ a time-slicing approximation, which is a mathematically rigorous method of defining the Feynman path integral. Our time-slicing approximation converges to a fundamental solution to the Schröidinger equation modified by the scalar curvature. The coefficient of the scalar curvature in the modified Schröidinger equation depends on the choice of the amplitude which appears in the definition of the time-slicing approximation.