Time-slicing approximation of Feynman path integrals on compact manifolds

TL;DR

This paper develops a method to approximate fundamental solutions of the Schrödinger equation on compact manifolds using time-slicing of Feynman path integrals, accounting for scalar curvature effects.

Contribution

It introduces a new approach to construct and analyze the convergence of Feynman path integral approximations on compact manifolds, considering scalar curvature modifications.

Findings

Convergence of the approximation to the fundamental solution in the uniform operator topology.

The method only requires considering short classical paths for the approximation.

Proved stability and consistency of the short-time approximate solutions.

Abstract

We construct fundamental solutions to the time-dependent Schr\"odinger equations on compact manifolds by the time-slicing approximation of the Feynman path integral. We show that the iteration of short-time approximate solutions converges to the fundamental solutions to the Schr\"odinger equations modified by the scalar curvature in the uniform operator topology from the Sobolev space to the space of square integrable functions. In order to construct the time-slicing approximation by our method, we only need to consider broken paths consisting of sufficiently short classical paths. We prove the convergence to fundamental solutions by proving two important properties of the short-time approximate solution, the stability and the consistency.