We construct fundamental solutions to the time-dependent Schrödinger 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 Schrödinger 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.

我们通过费曼路径积分的时间切片近似构造了紧致流形上的时间相关薛定谔方程的基本解。我们表明，短时近似解的迭代收敛到薛定谔方程的基本解，该方程由从 Sobolev 空间到平方可积函数空间的均匀算子拓扑中的标量曲率修正。为了通过我们的方法构建时间切片近似，我们只需要考虑由足够短的经典路径组成的断路。我们通过证明短时近似解的两个重要性质，稳定性和一致性来证明基本解的收敛性。