Abstract
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.
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Acknowledgements
The author thanks Professor Kenichi Ito, Professor Shu Nakamura and Professor Yoshihisa Miyanishi for a lot of valuable discussions, advices and comments. He also thanks to Professor Masaki Kawamoto for his encouragements.
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Communicated by Alain Joye.
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The author is supported by Leading Graduate Course for Frontiers of Mathematical Sciences and Physics (FMSP), at Graduate School of Mathematical Science, the University of Tokyo.
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Fukushima, S. Time-Slicing Approximation of Feynman Path Integrals on Compact Manifolds. Ann. Henri Poincaré 22, 3871–3914 (2021). https://doi.org/10.1007/s00023-021-01079-4
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DOI: https://doi.org/10.1007/s00023-021-01079-4