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Functional integrals for the Schrödinger equation on compact Riemannian manifolds. (English. Russian original) Zbl 1133.58018

Math. Notes 79, No. 2, 178-184 (2006); translation from Mat. Zametki 79, No. 2, 194-200 (2006).
Summary: In this paper, we represent the solution of the Cauchy problem for the Schrödinger equation on compact Riemannian manifolds in terms of functional integrals with respect to the Wiener measure corresponding to the Brownian motion in a manifold and with respect to the Smolyanov surface measures constructed from the Wiener measure on trajectories in the underlying space. The representation of the solution is obtained for the case of analytic (on some sets) potential and analytic initial condition under certain assumptions on the geometric characteristics of the manifold. In the proof, we use a method due to Doss and the representations via functional integrals of the solution to the Cauchy problem for the heat equation in a compact Riemannian manifold.

MSC:

58J32 Boundary value problems on manifolds
35J10 Schrödinger operator, Schrödinger equation
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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