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- W2079791108 abstract "We are concerned with rigorously defined, by time slicing method, Feynman path integral ∫ Ωx,y F (γ)eiνS(γ)D(γ) of a functional F (γ), cf. [6]. Here Ωx,y is the set of paths γ(t) in R starting from a point y ∈ R at time s and arriving at x ∈ R at time s′, S(γ) is the action of γ and ν = 2πh−1, with Planck’s constant h. If p(γ) is a vector field on the path space with suitable property, we prove the following integration by parts formula for Feynman path integral: ∫ Ωx,y DF (γ)[p(γ)]eiνS(γ)D(γ) = − ∫ Ωx,y F (γ)Div p(γ)eiνS(γ)D(γ)− iν ∫ Ωx,y F (γ)DS(γ)[p(γ)]eiνS(γ)D(γ). Here DF (γ)[p(γ)] and DS(γ)[p(γ)] are differentials of F (γ) and S(γ) evaluated in the direction of p(γ), respectively, and Div p(γ) is divergence of vector fields p(γ). This formula is an analogy to Elworthy’s integration by parts formula for Wiener integrals, cf. [1]. As an application, we prove a semiclassical asymptotic formula of the Feynman path integrals which gives us a sharp information in the case F (γ∗) = 0. Here γ∗ is the stationary point of the phase S(γ). 1 Feynman path integrals Feynman’s path integral introduced by [2] is a method to construct the fundamental solution k(s′, s; , x, y) of Schrodinger equation using Lagrangian of classical mechanics L(t, ẋ, x) = 1 2 |ẋ|2 − V (t, x). Here V (t.x) is the potential field and x is the position of the particle and ẋ is the velocity. Let [s, s′] be a time interval. Action of a path, γ : [s, s′] t → γ(t) ∈ R is" @default.
- W2079791108 created "2016-06-24" @default.
- W2079791108 creator A5006434906 @default.
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- W2079791108 creator A5057373270 @default.
- W2079791108 date "2015-08-19" @default.
- W2079791108 modified "2023-09-24" @default.
- W2079791108 title "Proceedings of the 40th Sapporo Symposium on Partial Differential Equations" @default.
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