The second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth

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  • second term of the semi classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth

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Abstract

Recently N. Kumano-go [15] succeeded in proving that piecewise linear time slicing approximation to Feynman path integral<br>\int F(\gamma)e^{i\ u S(\gamma)}\,\mathscr{D}[\gamma]<br>actually converges to the limit as the mesh of division of time goes to 0 if the functional F(γ) of paths γ belongs to a certain class of functionals, which includes, as a typical example, Stieltjes integral of the following form;<br>\begin{equation}\label{stieltjesint}%1F(\gamma) = \int_0^T f(t,\gamma(t)) \ ho(dt), \ ag{1}\end{equation}<br>where ρ(t) is a function of bounded variation and f(t, x) is a sufficiently smooth function with polynomial growth as |x| → ∞. Moreover, he rigorously showed that the limit, which we call the Feynman path integral, has rich properties (see also [10]).<br>The present paper has two aims. The first aim is to show that a large part of discussion in [15] becomes much simpler and clearer if one uses piecewise classical paths in place of piecewise linear paths.<br>The second aim is to explain that the use of piecewise classical paths naturally leads us to an analytic formula for the second term of the semi-classical asymptotic expansion of the Feynman path integrals under a little stronger assumptions than that in [15]. If F(γ) ≡ 1, this second term coincides with the one given by G. D. Birkhoff [1].

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