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

## 抄録

Recently N. Kumano-go [<b>15</b>] 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 <i>F</i>(γ) of paths γ belongs to a certain class of functionals, which includes, as a typical example, Stieltjes integral of the following form;<br>$$\label{stieltjesint}%1F(\gamma) = \int_0^T f(t,\gamma(t)) \ho(dt), \ag{1}$$<br>where ρ(<i>t</i>) is a function of bounded variation and <i>f</i>(<i>t</i>, <i>x</i>) is a sufficiently smooth function with polynomial growth as |<i>x</i>| → ∞. Moreover, he rigorously showed that the limit, which we call the Feynman path integral, has rich properties (see also [<b>10</b>]).<br>The present paper has two aims. The first aim is to show that a large part of discussion in [<b>15</b>] 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 [<b>15</b>]. If <i>F</i>(γ) ≡ 1, this second term coincides with the one given by G. D. Birkhoff [<b>1</b>].

## 収録刊行物

• Journal of the Mathematical Society of Japan

Journal of the Mathematical Society of Japan 58(3), 837-867, 2006-07-01

一般社団法人 日本数学会

## 各種コード

• NII論文ID(NAID)
10018381149
• NII書誌ID(NCID)
AA0070177X
• 本文言語コード
ENG
• 資料種別
ART
• ISSN
00255645
• NDL 記事登録ID
7987167
• NDL 雑誌分類
ZM31(科学技術--数学)
• NDL 請求記号
Z53-A209
• データ提供元
CJP書誌  NDL  J-STAGE

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