Titlebook: Rigorous Time Slicing Approach to Feynman Path Integrals; Daisuke Fujiwara Book 2017 Springer Japan KK 2017 Feynman path integral.Feynman

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Statement of Main Resultssemi-classical asymptotic formula called Birkhoff’s formula is proved from the standpoint of oscillatory integrals. In this chapter, these results as well as others are explained. Proofs will be given in subsequent chapters.

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Feynman Path Integral and Schr?dinger Equationtain the second term of the semi-classical asymptotic and prove that it satisfies the second transport equation. Our discussion of this is different from the usual method originated by Birkhoff (Bull Am Math Soc 39:681–700 (1933) [11]). Our method enables us to obtain the bound of the remainder term.

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Path Integrals and Oscillatory Integralsegral techniques and is given a definite value under some conditions. We give an example of a sufficient condition for that in Sect.?.. Furthermore, in such a case the stationary phase method, which is given by Theorem . in Sect.?., gives the value of the oscillatory integral asymptotically as ..

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Path Integrals and Oscillatory Integralse factor . oscillates rapidly and as a consequence there occurs a large scale of cancellation. Such an integral is commonly treated by oscillatory integral techniques and is given a definite value under some conditions. We give an example of a sufficient condition for that in Sect.?.. Furthermore, i

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Statement of Main Resultsies Assumption . and if the time interval is short, because it is an oscillatory integral that satisfies Assumption .. Furthermore, the time slicing approximation of Feynman path integrals converges to a limit as .. The limit turns out to be the fundamental solution of the Schr?dinger equation. The

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Feynman Path Integral and Schr?dinger Equationthe Schr?dinger equation. The main?tool is the .-boundedness theorem proof of which is left to Chap.?. in Part II. By the way we shall prove that the main term of the semi-classical asymptotic of the fundamental solution of the Schr?dinger equation satisfies the transport equations. At the end we ob

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Stationary Phase Method for Oscillatory Integrals over a Space of Large Dimensionm is given, which is independent of the dimension. This theorem enables us to discuss the time slicing approximation of Feynman path integrals when the dimension of the space goes to .. This was the central tool of our discussions in Sect.?5.4 of Chap.?..

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0921-3767 y the time slicing method, the method Feynman himself used, This book proves that Feynman‘s original definition of the path integral actually converges to the fundamental solution of the Schr?dinger equation at least in the short term if the potential is differentiable sufficiently many times and it

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