1999 Fiscal Year Final Research Report Summary
Semiclassical approximations and hyperasymptotics
| Project/Area Number |
10640148
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | MIYAGI UNIVERSITY OF EDUCATION |
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Principal Investigator |
HURUKI Yamada Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (00092578)
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| Co-Investigator(Kenkyū-buntansha) |
TAKEMOTO Hideo Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (00004408)
SHIRAI Susumu Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (30115175)
AZUMA Kazuoki Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (70005776)
TAKASE Koichi Faculty of Education, Miyagi University of Education Associate Professor, 教育学部, 助教授 (60197093)
URYU Hitoshi Faculty of Education, Miyagi University of Education Professor, 教育学部, 教授 (10139511)
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| Project Period (FY) |
1998 – 1999
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| Keywords | semiclassical approximation / hyperasymptotics / WKB approximation / resurgent function / Schroedinger equation |
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| Research Abstract |
In connection with the theory of semiclassical approximations, we studied the following problem. How can we treat the exponentially small terms in the asymptotic analysis. To do so, we studied some useful tools such as the method of hyperasymptotics, the theory of resurgent functions and the so-called exact WKB-method.
By using those tools, we got some hints and results as follows: (1) For 1-dimensional stationary Schroedinger equations with polynomial potentials, we treated the exponentially small terms as the influences of distant wells. In the research of this problem, we used our previous results for 1-dimensional stationary Schroedinger equations with piecewise constant potentials. In them, explicit representations of the quantum condition and the characterization of then by means of integral forms were shown. (2) We considered the asymptotic expansions of functions represented by some integral. The method to take out the exponentially small quantities as the influences of not nearby singular points of the phase function of the integral was searched. In so doing, we examined the method of steepest descent in a new viewpoint. The essential point is, if we deform the path of integration, it is essential not only to take the path through saddle points, but to take exactly the steepest paths.
Further, points of view essentially used were: (1) In considering the relations between asymptotic expansion and integral by means of Borel-Laplace transformation of the original expansion, the analysis of singularities of Borel transformed function in the complex domain is indispensable, and information of these singular points are include in the coefficients of original asymptotic expansions. (2) To characterize functions represented by asymptotic expansions as exact quantities, it is necessary to take the resummation on complex Lagrangian manifolds. Our researches are now in continuation and some of our results will be published in order.
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