2001 Fiscal Year Final Research Report Summary
Exact WKB analysis and microlocal analysis
| Project/Area Number |
11440042
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | KYOTO UNIVERSITY |
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Principal Investigator |
KAWAI Takahiro Research Institute for Mathematical Sciences, Kyoto University, Professor, 数理解析研究所, 教授 (20027379)
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| Co-Investigator(Kenkyū-buntansha) |
TAKEI Yoshitsugu Research Institute for Mathematical Sciences, Kyoto University, Associate Professor, 数理解析研究所, 助教授 (00212019)
AOKI Takashi School of Science and Engineering, Kinki University, Professor, 理工学部, 教授 (80159285)
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| Project Period (FY) |
1999 – 2001
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| Keywords | exact WKB analysis / microlocal analysis / Stokes geometry / exact steepest descent path / exact steepest descent method / virtual turning point / infra-red divergence / natural boundaries |
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| Research Abstract |
1°Concerning the Stokes geometry for higher order linear ordinary differential equations with a large parameter,
(1) we first made a concrete and detailed study of Laplace-type equations with the help of the ordinary steepest descent method ([AKT5]), and then by musing on the WKB-theoretic meaning of the obtained results reflectively from the viewpoint of the Borel resummation,
(2) we proposed in [AKT3] the exact steepest descent method that makes use of the newly invented notion "exact steepest descent paths" so that we may describe the Stokes geometry for general operators.
(3) Some concrete but delicate issues in the Stokes geometry are examined by the exact steepest descent method in [AkoT] and [KoT].
In view of the spiritual target of this project, the introduction of the exact steepest descent method into the exact WKB analysis is quite important, as it clearly exemplifies the complementary character of the exact WKB analysis and microlocal analysis, it shows that the global aspect o
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f the quantized Legendre transformation can be described in terms of the exact steepest descent paths.
2° Non-adiabatic transition probabilities for Landau-Zener type problems are calculated on the basis of microlocal analysis of operators with multiple characteristics ([AKT1]). Important in its own right is the concrete algorithm for detecting virtual turning points for the operators in question.
3° Microlocal structure of the S-matrix is studied ia [KS] when infra-red divergence is relevant.
4° Natural boundaries of solution of non-liner ordinary differential equations are studied in [K] from the viewpoint of WKB analysis. Microlocal study of natural boundaries of Dirichlet series was also made in [KStr].
5° Local theory of the exact WKB analysis for the infinite series of differential operators with a large parameter was developed in [AKKT] with the help of a quantized contact transformation, one of the basic notions in microlocal analysis.
6° [T1] constructed the exact WKB analysis for systems of differential equations, and we are currently (2002) trying to apply it to the study of higher order Painlev」 equations (Noumi equation etc.). Less
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