2002 Fiscal Year Final Research Report Summary
Research in Functional Analsys and Mathematical theory of Feynman path integrals
| Project/Area Number |
13640189
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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 | Gakushuin University |
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Principal Investigator |
FUJIWARA Daisuke Gakushuin Univ. Dept. of Math. Prof., 理学部, 教授 (10011561)
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| Co-Investigator(Kenkyū-buntansha) |
KATASE Kiyoshi Gakushuin Univ. Dept. of Math. Prof., 理学部, 教授 (70080489)
MIZUTANI Akira Gakushuin Univ. Dept. of Math. Prof., 理学部, 教授 (80011716)
KURODA Shigetoshi Gakushuin Univ. Dept. of Math. Prof., 理学部, 教授 (20011463)
KAKEUCHI Shingo Gakushuin Univ. Dept. of Math. Assist., 理学部, 助手 (00333021)
WATANABE Kazuo Gakushuin Univ. Dept. of Math. Assist., 理学部, 助手 (90260851)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Feynman path integrals / Oscillatory integrals / Schrodinger equation / Stationary phase / Selfajoint operator / Quantum mechanics / WKB-method / path integrals |
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| Research Abstract |
1. Fujiwara tried to give mathematically rigorous treatment of Feynman path integrals. He did not succeed in generalizing K.Ito's method. But he started the study of oscillatory integrals on abstract Wiener space with collaboration of Kazuo Watanabe, Itaru Mitoma and Naoto Kumanogo. A preliminary result was reported at the International symposium held at Univ. of Lisbon in June 2002.
2. S.T. Kuroda together with P. Kurasov of Stockholm University published a joint paper that tries to parameterize all self-adjoint operators in a Hilbert space through Resolvent equations. He and his student Nagatani applied the above mentioned method to study self-adjoint extension of Shrodinger operator with perturbation of point interaction type.
3. Mizutani together with Takshi Suzuki studied approximation by finite element method to degenerate nonlinear parabolic partial differential equations. They invented a scheme that preserves order and contraction property in L^1 and they succeeded in proving that their approximate solution actually converges to the true solution in L^1 space.
4. Watanabe together with P.kurasov of Stockholm University jointly studied H_4 realization of selfadjoint extension of operators. He studied point spectrum embedded in continuous spectrum which appear in the case of hamiltonians with potential of point interaction type. He also studied together with Takashi Suzuki smoothness of solution for Maxwell equations restricted to submanifold.
5. Shingo Takeuchi studied asymptotic behavior of solutions to logistic equations with degenerate dispersive term. He studied complex Ginzburg Landau equations too. In both cases, he succeeded in proving existence of global attractors.
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