2007 Fiscal Year Final Research Report Summary
Research on Functional Analysis and Mathematical theory of Feynman path integrals
| Project/Area Number |
17540170
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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 University, Dept. of Math, Prof (10011561)
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| Co-Investigator(Kenkyū-buntansha) |
YAJIMA Kenji Gakushuin Univ, Dept. of Math, Prof (80011758)
KAWASAKI Tetsuro Gakushuin Univ, Dept. of Math, Prof (90107061)
MIZUTANI Akira Gakushuin Univ, Dept. of Math, Prof (80011716)
WATANABE Kazuo Gakushuin Univ, Dept. of Math, Assist. Prof (90260851)
SHIMOMURA Akihiro Tokyo Metro Univ., Grad. Sch. Sci. & Eng., Assist. Prof (00365066)
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| Project Period (FY) |
2005 – 2007
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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 proved an improved remainder estimate of stationary phase method for oscillatory integrals over a space of large dimension. And he discovered and proved a new formula expressint the second term of the semi-classical asymptotics of Feynman path integrals.
2. Yajima discussed asymptotic behaviour as time t goes to ∞ of solution of Schrodinger equation on Rm under the assumption that the resolvent of the Hamiltoninan operator has sin- gularity at the infimum of its continuous spectrum. He got the following results:
(1) In the case m = 3, he proved the asymptotic behaviour in LP (Rm) of the solution of Schrodinger equation as t→∞
(2) The wave operators W± of the scattering is actually continuous as an operator on LP with m/2 < p < m/(m-2) if m > 5 or m =3. If m = 4, the same property holds under additional conditions.
3. Kawasaki discussed topological approach to Gauge theory.
4. Mizutani discussed finite element method for solution to some boundary value problem of ordinary differential equation of forth order.
5. Watanabe made research on solutions of PDE with dispersive type. He discussed decay as t →∞of solution of Schrodinger equation and also he discussed spectrum of Hamiltonians.
6. Shimomura discussed nonlinear dispersive partial differential equations with special emphasis on nonlinear Schrodinger equation.
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Research Products
(62 results)
