| Project/Area Number |
09440055
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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 |
解析学
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| Research Institution | University of Tokyo |
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Principal Investigator |
NAKAMURA Shu Graduate School of Mathematical Sciences, University of Tokyo, 大学院・数理科学研究科, 教授 (50183520)
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| Co-Investigator(Kenkyū-buntansha) |
KATO Keiichi Science University of Tokyo, Faculty of Sciences, 理学部, 助教授 (50224499)
OGAWA Takayoshi Kyushu University, Graduate School of Mathematics, 数理学研究院, 助教授 (20224107)
YAJIMA Kenji Graduate School of Mathematical Sciences, University of Tokyo, 大学院・数理科学研究科, 教授 (80011758)
堤 誉志雄 東北大学, 大学院理学系研究科, 教授 (10180027)
新井 仁之 東京大学, 大学院・数理科学研究科, 教授 (10175953)
小薗 英雄 名古屋大学, 大学院・多元数理科学研究科, 助教授 (00195728)
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| Project Period (FY) |
1997 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥13,600,000 (Direct Cost: ¥13,600,000)
Fiscal Year 2000: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1999: ¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 1998: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1997: ¥3,800,000 (Direct Cost: ¥3,800,000)
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| Keywords | Schrodinger operator / scattering theory / spectral theory / random Schrodinger operator / semiclassical limit / スヘクトル理論 / トンネル効果 / リフシッツ特異性 / 磁場 / シュレディンガー方程式 / 基本解 / 錯乱理論 |
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| Research Abstract |
The purpose of this project is to investigate the spectral and scattering theory for Schrodinger operators in general. Moreover, it is also intended to explore new area of problems in quantum physics and related topics. Quite a few reserch results has been obtained in the project, and only a selected results by the head investigator and collaborators are presented here.
1. By employing the theory of phase space tunneling, it is proved that the exponential decay rate of eigenfunctions for Schrodinger operator is larger in the semiclassical limit in the presence of constant magnetic field.
2. Semiclassical asymptotics of the scattrering is investigated. In particular, it is shown that the spectral shift function has a rapid jump (of the size 2π times integer) near each quantum resonance.
3. It is shown that the coefficients of the scattering matrix corresponding to the interaction between two nonintersecting energy surfaces decay exponentially in the semiclassical limit. A new method to analyze the phase space tunneling is developed and employed (joint work with A.Martinez, V.Sordoni).
4. The Lifshitz tail for the integrated density of states is proved for 2 dimensional discrete Schrodinger operators and continuous Schrodinger operators (arbitrary dimension) with Anderson-type random magnetic fields.
5. A new proof of the Wegner estimate based on the theory of the spectral shift function is developed. The Wegner estimate plays crucial role in the proof of Anderson localization for random Schrodinger operators (joint work with J.M.Combes, P.D.Hislop).
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