2000 Fiscal Year Final Research Report Summary
Spectral and Scattering Theory for Schrodinger Operators
| 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)
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| Project Period (FY) |
1997 – 2000
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| Keywords | Schrodinger operator / scattering theory / spectral theory / random Schrodinger operator / semiclassical limit |
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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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