1998 Fiscal Year Final Research Report Summary
Research on partial differential equations and selfadjoint operators of mathematical physics
| Project/Area Number |
09640158
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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 |
解析学
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| Research Institution | The University of Tokyo |
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Principal Investigator |
YAJIMA Kenji Graduate school of Mathematical Sciences, University of Tokyo, Full professor, 大学院・数理科学研究科, 教授 (80011758)
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| Co-Investigator(Kenkyū-buntansha) |
TSUTSUMI Yoshio Graduate school of Mathematical Sciences, University of Tokyo, Associate profess, 大学院・数理科学研究科, 助教授 (10180027)
NAKAMURA Shu Graduate school of Mathematical Sciences, University of Tokyo, Full professor, 大学院・数理科学研究科, 教授 (50183520)
TAMURA Hideo Department of Mathematics Okayama University, Full professor, 理学部, 教授 (30022734)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Schrodinger equation / Schrodinger operators / spectral theory / scattering theory / Fundamental solution / Pauli operator / semiclassical limit / Non-linear wave equation |
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| Research Abstract |
joint operators appearing in mathemmatical physics was carried out. Major attention was focused on the topics of linear and non-linears Schrodinger equations, non-linear wave equations and the spectral and scattering theory for Schrodinger operators and Pauli-operators. These problems were investigated by employing methods mainly from functional analysis, real-variable theory, Fourier analysis and micro-local analysis. As a result, the following new results were found :
1. The fundamental solution of time dependent Schrodinger equations is smooth and bounded for t * 0 if the potential subquadratic, whereas it is nowhere C^1 if the potential superquadratically increasing at infinity. The fundamental solution of pertubations of harmonic oscillator enjoy the recurrence of singularities if the perturbation are sublinear whereas it in general disappears if the perturbations are superlinear.
2. The fundamental solution remains continuous and bouned for a class of singular potentials including Coulomb potentials.
3. The asymptotic behavior of the number of eiegnvalues accumulating to zero of two dimensional Pauli operators with non-homogeneous magnetic fields has been established.
4. The low enegry limits of the scattering opertors for two dimensional Schrodinger opeartors with magnetic fields has been found. The asymptotic behavior of the scattering matrix when the magnetic field converges to so called magnetic string has been clarified.
5. The effect of the magnetic fields to the tunneling in semi-classical limit has been measured and it is found that it largely depends on the smoothness of the magnetic fields.
6. Semi-classical behavior of the spectral shift function for Schrodinger operators at the trapping energy has been clarified.
7. Strichartz type estimate is established for a system of non-linear wave equation with different propagation speeds and its relation to the well-posedness of critical non-linear wave equation has been clarified.
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