Project/Area Number  09640158 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  The University of Tokyo 
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)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1998 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1997 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  Schrodinger equation / Schrodinger operators / spectral theory / scattering theory / Fundamental solution / Pauli operator / semiclassical limit / Nonlinear wave equation / シュレーディンガー方程式 / シュレーディンガー作用素 / スペクトル理論 / 散乱理論 / 基本解 / Pauli作用素 / 準古典近似 / 非線形波動方程式 / シュレーディガー方程式 / 非線型波動方程式 / シュレデンガー方程式 / シュレデンガー作用素 / 特異性伝播 / 磁場 / スペクトル / 固有値漸進分布 / KleinGordonZakharov 
Research Abstract 
joint operators appearing in mathemmatical physics was carried out. Major attention was focused on the topics of linear and nonlinears Schrodinger equations, nonlinear wave equations and the spectral and scattering theory for Schrodinger operators and Paulioperators. These problems were investigated by employing methods mainly from functional analysis, realvariable theory, Fourier analysis and microlocal 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 nonhomogeneous 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 semiclassical limit has been measured and it is found that it largely depends on the smoothness of the magnetic fields. 6. Semiclassical 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 nonlinear wave equation with different propagation speeds and its relation to the wellposedness of critical nonlinear wave equation has been clarified.
