Research on partial differential equations and selfadjoint operators of mathematical physics

数学物理偏微分方程与自伴算子研究

基本信息

批准号：
09640158
负责人：
YAJIMA Kenji
金额：
$ 1.92万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

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.

进行了数学物理中出现的联合算子。主要注意力集中在线性和非线性薛定谔方程、非线性波动方程以及薛定谔算子和泡利算子的光谱和散射理论等主题上。主要采用泛函分析、实变量理论、傅立叶分析和微观局部分析等方法对这些问题进行研究。结果发现以下新结果： 1．如果势为二次方，则时间相关薛定谔方程的基本解是平滑的且有界于 t * 0，而如果势以超二次方无限增加，则它在任何地方都不是 C^1。简谐振子扰动的基本解如果是次线性扰动则奇点重现，而如果扰动是超线性奇点则一般消失。 2．对于包括库仑势在内的一类奇异势，基本解保持连续且有界。 3．建立了非均匀磁场二维泡利算子特征值个数累加为零的渐近行为。 4．已经发现了具有磁场的二维薛定谔算子的散射算子的低能量极限。阐明了当磁场收敛到所谓的磁弦时散射矩阵的渐近行为。5．测量了半经典极限下磁场对隧道效应的影响，发现其很大程度上取决于磁场的平滑度。 6．薛定谔算子在俘获能下谱移函数的半经典行为已得到阐明。7．针对不同传播速度的非线性波动方程组建立了Strichartz型估计，并阐明了其与临界非线性波动方程适定性的关系。