Research on partial differential equations and selfadjoint operators of mathematical physics

对数学物理中出现的联合算子进行了研究。主要研究了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谱和散射理论。本文主要采用泛函分析、实变量理论、傅立叶分析和微域分析等方法对这些问题进行了研究。结果发现：1.当势次二次时，含时薛定谔方程的基本解在t*0处光滑有界，而当势能在无穷远处增加时，基本解不存在C^1。如果微扰是次线性的，则谐振子的基本解存在奇点递归；如果微扰是超线性的，则基本解一般消失。对于包括库仑势在内的一类奇异位势，基本解保持连续且有界。建立了具有非均匀磁场的二维Pauli算子的零点累加个数的渐近行为。发现了具有磁场的二维薛定谔算子的散射算符的能量下限。阐明了当磁场收敛到所谓的磁弦时，散射矩阵的渐近行为。在半经典极限下测量了磁场对隧道效应的影响，发现这在很大程度上取决于磁场的平稳性。阐明了薛定谔算符谱移位函数在俘获能区的半经典行为。建立了一类具有不同传播速度的非线性波动方程组的Strichartz估计，并阐明了它与临界非线性波动方程适定性的关系。

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.