Functional Analysis and Partial Differential Equations

1.光谱理论。对B.Simon等人的分布秩一摄动理论进行了本质上的推广。系统地应用于点和面相互作用将是未来研究的主题。另一个研究课题是R.D.Yafaev修正散射矩阵的本征值分布理论，并证明了该理论适用于具有非球对称矢势的问题。通过积分和渐近分析。继续前人的研究，证明了在强拓扑下，时间分片近似收敛于路径积分。给出了熊野-Go-Taniguchi定理的一个简单直接的新证明。薛定谔方程的数值分析。从新的角度研究了含时薛定谔方程的有限步显式方法，提出了求解非线性薛定谔方程的算法。半线性椭圆型边值问题。结果表明，基于Nehari变分原理，可以用数值有效的方法构造某些不稳定解，并通过数值实验验证了该方法的有效性。退化的非线性抛物型方程。对有限差分近似进行了研究，并对一些具体问题证明了近似解收敛于真解。并进行了数值实验。实证法。作为实分析方法的应用，得到了具有一定位势的薛定谔算子逆的加权估计问题。在经典实分析领域中，得到了具有可分离变量的函数的Kakeya极大函数的范数估计。复杂的动力系统。得到了关于Mandelbrot集的六次曲线点的周期性和动力系统在Misiurewicz点附近的行为的新知识。

1. Spectral Theory. A substaintial generalization of the distributional rank-one perturbation theory due to B.Simon et al was made. Systematic applications to point and surface interactions will be the subject of future investigation. Another topics of research was the theory of eigenvalue distribution of modified scattering matrix due to R.D.Yafaev and it is shown that the theoryis applicable to problems with non-spherically symmetric vector potentials.2. Pass Integfral and asymptotic analysis. Continuing previous research, it was shown that time-slicing approximations converge to the path integral in the strong topology. A simple and direct new proof of theorems of Kumano-go-Taniguchi type was given.3. Numerical analysis of Schrodinger equations. Finite step explicit method for time-dependent Schrodinger equation was investigated from a new viewpoint and an algorithn for non-linear Schroodinger equation was proposed.4. Semi-linear elliptic boundary value problems. It was shown that based on the Nehari variational principle some kind of unstable solutions can be constructed in numerically efficient ways and the effectiveness of the method was confirmed by numeraicl experiments.5. Degenerate non-linear parabolic equations. Finite difference approximations were investigated and for some specific problems the convergence of approximate solution to the true solution was proved. Numerical experiments were also performed.6. Real analysis method. As applications of real analysis methods some problems of weighted estimates were obtained for the inverse of the Schrodinger operator with certain potentials. In the field of classical real analysis some norm estimates of Kakeya's maximal function were obtained for functions with separable variables.7. Complex dynamical system. New knowledge were obtained about the periodicity of heperbolic points of the Mandelbrot set and the behavior of the dynamical system around Misiurewicz points.