Research on Functional Analysis and Mathematical theory of Feynman path integrals.

1.藤原试图给数学严格的治疗费曼路径积分。他证明了高维空间上振荡积分的稳相法的一个改进的余项估计。并对Kumano-go关于Feynman路径积分收敛性的结果进行了讨论。他还证明了一个新的公式的第二项的半经典渐近费曼路径积分。2. Yajima等人对非相对论QED的简化模型纳尔逊模型的光谱和散射性质进行了研究。他还成功地证明了势函数增长阶为O的薛定谔方程的每一个解（|X| ^m），m > 2，在无穷大处，几乎在每一个t处，与其初始值相比，阶为1/m的可微性. Watanabe研究了色散型偏微分方程的解。在与T。Suzuki和T.小林还证明了Maxwell-Stokes方程组解的界面正则性。他还谈到了.4。Shimomura讨论了非线性薛定谔方程解的大时间行为。他成功地找到了一个很好的性质的解决方案，这是不能预见的仅仅从增长秩序的非线性项。下村还证明了带电磁势的非线性薛定谔方程的时间局部解的光滑效应。

1. Fujiwara tried to give mathematically rigorous treatment of Feynman path integrals. He proved an improved remainder estimate of stationary phase method for oscillatory integrals over a space of large dimension. And he make the discussion of Kumano-go's results on convergence of Feynman path integrals. He also proved a new formula for the second term of the semi-classical asymptotics of Feynman path integrals.2. Yajima got results on spectrum and scattering properties of Nelson model, which is a simplified model of non-relativistic QED. He also succeeded in proving that every solution of Schrodinger equation with potentials which grow of oder O(|x|^m), m > 2 at the infinity gains, at almost every t, differentiability of order 1/m compared with its initial value.3. Watanabe made research on solutions of PDE with dispersive type. In the joint with T. Suzuki and T. Kobayashi he also proved interface regularity of solutions to the system of Maxwell-Stokes equations. He also discussed.4. Shimomura discussed large time behaviour of solutions to non-linear Schrodinger equations. He succeeded in finding a quite a fine property of solutions which is not foreseen from mere growth order of nonlinear term. Shimomura also proved smoothing effects of time local solution to non-linear Schrodinger equation with electro-magnetic potential.