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

1. 藤原试图给出费曼路径积分的严格数学处理方法。他证明了大尺度空间上振荡积分的一种改进的定相法剩余估计。并对熊野吾关于费曼路径积分收敛性的结论进行了讨论。他还证明了费曼路径积分的半经典渐近第二项的一个新公式。Yajima得到了非相对论性QED的简化模型Nelson模型的光谱和散射特性的结果。他还成功地证明了薛定谔方程的每一个解，其势能增长为O阶（|x|^m），在无穷远处，几乎在每一个t处，与初始值相比，可微性都是1阶/m。Watanabe对分散型PDE溶液进行了研究。他还与铃木和小林共同证明了麦克斯韦-斯托克斯方程组解的界面正则性。他还讨论了……Shimomura讨论了非线性薛定谔方程解的大时间行为。他成功地发现了解的一个很好的性质，这是仅仅从非线性项的增长顺序无法预见的。Shimomura还证明了具有电磁势的非线性薛定谔方程的时间局部解的平滑效应。

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.