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

1.藤原试图对费曼路径积分进行严格的数学处理。他证明了高维空间上振荡积分驻相方法的一种改进的余项估计。并讨论了熊野戈关于Feynman路径积分收敛的结果。他还证明了Feynman路径积分的半经典渐近项的第二项的一个新公式。Yajima得到了Nelson模型的光谱和散射性质的结果，Nelson模型是非相对论QED的简化模型。他还成功地证明了势在无穷大处增长为O(|x^m)，m&gt；2的薛定谔方程的每一个解在几乎每一个t处都比其初值具有1/m阶的可微性。渡边对弥散型偏微分方程组的解进行了研究。在与T.Suzuki和T.Kobayashi的合作中，他还证明了Maxwell-Stokes方程组解的界面正则性。他还进行了讨论。Shimomura讨论了非线性薛定谔方程解的大时间性态。他成功地找到了解的一个相当好的性质，这是仅仅从非线性项的增长级所不能预见的。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光滑化效应。

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.