Research in Functional Analsys 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。他没有成功地推广伊藤的方法。但他与渡边一雄、三马一雄和熊野直人合作，开始了抽象维纳空间上振荡积分的研究。2002年6月在里斯本大学举行的国际研讨会上报告了初步结果。S.T. Kuroda和斯德哥尔摩大学的P. Kurasov发表了一篇联合论文，试图通过解析方程参数化Hilbert空间中的所有自伴随算子。他和他的学生Nagatani应用上述方法研究了具有点相互作用型摄动的Shrodinger算子的自伴随扩展。Mizutani和Takshi Suzuki一起研究了退化非线性抛物型偏微分方程的有限元逼近。他们发明了一个保持L^1空间的有序性和收缩性的方案并且他们成功地证明了他们的近似解实际上收敛于L^1空间的真解。Watanabe和斯德哥尔摩大学的P.kurasov共同研究了算子自伴随扩展的H_4实现。他研究了具有点相互作用型势的哈密顿量在连续谱中的嵌入点谱。他还与Takashi Suzuki一起研究了限于子流形的Maxwell方程解的光滑性。Takeuchi Shingo研究了具有退化色散项的logistic方程解的渐近行为。他也研究了复杂的金兹堡-朗道方程。在这两种情况下，他都成功地证明了全局吸引子的存在。

1. Fujiwara tried to give mathematically rigorous treatment of Feynman path integrals. He did not succeed in generalizing K.Ito's method. But he started the study of oscillatory integrals on abstract Wiener space with collaboration of Kazuo Watanabe, Itaru Mitoma and Naoto Kumanogo. A preliminary result was reported at the International symposium held at Univ. of Lisbon in June 2002.2. S.T. Kuroda together with P. Kurasov of Stockholm University published a joint paper that tries to parameterize all self-adjoint operators in a Hilbert space through Resolvent equations. He and his student Nagatani applied the above mentioned method to study self-adjoint extension of Shrodinger operator with perturbation of point interaction type.3. Mizutani together with Takshi Suzuki studied approximation by finite element method to degenerate nonlinear parabolic partial differential equations. They invented a scheme that preserves order and contraction property in L^1 and they succeeded in proving that their approximate solution actually converges to the true solution in L^1 space.4. Watanabe together with P.kurasov of Stockholm University jointly studied H_4 realization of selfadjoint extension of operators. He studied point spectrum embedded in continuous spectrum which appear in the case of hamiltonians with potential of point interaction type. He also studied together with Takashi Suzuki smoothness of solution for Maxwell equations restricted to submanifold.5. Shingo Takeuchi studied asymptotic behavior of solutions to logistic equations with degenerate dispersive term. He studied complex Ginzburg Landau equations too. In both cases, he succeeded in proving existence of global attractors.