Research in Functional Analsys 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。他没有成功地推广伊藤的方法。但他与渡边和夫、三生太郎和熊野直人合作，开始了抽象Wiener空间上的振荡积分的研究。在大学举行的国际专题讨论会上报告了初步结果。2002.2年6月，里斯本。S.T.黑田东彦和斯德哥尔摩大学的P.Kurasov发表了一篇联合论文，试图通过Resolent方程将Hilbert空间中的所有自伴算子参数化。他和他的学生Nagatani应用上述方法研究了具有点相互作用型扰动的Shrodinger算子的自伴延拓。Mizutani和Takshi Suzuki研究了退化非线性抛物型偏微分方程解的有限元逼近。他们发明了一种在L^1空间中保持序和压缩性质的方案，并成功地证明了他们的近似解实际上收敛于L^1空间中的真解。渡边与斯德哥尔摩大学的P.Kurasov共同研究了算子自伴扩张的H_4实现。他研究了嵌入在连续谱中的点谱，这种谱出现在具有点相互作用势的哈密顿量的情况下。他还与铃木隆一起研究了限于子流形的Maxwell方程解的光滑性。竹内信吾研究了具有退化色散项的Logistic方程的解的渐近行为。他还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性。

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.