DEVELOPEMENT OF RENORMALIZATION GROUP METHODS AS TOOLS OF ANALYSIS AND THEIR APPLICATIONS TO DYNAMICAL SYSTEMS

1.伊藤和田村(金泽大学)用重整化群(块自旋变换)方法研究了经典的O(N)对称自旋模型。在第一阶段，他们证明了泛函行列式det^&lt；N/2&gt；(1+2iG_ψ/√&lt；N&gt；)关于ψ的可积性，其中ψ是真正引入的傅里叶变换域。利用聚合物(团簇)膨胀技术，他们证明了反临界温度β_c在二维上服从约束β_c&gt；N log N，这意味着存在强偏差。(尺寸大于或等于3，则为β_c~N。)在本模型中，β_c=∞。为了建立这一猜想，他们递归地将BST应用于模型，将行列式分解为来自不同距离尺度的波动的许多行列式的乘积。它们表明，递归关系的主要部分相当简单，并且再现了威尔逊-戴森型递阶逼近的流。剩下的就是…了更多地控制非局部小项的递归公式，彻底解决问题。伊藤和广岛(Setsunan University.)研究经典量子电动力学(QED)中的Pauli-Fierz模型。尽管如果没有引入动量截断，QED被认为是微不足道的，但Pauli-Fierz模型可能并非如此。他们将重整化群类型的论证应用于Pauli-Fierz模型(这个想法最初是由J.Froelich(ETH)提出的)。但他们的分析还有待观察。Teramoto考虑了旋转圆柱间Couette流摄动的Couette-Taylor问题，并证明了当Taylor数增大时，会出现定常分叉或Hopf分叉。他用计算机数值分析的方法证明了这一点。Teramoto和Ito研究了湍流的性质，其中包括关于能量耗散和偏离能量的柯尔莫戈洛夫定律。他们试图从纳维-斯托克斯方程推导出偏差，但今年未能得到具体结果。Shimada用原点的边界条件刻画了Aharonov-Bohm哈密顿算子的所有可能的自伴扩张，这些条件将Aharonov和Bohm处理的算子与其他可能的算子区分开来。他发展了它们的算符的散射理论，并得到了相移公式。较少

1. Ito and Tamura (Kanazawa Univ.) studied classical O(N) symmetric spin model by renormalization group (block spin transformation) method. In the first stage, they argued the integrability of the functional determinent det^<N/2>(1+2iG_ψ/√<N>) with respect to ψ, where ψ is the auxially field introcuced for Fourier Transformation. Using the technique called polymer (cluster) expansion, they showed that the inverse critical temperature β_c obeys the bound β_c>N log N in two dimensions, which implies the existence of strong deviation. (β_c〜N for the dimension more than or equal to 3.) It is believed that β_c=∞ in the present model. To establish this conjecture, they recursively apply the BST to the model to decompose the determinant into product of many determinants which comes from fluctuations of various distance scales. They showed that the main part of the recursion relations is quite simple, and reproduces the flow of the hiererchical approximation of Wilson-Dyson type. It remains to … More control small non-local terms of the recursion formulas to solve the problem completely.2. Ito and Hiroshima (Setsunan Univ.) investigate the Pauli-Fierz Model which is rgarded as a classical Quantum Electrodynamics (QED). Though QED is believed to be trivial if no momentum cutoff is introduced, the Pauli-Fierz model may not. They apply the renormalization group type argumemt to the Pauli-Fierz model (this idea is originally due to J.Froelich(ETH)). But their analysis remains to be seen.3. Teramoto considere Couette-Taylor problems of the perturbation to the Couette flow between two rotating cylinders, and shows that the stationary bifurcation or Hopf bifurcation occurs when the Taylor number increases. He proved it with the help of numerical analysis by computer.4. Teramoto and Ito investigated properties of turbulence, among them, the Kolmogorov law about the dissipation of energy and deviation from it. They tried to derive the deviation from the Navier-Stokes equation but they could not obtain concrete results this year.5. Shimada characterized all possible selfadjoint extensions of Aharonov-Bohm hamiltonian in terms of boundary conditions at origin which distinguish the operator treated by Aharonov and Bohm from other possible ones. He developed scattering theory for their operators and obtained the phase shift formula. Less