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

1. Ito和Tamura（金泽大学）利用重整化群（块自旋变换）方法研究了经典O(N)对称自旋模型。在第一阶段，他们论证了泛函行列式det^<N/2>(1+2iG_ψ/√<N>)对ψ的可积性，其中ψ是傅里叶变换引入的固有场。利用聚合物（簇）膨胀技术，他们证明了逆临界温度β_c在二维上服从束缚β_c>N log N，这意味着存在强偏差。（β_c ~ N为大于或等于3的维数。）我们认为在该模型中β_c=∞。为了建立这一猜想，他们递归地将BST应用到模型中，将行列式分解为来自不同距离尺度波动的许多行列式的乘积。他们证明了递归关系的主要部分是相当简单的，并再现了威尔逊-戴森型的层次近似的流程。要彻底解决这一问题，还需要进一步控制递归公式中的小的非局部项。伊藤和广岛（Setsunan大学）研究Pauli-Fierz模型，该模型被认为是经典的量子电动力学（QED）。虽然如果不引入动量截断，QED被认为是微不足道的，但Pauli-Fierz模型可能不是。他们将重整化群类型参数应用于Pauli-Fierz模型（这个想法最初是由J.Froelich（ETH）提出的）。但是他们的分析还有待观察。Teramoto考虑了两个旋转圆柱间的Couette流的扰动的Couette-Taylor问题，并表明当Taylor数增加时，会出现平稳分岔或Hopf分岔。他借助计算机的数值分析证明了这一点。Teramoto和Ito研究了湍流的性质，其中包括关于能量耗散和偏离的Kolmogorov定律。他们试图从Navier-Stokes方程中推导出偏差，但今年未能得到具体的结果。Shimada用原点的边界条件刻画了所有可能的Aharonov-Bohm哈密顿算子的自伴随扩展，这些边界条件将Aharonov和Bohm处理的算子与其他可能的算子区别开来。他发展了算符的散射理论，得到了相移公式。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