STUDY OF DIFFERENTIAL EQUATIONS BY RENORMALIZATION

(1)在与金泽大学田村教授的联合研究中，Ito在低维O(N)自旋模型的相变问题上获得了几个新的结果。虽然人们推测这种模型中不会发生相变，但没有人严格地证明这一点。通过几个步骤的重整化，他们可以得到一个更好的逆临界温度的估计。(2)Teramoto研究了沿倾斜平面下降的Navier-Stokes流问题定常解的稳定性，证明了在一定条件下存在Hopf分支。为了应用分歧理论，他必须证明在这个问题中出现的线性算子生成一个解析半群。Ito和Teramoto目前正在准备一种新的方法来研究上面获得的滚波之间的相互作用，并试图对向湍流过渡的机制提供新的见解。(3)推测Nakawa ki研究中“反导数”的多样性与重整化群有关。(4)岛田研究了量子力学中Aharonov-Bohm效应的数学散射理论。他得到了与此问题有关的算子的母函数，并构造了波算子、散射矩阵和散射幅度的表示式。这项工作为研究Aharonov-Bohm效应提供了数学基础。

(1) In the joint study with Prof. Tamura in Kanazawa University, Ito obtained several new results on the problems of phase transition of low dimensional O(N) spin models. Though it has been conjectured that he phase transition never occurs in such models, no one proves this rigourously. They could get a better estimate for the inverse critical temperature, by use of several steps of renormalization.(2) Teramoto studied the stability of stationary solution to the problem of The Navier-Stokes flow down an inclined plane and showed that under certain conditions there occurs Hopf bifurcation. To apply the bifurcation theory he must show that the linear operator arising in this problem generates an analytic semigroup. Ito and Teramoto are now preparing a new method to study the interactions between the roll waves obtained above and try to give new insights to the mechanism of transition to turbulence.(3) It is conjectured that the difficulty of diversity of the "antiderivatives" in the study of Nakawaki has something to do with renormalization group.(4) Shimada studied the mathematical scattering theory on the Aharonov - Bohm effect in the quantum mechanics. He obtained the engenfunctions of the operator associated with this problem and constructed the wave operator, the scattering matrix and the representation of scattering amplitude. This work gives the mathematical foundation on the study of the Aharonov - Bohm effect.