Algebro-analytic study of partial differential equations

A.金子研究了。与国外研究人员讨论了回火分布空间中卷积方程的可解性，并得到了一个新的必要性结果。然后与研究生合作，发展了用有限差分法求解卷积方程的方法。K.Namba继续研究了关于代数雕刻的Zeta函数的零点的奇异分布的Sato猜想，最近发现了一种有趣的现象，关于一类代数曲线，它可以用超几何函数的有限域模拟来描述。E.Horikawa对Aomoto Gelfand的超几何函数的邻接关系提出了新的观点，澄清了与李代数的关系，使理论变得透明。K.Yajima证明了在有限运动电荷诱导的电磁场中Dirac粒子Dirac方程的解是唯一的，并且在电荷速度有界的条件下，传播子守恒为1阶Sobolev空间。他还讨论了具有长程势的一维Stark哈密顿量的散射理论，并用代表经典轨道的修正传播子证明了修正波算符的存在性和完备性，从而澄清了经典散射理论与量子散射理论之间的悖论。利用ENSS的微局部渐近估计，H.Kitada证明了短程多粒子薛定谔算子的波算子的渐近完备性。在此基础上，给出了长程多粒子情况下的渐近完备性的充要条件。证明了关于SL(2，Z)的mod3子群存在满足某些算术条件的指数映射，并将其应用于虚二次域的算术中。

A. Kaneko studied the. solvability of convolution equation in the space of tempered distributions with a foreian research visitor, and got anew result for the necessity. Then in colaboration with graduate students developed the method of solution of convolution equations by finite difference method. K. Namba continued to study the Sato conjecture on the ancular distribution of the zeros of the zeta functions of algebraic carves, and recently found an interesting phenomenom concerning a kind of alaebraic curves which can be described in terms of a finite field analogue of the hypergeometric function. E. Horikawa rave a new viewpoint to the contiguity relation of the hypergeometric functions of AomotoGelfand, clarified relation with Lie algebras and made the theory transparent. K. Yajima proved that the solution of the Dirac equation for the Dirac particles in the electromagnetic field induced by finitely maly moving charges exists uniquely and the propagator conserves the Sobolev space of order 1. under a condition of bound for the velocities of the charges. He also treated the scattering theory of the one dimensional Stark Hamiltonian with long range potential, and showed that the existence and completeness of modified wave operator by means of modified propagator representing the classical trajectories asymptotically, thus clarifying the paradox between the classical scattering theory and the quantum one which has long been a theme of discussion. H. Kitada showed the asymptotic completeness of the wave operator for the Schroedinger operator with short range many particles employing only the microlocal asymptotic estimate of Enss. Further he gave a necessary and sufficient condition for the asymptotic completeness in the case of long range many particles. H. Ito gave a proof to the existence of an index map satisfying some arithmetic conditions in relation to subgroups mod 3 of SL(2, Z), and gave applications to the arithmetics of imaginary quadratic fields.