Algebro-analytic study of partial differential equations

A. Kaneko研究了。与国外学者共同研究了缓变分布空间中卷积方程的可解性，并得到了新的结果。随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继续研究关于代数雕刻的zeta函数零点角状分布的佐藤猜想，最近发现了一类可以用超几何函数的有限域模拟来描述的代数曲线的有趣现象。E. Horikawa对AomotoGelfand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证明了在有限多运动电荷诱导的电磁场中，Dirac粒子的Dirac方程解唯一存在，传播子守恒1阶Sobolev空间。在电荷速度受限的条件下。他还处理了一维Stark hamilton长范围势的散射理论，并通过修正传播子渐近地表示经典轨迹，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长期以来争论不休的悖论。H. Kitada只用Enss的微局部渐近估计证明了具有短程多粒子的薛定谔算子的波算子的渐近完备性。进一步给出了长范围多粒子渐近完备性的充分必要条件。在SL（2， Z）的子群mod 3上证明了满足若干算术条件的索引映射的存在性，并给出了在虚二次域的算术中的应用。

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.