Algebro-analytic study of partial differential equations

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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.