Scattering and Inverse Scattering for Linear and Nonlinear Wave Propagations

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In this project we have been studying several kind of wave propagation phenomena for the fundamental equations of physics like Maxwell equations and Schr"odinger equations. Our main subjects are in the scattering theory and inverse scattering theory, and these problems are important not only in the fields of mathematical analysis but also in those of applied sciences.We shall summarize the results of the head investigator.In [1] is developed the scattering theory for wave equations in exterior domain with coefficients depending on both space and time. The space-time weighted energy estimates play the crucial role in this problem. In [5] we studied similar problems for Schr"odinger equations with time dependent potentials. The space time LAp-LAq estimates becomes important in this problem. In [2] is obtained a Rellich type asymptotic estimates for generalized eigenfunctions for the Schr"odinger operator with oscillating longrange potentials. The results are applied to show the principle of limiting absorption for this operator. [3] is the joint work with Prof. M. Nakao, and a standard decay condition of energy is shown for wave equations in exterior domain with dissipation which is effective only near infinity. In [4] is studied the inverse scattering for Schr"odinger equations on graphs. These problems are recently recognized to be important not only in the classical field of nano-scale technology (like solid state physics and electrionics) but also in the filed of information stechnology. [4] is obtained as a joint work with Profs V. Marchenko and I. Trooshin, and is restricted to the simplest graph including a compact part. Our joint works will proceed into more general graphs including compact parts. Another work is done on the scattering inverse problem for non-selfadjoint wave equation as a continuation of the former work of the same title (Kluwer Academic Publishers, ISAAK 10 (2003), 303-316).