A stationary approach to long-range scattering

Abstract

This paper deals with a stationary construction of modified wave operators for long-range scattering. The time dependent construction of modified wave operators for longrange scattering, the study of which was begun by Dollard [4] for the Schrϋdinger operator with pure Coulmob potential, has been rather well established recently by Buslaev-Matveev [3], Alsholm-Kato [1], Alsholm [2] and others. But it seems that the time independent (or stationary) approach has not been tried yet. In the theory of short-range scattering, however, the stationary approach has played an important role (see e.g. [7], [8], [9] and [10]). So it is not too ridiculous to conceive that the stationary approach may be useful also in studying the long-range scattering. In fact, using our method developed below, we can prove that the invariance principle for modified wave operators holds under Assumption 1.1 (see §4). But the proof of this result will not be discussed in this work. It will be discussed elsewhere. In this paper we shall construct the modified wave operators by a stationary method essentially following the line established by Kato and Kuroda [7] and [8]. But some modifications will be necessary (see the proof of Theorem 2.8). Now we describe the outline of this paper and at the same time give a heuristic explanation of some notations which will be used in this paper. Consider the time dependent modified wave operator for long-range scattering: