On a limiting case of a theorem of Kato and Kuroda

Abstract

Cook [5] and Jauch [6] introduced an abstract version of the two characteristic matrices of Mijller [2] and called them wave operators. At the same time Jauch [6] emphasized that wave operators corresponding to physically realistic systems have an additional property that he called completeness. Several different authors formulated criteria for the existence and completeness of wave operators for various abstract and concrete classes of operators [3, 4, 7, 8, 10, 12, 15-17, 20, 22-241. For the pair of operators (d, d + p) recently this problem was taken up again by Kato and Kuroda [25, 26, 33, 341. Here, as usual, il denotes the negative Laplacian acting in free space and p denotes a potential. They formulated general criteria on the potential for the existence and completeness of the wave operators. According to their abstract theory it also follows that the absolutely continuous parts of the perturbed and unperturbed operator are unitarily equivalent. In this paper, we consider a limiting case of their cnndition. Roughly speaking, we treat potentials with decay exponent equal to one and satisfying an additional smallness condition at infinity. This additional smallness condition is in accordance with a result of Dollard [14] which says that, for the Coulomb potential, the wave operators do not exist. At the same time in this paper, we illustrate that an estimate formulated elsewhere [27] in conncction with a different problem [18] can be applied to study wave operators. In Section 2, our condition is formulated and the corresponding Theorem 2.1 is stated. In Section 3, we state a version of an abstract theorem of Kato and Kuroda. This theorem is implicit in their work [25] and we state it for completeness only. In Section 4 we verify the assumptions of this abstract theorem for potentials satisfying the assumptions of Section 2.