On a Stationary Approach to Scattering Problem

Here, T(^p) denote^ the trace class of completely continuous operators in § and p(Hp) the resolvent set of Hp. The perturbation theory of absolutely continuous (abbr. a.c.) parts of Hp as well as the theory of wave and scattering operators has recently been studied independently by de Branges [2], Birman and Kreïn [ l ] , and Kato [3], In [ l ] and [3] the problem was considered from the viewpoint of the scattering theory. In particular, the wave operators W± were proved to exist and hence to be partially isometric operators which give the unitary equivalence of a.c. parts of Ho and H\. In [2], on the contrary, similar partially isometric operators W± were constructed somewhat explicitly and without referring to the limit of wave operator type. The purpose of the present note is to study the latter approach from a viewpoint of the scattering theory and to see that the so-called time-independent or stationary approach to the theory of wave and scattering operators can be made possible under the condition (1). In a simpler case, a similar study was made in [4], Our construction of the operator similar to ÎY+, i.e. the operator given by the right side of (9), is similar to but slightly different from that given in [2]. In particular, the use of the auxiliary operator I in [2] is avoided. Furthermore, the construction of the operators WQ and TT\ in 3 might be a little more explicit than that of the corresponding operators given in [2].