Wave operators and unitary equivalence

This paper is concerned with the wave operators W = W(H l9 Ho) associated with a pair H o , Hi of selfadjoint operators. Let (M) be the [set of all real-valued functions on reals such that the interval (-00, 00) has a partition into a finite number of open intervals I k and their end points with the following properties: on each I k , is continuously differentiable, 1 0 and ' is locally of bounded variation. Theorem 1 states that, if Hi = H o + V where V is in the trace class T, then W' W{{H 1 \ (H 0 )) exist and are complete for any e(M); moreover, M' are "piecewise equal" to W (in the sense to be specified in text). Theorem 2 strengthens Theorem 1 by replacing the above assumption by the condition that n (H0 = n(H 0 ) + Vnt V n eT, where n e(M) and n is univalent on (-n, n) for n = 1, 2, 3,.... As corollaries we obtain many useful sufficient conditions for the existence and completeness of wave operators.