Vol. 15, No. 1, 1965

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Wave operators and unitary equivalence

Tosio Kato

Vol. 15 (1965), No. 1, 171–180

This paper is concerned with the wave operators W± = W±(H1,H0) associated with a pair H0, H1 of selfadjoint operators. Let (M) be the set of all real-valued functions ϕ on reals such that the interval (,) has a partition into a finite number of open intervals Ik and their end points with the following properties: on each Ik, ϕ is continuously differentiable, ϕ0 and ϕ is locally of bounded variation. Theorem 1 states that, if H1 = H0 + V where V is in the trace class T, then W±± W±(ϕ(H1),ϕ(H0)) exist and are complete for any ϕ (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 ψn(H1) = ψn(H0) + V n,V n T, where ψn (M) and ψn 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.

Mathematical Subject Classification
Primary: 81.47
Received: 8 January 1964
Published: 1 March 1965
Tosio Kato