#### Vol. 15, No. 1, 1965

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Wave operators and unitary equivalence

### Tosio Kato

Vol. 15 (1965), No. 1, 171–180
##### Abstract

This paper is concerned with the wave operators ${W}_{±}={W}_{±}\left({H}_{1},{H}_{0}\right)$ associated with a pair ${H}_{0}$, ${H}_{1}$ of selfadjoint operators. Let $\left(M\right)$ be the set of all real-valued functions $\varphi$ on reals such that the interval $\left(-\infty ,\infty \right)$ has a partition into a finite number of open intervals ${I}_{k}$ and their end points with the following properties: on each ${I}_{k}$, $\varphi$ is continuously differentiable, ${\varphi }^{\prime }\ne 0$ and ${\varphi }^{\prime }$ is locally of bounded variation. Theorem 1 states that, if ${H}_{1}={H}_{0}+V$ where $V$ is in the trace class $T$, then ${W}_{±}^{\prime }±{W}_{±}\left(\varphi \left({H}_{1}\right),\varphi \left({H}_{0}\right)\right)$ exist and are complete for any $\varphi \in \left(M\right)$; moreover, ${M}_{±}^{\prime }$ 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 ${\psi }_{n}\left({H}_{1}\right)={\psi }_{n}\left({H}_{0}\right)+{V}_{n},\phantom{\rule{1em}{0ex}}{V}_{n}\in T$, where ${\psi }_{n}\in \left(M\right)$ and ${\psi }_{n}$ is univalent on $\left(-n,n\right)$ for $n=1,2,3,\dots$. As corollaries we obtain many useful sufficient conditions for the existence and completeness of wave operators.

Primary: 81.47