Wave operators and unitary equivalence

Tosio Kato

Vol. 15 (1965), No. 1, 171–180

DOI: 10.2140/pjm.1965.15.171

Abstract

This paper is concerned with the wave operators $W_{\pm }=W_{\pm }\left(H_1,H_0\right)$ 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 $W_{\pm }^{\prime }\pm W_{\pm }\left(\varphi \left(H_1\right),\varphi \left(H_0\right)\right)$ exist and are complete for any $\varphi \in \left(M\right)$; moreover, $M_{\pm }^{\prime }$ are “piecewise equal” to $W_{\pm }$ (in the sense to be specified in text). Theorem 2 strengthens Theorem 1 by replacing the above assumption by the condition that ${\psi }_n\left(H_1\right)={\psi }_n\left(H_0\right)+V_n,V_n\in T$, 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.

Mathematical Subject Classification

Primary: 81.47

Milestones

Received: 8 January 1964

Published: 1 March 1965