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
realvalued 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
${\psi}_{n}\left({H}_{1}\right)={\psi}_{n}\left({H}_{0}\right)+{V}_{n},\phantom{\rule{1em}{0ex}}{V}_{n}\in T$,
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.
