This is the abstract of the paper
  "A note on the existence of wave operators"
by Jan A. Van Casteren.

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Abstract:
Let, for $j=0$, 1, $H_j=H_j^{\ast}\geq-\omega_jI$,
$\omega_j>-\infty$, be a self-adjoint operator
in the Hilbert space $\Hcal_j$. Let $J:\Hcal_1\rightarrow\Hcal_0$
be a continuous linear operator with domain $\Hcal_1$ and range in
$\Hcal_0$.
Write $H_j=\int_{-\omega_j}^{\infty}\xi E_j(d\xi)$, $j=0$, 1.
Then the wave operators
$$\Omega_{\pm}\left(H_0,J,H_1\right)f:=\hbox{s-}\lim_{t\rightarrow\pm\infty}
\exp\left(itH_0\right)J\exp\left(-itH_1\right)f,\quad
f\in P_{\hbox{ac}}\left(H_1\right){\Cal H}_1,$$
exist if and only if
for every $f\in P_{\hbox{ac}}\left(H_1\right){\Cal H}_1$
(absolutely continuous subspace of $H_1$)
both strong limits
$$\Omega_{\pm}\left(H_0,E_0\left(A_0\right)JE_1\left(A_1\right),H_1\right)f
:=\hbox{s-}\lim_{t\rightarrow\pm\infty}
\exp\left(itH_0\right)E_0\left(A_0\right)JE_1\left(A_1\right)
\exp\left(-itH_1\right)f,$$
exist for every pair of bounded intervals $A_0$ and $A_1$ in $\rz$, and
the function
$$t\mapsto\exp\left(itH_0\right)J\exp\left(-itH_1\right)f$$
is slowly oscillating at $\pm\infty$.
Moreover the wave operators
$\Omega_{\pm}\left(H_0,E_0\left(A_0\right)JE_1\left(A_1\right),H_1\right)$,
where $A_0$ and $A_1$ are arbitrary bounded intervals, exist if and only
if for
every $\varepsilon>0$ the wave operators
$\Omega_{\pm}\left(H_0,J_{\varepsilon},H_1\right)$ exist. Here
%$J_{\varepsilon}$ is the operator
$J_{\varepsilon}=\int_0^1\exp\left(i\tau\varepsilon H_0\right)J
\exp\left(-i\tau\varepsilon H_0\right)d\tau$.



Jan van Casteren
Department of Mathematics and Computer Science
University of Antwerp (UIA)
Universiteitsplein 1
2610 Antwerp/Wilrijk
Belgium

Telephone:  +32 3 820 24 02 (office);
            +32 3 820 24 01 (department);
            +32 3 239 77 33 (home).
Fax:        +32 3 820 24 21 (department).
e-mail:     vcaster@uia.ua.ac.be
Web site:   http://win-www.uia.ac.be/hpwisinf/index.html

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