C \norm{\psi}^2;\quad  Q_1=Q_1(a,j)=\xi^+_j( x) G_{d_j} .
\end{align*}

\begin{align*}
\int^\infty_{-\infty} \,\norm[\big]{Q_2^\pm f_1 (H)\psi(t)}^2\,\d t\leq
C \norm{\psi}^2;\quad Q_2^\pm=r^{-1/2}\sqrt{(\chi^2_+)'}\parbb{\pm B/\epsilon_0}.
\end{align*}

*  Let $ T_1^a= \sqrt{-\chi'_-}\parb{\pm B_{\delta,\rho_1}^a}{\chi_+}(\pm
B/\epsilon_0)$ and

$\vH^a={\parb{\mathop{\mathrm{Hess}  }r^a}\parb{x^a/r^\delta}} $.

\begin{align*}
\int^\infty_{-\infty} \,&\norm[\big]{ Q_3^\pm f_1 (H)\psi(t)}^2\,\d t\leq
C \norm{\psi}^2;\\&\quad Q_3^\pm=Q_3^\pm(a)=2\parb{\vH^a}^{1/2}\parb{p^a\mp\tfrac \delta 4\tfrac
B}\tilde{f}_2(H_a) r^{(\rho_1-\delta)/2}T_1^aM_a.
\end{align*}

*  Let $\eta(b)=\sqrt{b}\chi_+(
b/\epsilon_0)$, $\zeta_1(b)=\sqrt{-b\chi'_-(
b)}$  and $T_{2}^a={\zeta_1}\parb{\pm B_{\delta,\rho_1}^a}{\eta}(\pm
B)$. Then

\begin{align*}
\int^\infty_{-\infty}
\,&\norm[\big]{Q_4^\pm f_1 (H)\psi(t)}^2\,\d t\leq
C \norm{\psi}^2;\\&\quad Q_4^\pm=Q_4^\pm(a)=\rho_1^{1/2} r^{-1/2}T^a_2M_a.
\end{align*}

*  Let $T_{1+}^a= \sqrt{-\chi'_-}\parb{\pm B_{\delta,\rho_1}^a}\chi_+\parb{r^{\rho_2-1}r_\delta^a
}{\chi_+}(\pm B/\epsilon_0)$ and  $\vH^a$ be given as in <ref>. Then

\begin{align*}
\int^\infty_{-\infty} \,&\norm[\big]{Q_5^\pm f_1 (H)\psi(t)}^2\,\d t\leq
C \norm{\psi}^2;\\&\quad Q_5^\pm=Q_5^\pm(a)=2\parb{\vH^a}^{1/2}\parb{p^a\mp\tfrac \delta 4\tfrac
\end{align*}

*  Let $\eta(b)=\sqrt{b}\chi_+(
b/\epsilon_0)$,   $\zeta_1(b)=\sqrt{-b\chi'_-(
b)}$ (as in <ref>) and $T_{2+}^a= {\zeta_1}\parb{\pm B_{\delta,\rho_1}^a}\chi_+\parb{r^{\rho_2-1}r_\delta^a
}{\eta}(\pm B)$. Then

\begin{align*}
\int^\infty_{-\infty}
\,&\norm[\big]{Q_6^\pm f_1 (H)\psi(t)}^2\,\d t\leq
C \norm{\psi}^2;\\&\quad Q_6^\pm=Q_6^\pm(a)=\rho_1^{1/2} r^{-1/2}T_{2+}^aM_a.
\end{align*}

*  Let $\zeta_2(b)=\sqrt{-b \chi_+(-b)}$,
$T_3^a =\zeta_2\parb{\pm B_{\delta,\rho_1}^a}\tilde{f}_2(H_a)\chi_+(
\pm B/\epsilon_0)$ and $\chi_{1+}(s)=2\chi_+(s)\chi_+'(s)$. Then

\begin{align*}
\int^\infty_{-\infty} \,&\norm[\big]{Q_7^\pm f_1 (H)\psi(t)}^2\,\d t\leq
C \norm{\psi}^2;\\&\quad Q_7^\pm=Q_7^\pm(a)={r^{(\rho_2-\rho_1-1)/2}\sqrt{\chi_{1+}}\parb{r^{\rho_2-1}r_\delta^a }
\end{align*}

*  Let $\eta(b)=\sqrt{b}\chi_+(
b/\epsilon_0)$ (as in <ref>),
$T_4^a=\chi^{1/2}_-\parb{\pm B_{\delta,\rho_1}^a}\eta(\pm
B)$ and  $\chi_{2+}(s)=2s\chi_+(s)\chi_+'(s)$. Then

\begin{align*}
\int^\infty_{-\infty} \,&\norm[\big]{Q_8^\pm f_1 (H)\psi(t)}^2\,\d t\leq
C \norm{\psi}^2;\\&\quad Q_8^\pm=Q_8^\pm(a)=(1-\rho_2)^{1/2} {r^{-1/2}\sqrt{\chi_{2+}}\parb{r^{\rho_2-1}r_\delta^a }
\end{align*}

We only consider the upper sign case. The lower sign case assertions can be
demonstrated similarly.

We choose  $m=m_j$ conforming with (<ref>) and let $M=M_j$ be the
corresponding operator  defined in  (<ref>). Applying
(<ref>) and (<ref>) we can then estimate

\begin{align*}
&{\i f_1(H) [g(H),  M] f_1(H)}\\& \simeq_1
p\cdot \parb{\chi_+(\abs{x})\mathop{\mathrm{Hess}}
\\& \geq 4f_1(H) G^*_{d_j}\chi_+(\abs{x})\xi^+_j( x)^2
G_{d_j} f_1(H).
\end{align*}
\begin{align*}
\Psi_{1,j}=  f_1(H) M_j f_1(H)
\end{align*}
Since  $\Psi_1=\Psi_{1,j}$ is   bounded, <ref> follows from
integration of $\tfrac{\d}{\d t}\inp{\Psi_1}_t=\inp{\i[g(H),\Psi_1]}_t$ using the above
estimation   and

<ref>:  Let
\begin{align*}
\Psi_2=  f_1(H) {\chi^2_+}(B/\epsilon_0) f_1(H).
\end{align*}
We obtain <ref> by integrating  $\tfrac{\d}{\d
t}\inp{\Psi_2}_t=\inp{\i[g(H),\Psi_2]}_t$ using (<ref>), (<ref>)
and the boundedness of  $\Psi_2$.  Here
(<ref>) is used with  $f=\tilde f_2$. Upon    multiplying   both
sides of that  estimate from the left and from the right by a factor
$f_1(H)$  the second term to the right gets the good form $\simeq_1 0$.

<ref>, <ref>:
Let  $A_1=\chi_+(
B/\epsilon_0)$, $A_3^a=\chi_-\parb{B_{\delta,\rho_1}^a}$ (as in
(<ref>) for the `plus case') and

\begin{align*}
\Psi_3=  f_1(H) M_aA_1A_3^aA_1M_af_1(H).
\end{align*}
We obtain <ref> and  <ref> by integrating  $\tfrac{\d}{\d
t}\inp{\Psi_3}_t$ using   <ref>, <ref>,
the boundedness of  $\Psi_3$ and by using (<ref>)–(<ref>)
(as well as  the arguments for the latter  assertions and the
accompanying   comments).
By the  product rule for commutation  there are several
terms. Let us first treat the contribution
from $\i[g(H), M_a]$: We write using
(<ref>), (<ref>)  and  Remark <ref>

\begin{align*}
&2f_1(H) \Re\parb{\i [g(H),  M_a] A_1A_3^aA_1M_a}f_1(H)\\
& \simeq_1
p\cdot \parb{\mathop{\mathrm{Hess}}
A_1A_3^aA_1 M_a}
\\& \simeq_1 8\Sigma_{j\leq J}f_1(H) \Re\parbb{ \,
G^*_{d_j} \xi^+_j\vG_j \parb{\tilde{f}_1(H)\xi^+_jG_{d_j}\tilde{f}_1(H)}  A_1A_3^aA_1
M_a} f_1(H)
\\& \simeq_1 \Sigma_{j\leq J}\,T_j;\quad T_j={ f_1(H) \, G^*_{d_j}\xi^+_j{\vG'_j}\xi^+_j G_{d_j} f_1(H)}\text{ with } {\vG'_j}\text{ bounded}.
\end{align*}
Indeed the last expression arises by commuting the
factor $\tilde{f}_1(H)\xi^+_jG_{d_j}\tilde{f}_1(H)$ to the right producing an error on the
form $\vO(r^{\rho_1-\delta-1})=\vO(r^{(-1)_-})$ along the pattern of  Remark <ref>.
Now we   apply and <ref> to each term $T_j$ on the
right-hand side.

As for the contribution
from $\i[g(H), A_1]$ we use (<ref>),  writing

\begin{align*}
&2f_1(H) \Re\parb{\i \tilde f_2(H) M_a[g(H),  A_1] A_3^aA_1M_a\tilde
f_2(H) }f_1(H) \\& \simeq_1
2f_1(H) \Re\parb{\i M_a\tilde f_2(H) [g(H),  A_1]\tilde f_2(H)  A_3^aA_1M_a}f_1(H) \\& \simeq_1
f_1(H)\sqrt{(\chi^2_+)'}\parbb{\pm B/\epsilon_0} \vO(r^{-1})\sqrt{(\chi^2_+)'}\parbb{\pm B/\epsilon_0}f_1(H).
\end{align*}

The right-hand side is treated by  <ref>.

Next let us elaborate on how
(<ref>)–(<ref>) are  used for  treating  the contribution
from $\i[g(H), A_3^a]$. We need to compute

\begin{align*}
f_1(H) M_aA_1\i
\end{align*}

Modulo a term in $\vO(r^{(-1)_-})$ this
quantity is given by

\begin{align*}
&f_1(H) \tilde\xi_a^+M_aA_1\i
&\simeq_1 f_1 (H)M_aA_1 \tilde f_2(H_a)\i
[g(H_a),A_3^a] \tilde f_2(H_a)A_1M_af_1(H).
\end{align*}
Note that any derivative of $\tilde\xi_a^+$ will
contribute by an error on the form  $\vO(r^{-\infty})$ due to
(<ref>). Then we apply the computations of Subsection <ref>, and by the same argument as above
(i.e.  using that $f_1\prec \tilde f_2$) we
can then remove  appearing  factors of $\tilde f_2(H_a)$.

<ref>–<ref>: With $A_1$ and $A_3^a$ as above we

\begin{align*}
\Psi_4=  f_1(H) M_aA_1\chi_+\parb{r^{\rho_2-1}r_\delta^a }A_3^a\chi_+\parb{r^{\rho_2-1}r_\delta^a }A_1M_af_1(H).
\end{align*}
We   compute and integrate  $\tfrac{\d}{\d
t}\inp{\Psi_4}_t$, using again
<ref>, <ref> and   (<ref>) as well as the
boundedness of  $\Psi_4$. We got from the above  computation of  $\i[g(H),A_3^a]$ effectively negative
main terms. There is an additional  contribution, more
precisely from the two appearances of
$\i[g(H),\chi_+\parb{r^{\rho_2-1}r_\delta^a }]$  and this is also
effectively negative (contributing by two  additional negative terms), cf. the treatment of
$\i[g(H),A_2^a]$ in  Subsection
<ref>. Since the  four non-trivial  contributing terms
come out with the same sign we conclude

*  Due to
(<ref>) the appearing factor of
$ \tilde{f}_2(H_a) $ in <ref>,  <ref>  and <ref> can  be
replaced by  $ \tilde{f}_2(H) $ (or by  $ \tilde{f}_2(\breve H_a)
$, cf. <ref> below). (This was actually implicitly used in
the above proof.) The factor  serves as a regularization indeed
making $Q^\pm_3(a)$, $Q^\pm_5(a)$  and  $Q^\pm_7(a)$ bounded relatively to  $H$
and can  (presumably) not  be removed. By the same argument one
could  make all of the `$Q$-operators' bounded, if preferred. Note
for example that  in all cases $Q \tilde f_2 (H)=\vO(r^{(\rho_1-\delta)/2})$
in the precise sense of  (<ref>).

There are very  similar
bounds  for $\brH_a$. So denote  $\varphi(t)=\e^{-\i t \brH_a } \varphi$ for
$\varphi\in \vH$.  Using  (<ref>) and
(<ref>) we may replace $f_1 (H)\psi(t)$ in
<ref>–<ref> by
$f_1(\brH_a)f_1(\brh_\alpha)\varphi(t)$ and the expression
$\norm{\psi}^2$ to right of the estimates by $\norm{\varphi}^2$.
The proof of these
modifications of
<ref>–<ref>  are essentially the same. Note that the analogue of (<ref>) reads

\begin{align}\label{eq:smoothBnd22}
\int^\infty_{-\infty} \,\norm{Q_0f_1(\brH_a)f_1(\brh_\alpha)\varphi(t)}^2\,\d t\leq
C_\epsilon \norm{\varphi}^2;\quad Q_0=r^{-1/2-\epsilon},\,\epsilon>0.
\end{align}

When modified by replacing the factors of $f_1 (H)$ by factors of
$f_1(\brH_a)f_1(\brh_\alpha)$ the  propagation observables
$\Psi_{1,j},j\leq J,\Psi_2,\Psi_3$ and $ \Psi_4$ may be denoted
$\Phi_{1,j},$ $j\leq J, \Phi_2,\Phi_3$ and $\Phi_4$, respectively (to
be  useful  in the proof of Lemma

*  The structure of the bounds
<ref>–<ref>  and the analogous ones  mentioned in <ref> is given as

\begin{align*}
&\int^\infty_{-\infty} \,\norm[\big]{Q^\pm f_1 (H)\psi(t)}^2\,\d t\leq
C \norm{\psi}^2,\\
&\int^\infty_{-\infty} \,\norm[\big]{Q^\pm f_1(\brH_a)f_1(\brh_\alpha)\varphi(t)}^2\,\d t\leq
C \norm{\varphi}^2,
\end{align*}
respectively, where $Q^\pm $ may (or may not)  depend on
$a$ and $j\leq J=J(a)$. In addition we also have such bounds with
cf. (<ref>) and (<ref>). We  give the
$Q$'s  (including $Q=\vO(r^{(-1/2)_-})$) some uniform index to distinguish them, say
$Q^\pm =Q^\pm(a,k)$ (not specifying $k$). By the Kato smoothness
theory [17] a consequence of the above bounds are the following resolvent
bounds, valid for any combination of the indices,

\begin{align}\label{eq:kato1}
\quad \sup _{\Im z>0}\norm{(2\i\pi)^{-1}Q^+(b,l)f_1 (H)\parb{R(z)-R(\bar z)}f_1
(H)Q^-(a,k)^*}< \infty.
\end{align}
In particular, recalling (<ref>),
(<ref>) and the surrounding discussion, there exist weak limits as $z\to \lambda\in\R$ from above and these limits define a   weakly continuous
function in
$\lambda$. Whence we can record that the  limiting operators

\begin{align}\label{eq:kato2}
\begin{split}
& \quad Q^+(b,l))f_1 (H)\delta(H-\lambda)f_1 (H)Q^-(a,k)^* \in
\vL(\vH)\\&  \quad\quad \text{ exist with a weakly continuous
dependence of }\lambda\in\R.
\end{split}
\end{align}

*  We do not show the existence of

\begin{align*}
\wlim _{\epsilon\to 0_+} \,Q^+(b,l))f_1 (H)R(\lambda+ \i \epsilon)f_1 (H)Q^-(a,k)^*,
\end{align*}
although it is likely  doable by mimicking
<cit.>.  Up to Subsection <ref> it is more
to deal with the delta function only. However in
Subsection <ref>  we need the resolvent bounds

\begin{align}\label{eq:kato10}
\quad \sup _{\Im z\neq 0}\norm{Q^\pm(b,l)f_1 (H){R(z)}}_{\vL(\vB,\vH)}< \infty,
\end{align}
which follow by combining our commutator
bounds, the Besov space bound (cf. (<ref>))
and the proof
of <cit.>.

§.§ Integrability of  $T^\pm_\alpha$

We shall show the assertion (<ref>) in a standard fashion  verifying `integrability of the

The operators $\Omega^\pm_\alpha$ are well-defined.

The existence of the time-limits  will be verified by means of  Section
<ref>, (<ref>), Lemma <ref> and
Remarks <ref> <ref>,  computing  and estimating the quantity

\begin{align*}
T^\pm_\alpha&=\i f_1(H) g(H)  M_a N^a_\pm M_a f_1(\brH_a)f_1(\brh_\alpha)-\i f_1(H)  M_a N^a_\pm M_a g(\brH_a) f_1(\brH_a)f_1(\brh_\alpha)\\& =
{f_1(H) \i [g(H),  M_a N^a_\pm M_a] f_1(\brH_a)f_1(\brh_\alpha)}\\&\quad\quad
M_a\tilde{\xi}_a^+\parb{g(H)-g(\brH_a)}f_1(\brH_a)f_1(\brh_\alpha)=: T_1+T_2.
\end{align*}
For convenience we treat only the `plus case'.
The calculation is  modulo  terms on the form
$T=f_1(H)\vO(r^{(-1)_-})f_1(\brH_a)f_1(\brh_\alpha)$. Let us abbreviate this
property as $T\simeq_2 0$ (which is a different relation than the one
used in Subsection <ref>). Any such `error'
can be treated by
the smoothness bounds
(<ref>) and (<ref>).

We aim at verifying integrability of $T^+_\alpha$  in the precise
sense  stated below  as
(<ref>) ($T^-_\alpha$ can be
treated similarly).
Most of the work is
essentially  already done in the proof of  Lemma
<ref>. However the
computation and estimation of the  term $T_2$ to the right is different. We claim
that this term $\simeq_2 0$.  To see this we
note that on the support of the factors $A_2^a$ in the definition of

\begin{align}\label{eq:abndindre}
\inp{x^a}\leq C {r}^{1-\rho_2}.
\end{align}
Next we
use a Taylor formula like the one used in (<ref>), however used
We write and estimate  for any $b\not\leq a$, on the support of
$\tilde{\xi}_a^+$, with (<ref>)  and for $r$ large

\begin{align*}
\abs[\big]{V_{\rm lr}^b(x)- V_{\rm lr}^b(x_a)}&=\abs[\Big]{\int_0^1 \pi^bx^a\cdot (\nabla
V_{\rm lr}^b)\parb{\pi^b(x-tx^a)}\,\d
t}\\&\leq C_1\inp{x^a}\parb {\abs{x^b}-C_2\abs{x^a}}^{-(1+\mu)}
\\&\leq C_3{r}^{1-\rho_2}\parb {\abs{x}-C_4{r}^{1-\rho_2}}^{-(1+\mu)}
\\&\leq C_5{r}^{-\rho_2-\mu}.
\end{align*}

Similarly,  under the same
conditions as above,

\begin{align*}
\abs{\pi^bx_a}\geq \abs{\pi^bx}-\abs{\pi^bx^a}\geq c_1 r\geq
c_2\abs{x_a};\quad c_1>c_2>0.
\end{align*}

For the case where $I^{\rm sr}_a=0$ we can use the above bounds and
the fact that $\rho_2+\mu>1$ to
argue, abbreviating $\chi_-\parb{\inp{x^a}

\begin{align*}
T_2&\simeq_2 f_1(H)\vO(r^0)\chi_-(\cdot)\tilde{\xi}_a^+\parb{g(H)-g(\brH_a)}f_1(\brH_a)f_1(\brh_\alpha)\\
&\simeq_2 \int_{\C}
I_a}(\brH_a-z)^{-1}f_1(\brH_a)f_1(\brh_\alpha)\,\d \mu_g(z)\\
&=f_1(H)\vO(r^{-\rho_2-\mu})f_1(\brH_a)f_1(\brh_\alpha)\,\simeq_2 0.
\end{align*}
If $I^{\rm
sr}_a\neq 0$ there is an extra term on the form $\vO(r^{-1-\mu})$ in
the above integral, which obviously yields the same conclusion.

As for the term $T_1$ we use  as in Section <ref> the functions $ f_1\prec \tilde f_2\prec\tilde{f_1}$. We may write $T_1$ as

\begin{align}\label{eq:comuse}
T_1\simeq_2 f_1(H) \tilde f_2(H)\i [g(H),  M_a N^a_\pm M_a] \tilde f_2(H)f_1(\brH_a)f_1(\brh_\alpha).
\end{align}

Applying   the  product rule for commutation  several
terms arise,
essentially all treated in Section
<ref> (see also the proof of Lemma <ref>). Let us show how  to treat the contribution from the factors
of $\i [g(H), A_1]$. (The contribution from the factors
of $\i [g(H), M_a]$ may be  treated similarly, cf. the proof of  Lemma

By (<ref>) and Remark <ref>  we can write it as

\begin{align*}
&2f_1(H) \tilde f_2(H) M_a\Re\parbb{\i [g(H),
A_1]A_2^aA_3^aA_2^aA_1}M_a\tilde f_2(\brH_a)f_1(\brH_a)f_1(\brh_\alpha)\\
&\simeq_2 f_1(H)
(Q^+_2)^*\vO(r^{-0})Q^+_2f_1(\brH_a)f_1(\brh_\alpha);\quad Q^+_2=r^{-1/2}\sqrt{(\chi^2_+)'}\parbb{\pm B/\epsilon_0}.
\end{align*}

Next we apply estimating

\begin{align*}
&\int^\infty_{-\infty} \abs[\big]{\inp{f_1(H)\psi(t),
(Q_2^+)^*\vO(r^{-0})Q_2^+f_1(\brH_a)f_1(\brh_\alpha)\varphi(t)}}\,\d t\\
&\leq \int^\infty_{-\infty}
\,\norm{Q_2^+{f_1(H)\psi(t)}}\,\norm{Q_2^+f_1(\brH_a)f_1(\brh_\alpha)\varphi(t)}\,\d t\\
&\leq \parbb{\int^\infty_{-\infty} \,\norm{Q_2^+{f_1(H)\psi(t)}}^2\,\d t}^{1/2}\,\parbb{\int^\infty_{-\infty} \,\norm{Q_2^+f_1(\brH_a)f_1(\brh_\alpha)\varphi(t)}^2\,\d t}^{1/2}.
\end{align*}
The first factor is finite due to Lemma <ref>
<ref>, in turn  with a bound proportional to
$\norm{\psi}$. Similarly, thanks to   Remark <ref>
<ref>, the
second factor is finite too (although not needed with a bound proportional with

We may argue similarly for the other commutators arising from expanding
the commutator in  (<ref>). More precisely we  first compute

\begin{align*}
&f_1(H) M_aA_1\i
[g(H),A_2^aA_3^aA_2^a]A_1M_af_1(\brH_a)f_1(\brh_\alpha)\\&\quad\simeq_2 f_1(H) M_aA_1\i
\Big [g(H),\parbb {A_3^a-\chi_+\parb{r^{\rho_2-1}r_\delta^a
}A_3^a\chi_+\parb{r^{\rho_2-1}r_\delta^a }}\Big ]A_1M_af_1(\brH_a)f_1(\brh_\alpha)\\&\quad
\simeq_2 f_1(H) \parbb{-(Q_{3}^+)^2-Q_{4}^+)^2+(Q_{5}^+)^2+Q_{6}^+)^2 +(Q_{7}^+)^2+Q_{8}^+)^2} f_1(\brH_a)f_1(\brh_\alpha).
\end{align*}
We integrate, estimate by  and  invoke  Remarks <ref>
<ref> and <ref>.

These estimates  lead
to the conclusion that

\begin{align*}  \int^\infty_{-\infty} \abs[\big]{\inp{\psi(t),
T^+_\alpha\varphi(t)}}\,\d t\leq C_\varphi\norm{\psi},
\end{align*}
and by the same arguments,  that

\begin{align}\label{eq:intCau}
\forall \epsilon>0\,\,\exists t_\epsilon=t_\epsilon(\varphi)>0:\quad \int^\infty_{t_\epsilon} \abs[\big]{\inp{\psi(t),
T^+_\alpha\varphi(t)}}\,\d t\leq \epsilon \norm{\psi}.
\end{align}

Clearly the existence of
$\Omega^+_\alpha$ follows from (<ref>). The existence of
$\Omega^-_\alpha$ can be shown similarly.

§ FORMULA FOR $\WIDETILDE

Recall from Section <ref>,

\begin{align}\label{eq:formwOp}
\widetilde W^\pm_\alpha= \Omega^\pm_\alpha J_\alpha \breve
w_a^{\pm}f_2(k_\alpha)\mand \widetilde S_{\beta\alpha}=\parb{\Omega^+_\beta J_\beta \breve
w_b^{+}f_2(k_\beta}^*\Omega^-_\alpha J_\alpha \breve
\end{align}

We show  in Appendix <ref> the following representation
of  $\widetilde
S_{\beta\alpha}(\lambda)$, formally given as

\begin{align}\label{eq:SreP}
\widetilde
S_{\beta\alpha}(\lambda)=(2\pi \i)^2f^2_2(\lambda){\gamma_b^+(\lambda_\beta)}J^*_\beta
\parb{T^+_\beta}^* \delta(H-\lambda)T^-_\alpha
J_\alpha{\gamma_a^-(\lambda_\alpha)}^*;\quad \lambda_\alpha:=
\lambda-\lambda^\alpha.
\end{align}

Parallel to Section <ref> we call the appearing
$\lambda$-depending operators
$J_\beta{\gamma_b^+(\lambda_\beta)}^*$ and $J_\alpha{\gamma_a^-(\lambda_\alpha)}^*$ future
and past auxiliary
channel wave matrices, respectively. In Subsections <ref> and <ref> we examine the well-definedness  of
the right-hand side of (<ref>) upon substituting
expressions for $T^+_\beta$ and $T^-_\alpha$
derived in the proof of Lemma
<ref>. With the resulting  final
representation we then derive the weak
continuity of $\widetilde S_{\beta\alpha}(\cdot)$  from
which Theorem <ref> follows.

§.§ Phase space estimates of  auxiliary channel wave matrices

We aim at   bounding  the operators (recall the index convention of
Remark <ref> <ref>)

\begin{align*}
\vH}
\end{align*}
as well as showing  a continuity property. This  will
along with (<ref>) be
crucial  for our usage of (<ref>).

For all  $a\in \vA'$ and indices $k$

\begin{align}\label{eq:Tbnd}
\sup_{\lambda\in\Lambda}\,\,\norm{Q^\pm(a,k)f_1(\brH_a)f_1(\brh_\alpha)J_\alpha{\gamma_a^\pm(\lambda_\alpha)}^*}<\infty,
\end{align}

and  the operator-valued functions

\begin{align*}
\vH}
\end{align*}
are strongly continuous in $\lambda\in
\Lambda$.

It suffices     to consider   the cases $Q^\pm(a,k)\neq Q_0$ since
$ Q_0$ (as given by (<ref>)) the assertions  reduce to  properties of
${\gamma_a^\pm(\lambda_\alpha)}^*$ stated in Section
<ref>. Whence from this point we assume
$Q^\pm(a,k)\neq Q_0$.
The  uniform
bounds (<ref>) will be proved in Step I below. The stated
strong continuity assertions will then be shown in  Steps
II–V. Note that  weak continuity is an immediate consequence of
(<ref>) and the strong continuity for  $Q^\pm(a,k)= Q_0$ (to
be used in Steps IV and V).

I. By `the $T^*T$ argument' is suffices for  (<ref>)  to show that

\begin{align*}
\sup_{\lambda\in\Lambda}\,\,\norm{{\gamma_a^\pm(\lambda_\alpha)}J_\alpha^*f_1(\brh_\alpha)f_1(\brH_a)Q^\pm(a,k)^*Q^\pm(a,k)f_1(\brH_a)f_1(\brh_\alpha)J_\alpha{\gamma_a^\pm(\lambda_\alpha)}^*}<\infty.
\end{align*}
We pick, using quantities from Remark
<ref> <ref>,

\begin{align}\label{eq:probndbbs}
\begin{split}
\set{\Phi_{1,j},\Phi_2,\Phi_3,\Phi_4\,|\,j\leq J}:\\&\quad
\i[\brH_a,\Phi]\geq f_1(\brh_\alpha)f_1(\brH_a)Q^\pm(a,k)^*Q^\pm(a,k)f_1(\brH_a)f_1(\brh_\alpha)\\&\quad\quad+f_1(\brh_\alpha)f_1(\brH_a)\vO(r^{(-1)_-})f_1(\brH_a)f_1(\brh_\alpha).
\end{split}
\end{align}

For $\rho>1$ we let $\chi_\rho=\chi_-(r/\rho)$ and compute
using   the notation $\inp{T}_\phi=\inp{\phi,T\phi}$ with
g\in L^2(C_a)$,

\begin{align*}
0&=\inp{\i [\brH_a-\lambda,\chi_\rho\Phi\chi_\rho]}_\phi\\
&=\inp{\chi_\rho\i [\brH_a,\Phi]\chi_\rho}_\phi+2\Re{\inp{\i
[\brH_a,\chi_\rho]\Phi \chi_\rho}_\phi}.
\end{align*}

For the last term to the right we compute and insert the expression

\begin{align*}
\i
\end{align*}
commute and  invoke the dual versions of
(<ref>) allowing us to  conclude that this term is
bounded by $C\norm{g}^2$ with  a constant $C>0$ being  independent of
$\rho>1$. For the first term we use (<ref>),
(<ref>) and  a commutation,  yielding the bound

\begin{align}\label{eq:Tbndmm}
\sup_{\lambda\in\Lambda, \rho>1}\,\,\norm{\chi_\rho Q^\pm(a,k)f_1(\brH_a)f_1(\brh_\alpha)J_\alpha{\gamma_a^\pm(\lambda_\alpha)}^*}<\infty.
\end{align}

(<ref>) follows from  (<ref>)  and
Lebesgue's monotone convergence theorem (by taking $\rho\to \infty$).

II.  For
$Q_1(a,j)=\xi^+_jG_{d_j}$ (as  in   Lemma
<ref> <ref>) the strong continuity assertion follows from
(<ref>) if  $ d_j= a$ due to the representation

\begin{align*}
\xi^+_jG_{d_j}f_1(\brH_a)f_1(\brh_\alpha)J_\alpha{\gamma_a^\pm(\lambda_\alpha)}^*= f^2_1(\lambda)\xi^+_jJ_\alpha
\end{align*}

For   $ d_j\neq a$ we have $ d_j \lneq a$ (since $ d_j\leq a$). The fact
that the factor
$\xi^+_j$ is supported in $Y_{d_j}(\delta_j)$ then yields  the estimate
$\abs{x^a}\geq \epsilon\abs{x}$ on $\supp \xi^+_j$ for some
We decompose  for
(large) $\rho>1$

\begin{align*}
\xi^+_jG_{d_j}&=\chi_1\xi^+_jG_{d_j}+\chi_2\xi^+_jG_{d_j},\\&\chi_1=\chi^2_+\parb{\abs{x^a}/\rho},\quad \chi_2=\chi^2_-\parb{\abs{x^a}/\rho},
\end{align*}
and write correspondingly

\begin{align*}
\xi^+_jG_{d_j} J_\alpha{\gamma_a^\pm(\lambda_\alpha)}^*g
\end{align*}

For the first term we estimate using (<ref>), cf. the proof of

\begin{align*}
\norm{\chi_1\xi^+_jG_{d_j}&J_\alpha{\gamma_a^\pm(\lambda_\alpha)}^*g}^2\leq
C_1\int_{\abs{x^a}\geq \rho}\d
x^a \parb{\abs{u^\alpha}^2+\abs{p^au^\alpha}^2}\\&\abs{x^a}^{-1}\int_{\abs{x_a}\leq
\epsilon^{-1}\abs{x^a}} \parb{\abs{\gamma_a^\pm(\lambda_\alpha)^*g}^2+\abs{p_a{\gamma_a^\pm(\lambda_\alpha)}^*g}^2}\,\d
C_2\int_{\abs{x^a}\geq \rho}\,\parb{\abs{u^\alpha}^2+\abs{p^au^\alpha}^2}\,\d
x^a ;
\end{align*}
here the constant $C_2$ is independent of  $\lambda\in
\Lambda$. Consequently the right-hand side can be taken arbitrary
small uniformly in $\lambda$ by taken $\rho>1$ big enough (now fixed).

For the the second term  note that  $\abs{x}\leq
2\epsilon^{-1}\rho$ on the   support of $\chi_2$. Hence this term is continuous in
$\lambda\in \Lambda$
thanks to the established continuity for $Q^\pm(a,k)= Q_0$.
Whence we have treated  the case
$Q_1(a,j)=\xi^+_jG_{d_j}$ for  any $j \leq J$.

III. For  $Q_2^+$ (as  in   Lemma
<ref> <ref>, leaving out a similar discussion
for $Q_2^-$)  it suffices by  (<ref>) to show continuity in
$\lambda$ of

\begin{align*}
g \in C^\infty (C_a).
\end{align*}
For $\lambda$ close to a fixed $\lambda'\in \Lambda$ we have
correspondingly that $\lambda_\alpha=\lambda-\lambda^\alpha$ is close to
$\lambda'_\alpha=\lambda'-\lambda^\alpha$, and we can to some extent
use  that for $s>1/2$

\begin{align*}
\norm{\inp{x_a}^{-s}\phi(\lambda)}\to 0\text{ for }\lambda\to \lambda';\quad \phi(\lambda):={\gamma_a^+(\lambda_\alpha)}^*g-{\gamma_a^+(\lambda'_\alpha)}^*g.
\end{align*}

By  arguments from Step II, in particular    it suffices to show that   for some

\begin{align*}
\norm{Q_2^+\chi^2_-\parb{\abs{x^a}/(\epsilon
J_\alpha\phi(\lambda)}\to 0
\text{ for }\lambda\to \lambda'.
\end{align*}
In turn, possibly seen  by using one-body versions of Lemmas <ref> and
<ref> and <cit.>
(more precisely by using resolvent bounds), it suffices to
show that  $\norm
{\tilde{\varphi}(\lambda)}=o(\abs{\lambda-\lambda'}^0)$, where for
$\epsilon, \epsilon_0>0$ taken  small

\begin{align*}
\tilde{\varphi}(\lambda)&:=\tilde{\varphi}_+(\lambda)+\tilde{\varphi}_-(\lambda),\\
\tilde {\varphi}_\pm(\lambda)&:=
\tilde f_1(\brH_a)\sqrt{(\chi^2_+)'}(
B/\epsilon_0)\tilde f_1(\brH_a)r^{-1/2}T^\pm_\epsilon
f_1(\brH_a)f_1(\brh_\alpha)J_\alpha\phi(\lambda),\\\quad  T^\pm_\epsilon&:=\chi^2_-\parb{\abs{x^a}/(\epsilon
\chi_+\parb{\pm{\Re{\parb{(x_a/r)\cdot
\end{align*}
(Recall from the discussion at the end of Subsecton
<ref> that we allow ourselves
the freedom of considering only sufficiently small values of $\epsilon_0$.)
We can represent

\begin{align}\label{eq:Bdecomp}
\begin{split}
B&=2\Re{\parb{(x/r)\cdot p+\vO(r^{-1})\cdot
p}}\\&=2\Re \parb{{(x_a/r)\cdot p_a}+{(x^a/r)\cdot p^a}+{\vO(r^{-1})\cdot
\end{split}
\end{align}
and by using this
and   commutation (cf.  Remark
<ref>),   we estimate (with $\inp{T}_\varphi=\inp{\varphi,T\varphi}$)

\begin{align}\label{eq:commsmall}
\begin{split}
&\tfrac12 \norm {\tilde{\varphi}_\pm(\lambda)}^2\\
&\leq \inp{I\mp {B}/(4\epsilon_0)}_{\tilde{\varphi}_\pm(\lambda)}+o(\abs{\lambda-\lambda'}^0)\\
&\leq \inp{(1+C\epsilon) I\mp\Re{\parb{(x_a/r)\cdot
\inp{(C\epsilon-1)
\\&\leq o(\abs{\lambda-\lambda'}^0).
\end{split}
\end{align}

IV. As for $Q_3^\pm(a)$ and $ Q_4^\pm(a)$ we introduce
$Q^\pm:=\sqrt{\abs{Q_3^\pm(a)}^2+\abs{Q_4^\pm(a)}^2}$. It  suffices (thanks to
(<ref>)  and the
weak continuity) to
show that

\begin{align}
\label{eq:weakStrong}
\begin{split}
\norm{Q^\pm
;\\&\quad \quad \quad\quad
g\in C^\infty(C_a).
\end{split}
\end{align}

Letting  $\chi_\rho=\chi_-(\abs{x_a}/\rho)$ for $\rho>1$ and $\phi^\pm(\lambda)=J_\alpha{\gamma_a^\pm(\lambda_\alpha)}^*g$,
we compute as
in Step I
(now with $\Phi^\pm_3=\Phi^\pm_3(a)$)

\begin{align*}
0&=\inp{\i [\brH_a,\chi_\rho\Phi^\pm_3\chi_\rho]}_{\phi^\pm(\lambda)}\\
&=\inp{\i [\brH_a,\Phi^\pm_3]}_{\chi_\rho\phi^\pm(\lambda)}+2\Re{\inp{\i [\brH_a,\chi_\rho]\Phi^\pm_3\chi_\rho}_{\phi^\pm(\lambda)}}.
\end{align*}
We take the limit $\rho\to \infty$. For the first term we
get in the limit $\mp\norm{f^2_1(\lambda)Q^\pm\phi^\pm(\lambda)}^2$ plus a term that we
know is  continuous in $\lambda$ by Steps II and III. Here we used
commutation to replace $\norm{f^2_1(\lambda)Q^\pm\chi_\rho\phi^\pm(\lambda)}^2$ by

\begin{align*}
\inp{Q_3^\pm(a)^*\chi^2_\rho Q_3^\pm(a)}_{f_1(\brH_a)f_1(\brh_\alpha)\phi^\pm(\lambda)}+\inp{Q_4^\pm(a)^*\chi^2_\rho Q_4^\pm(a)}_{f_1(\brH_a)f_1(\brh_\alpha)\phi^\pm(\lambda)}
\end{align*}
before taking the limit.

As for the second term we compute  as follows,   using again
one-body versions of  Lemma
<ref> and <cit.>,  and
using in
the second step also  <ref>
and <ref> from Section <ref>. (The
computation overlaps partially one in Appendix <ref>.) A
further elaboration is needed, it is given after the computation.  We compute for any
$\epsilon,\epsilon_0>0$ taken small

\begin{align}\label{eq:cruci}
&\lim_{\rho\to \infty}2\Re\inp{\i
&=\pm 2\sqrt{\lambda_\alpha}f^4_1(\lambda)\lim_{\rho\to
\infty}\rho^{-1}\Re\inp [\big]{(\chi^2_-)'\parb{\tfrac{\abs{x_a}}\rho}\chi_-\parb{\tfrac{\abs{x^a}}{\epsilon
&=\pm \parbb{2\sqrt{\lambda_\alpha}}^3f^4_1(\lambda)\lim_{\rho\to
\infty}\rho^{-1}\inp [\big]{(\chi^2_-)'\parb{\tfrac{\abs{x_a}}\rho}}_{m_a(\hat x_a)\chi_+\parb{\pm{\Re{\parb{(x_a/r)\cdot
&=\mp\tfrac {2\lambda_\alpha}\pi
\end{align}

Note that in the first step of (<ref>) we used  that
effectively the factor $A_3^a=A_{3\pm}^a$ of $\Phi^\pm_3$ can be removed. This is
essentially due to the fact that
$B_{\delta,\rho_1}^a=0$ in $\set{\abs{x^a}<c r^\delta}$,
cf. <ref> of Section <ref>, since for the complement we have

\begin{align*}
\infty} \rho^{-1}\Re\inp [\big]{p_a\cdot \hat x_a(\chi^2_-)'\parb{\tfrac{\abs{x_a}}\rho}\chi_-\parb{\tfrac{\abs{x^a}}{\epsilon
r}}\chi^2_+\parb{\tfrac{2\abs{x^a}}{c r^\delta}}\Phi^\pm_3}_{\phi^\pm(\lambda)}
\end{align*}
In fact (using commutation and   Remark

\begin{align*}
\infty} \rho^{-1}\Re\inp [\big]{p_a\cdot \hat x_a(\chi^2_-)'\parb{\tfrac{\abs{x_a}}\rho}\chi_-\parb{\tfrac{\abs{x^a}}{\epsilon
r}}\chi^2_-\parb{\tfrac{2\abs{x^a}}{c r^\delta}}\Phi^\pm_3}_{\phi^\pm(\lambda)}
\\&=2 f^4_1(\lambda)\lim_{\rho\to
\infty} \rho^{-1}\Re\inp [\big]{p_a\cdot \hat x_a(\chi^2_-)'\parb{\tfrac{\abs{x_a}}\rho}\chi_-\parb{\tfrac{\abs{x^a}}{\epsilon
r}}\chi^2_-\parb{\tfrac{2\abs{x^a}}{c r^\delta}}A_3^aM_a\tilde f_2(\brH_a)^2A_1^2M_a}_{\phi^\pm(\lambda)},
\end{align*}

\begin{align}\label{eq:aligeen}
\chi^2_-\parb{\tfrac{2\abs{x^a}}{c r^\delta}}\parb{A_3^a-I}= \pm
\int _{\C}
\chi^2_-\parb{\tfrac{2\abs{x^a}}{c r^\delta}}B_{\delta,\rho_1}^a(\pm B_{\delta,\rho_1}^a-z)^{-1}\,z^{-1}\mathrm
\end{align}
Once  $A_3^a$ is removed we can also remove the remaining factor
$\chi^2_-\parb{\tfrac{2\abs{x^a}}{c r^\delta}}$. Similarly  in the first step we also replaced   the factors of
$ A_1= A_{1\pm}$ by   factors of $\chi_+\parb{\pm{\Re{\parb{(x_a/r)\cdot
p_a}}}/(4\epsilon_0)}$. Using the one-body version of <cit.> as in Step III this is
justified by the assertions

\begin{align*}
\infty} \rho^{-1}\Re\inp [\big]{p_a\cdot \hat x_a(\chi^2_-)'\parb{\tfrac{\abs{x_a}}\rho}\chi_-\parb{\tfrac{\abs{x^a}}{\epsilon
r}} M_a\tilde f_2(\brH_a)\parb{A_1^2-I}\tilde f_2(\brH_a)M_a}_{ \phi_1^\pm}=0,\\
&2 f^4_1(\lambda)\lim_{\rho\to
\infty} \rho^{-1}\Re\inp [\big]{p_a\cdot \hat x_a(\chi^2_-)'\parb{\tfrac{\abs{x_a}}\rho}\chi_-\parb{\tfrac{\abs{x^a}}{\epsilon
r}}M_a\tilde f_2(\brH_a)A_1^2\tilde f_2(\brH_a)M_a}_{\phi_2^\pm}=0;\\&\quad \quad \phi_1^\pm=\chi_+\parb{\pm{\Re{\parb{(x_a/r)\cdot
p_a}}}/(4\epsilon_0)}\phi^\pm(\lambda),\\&\quad\quad  \phi_2^\pm=\chi_+\parb{\mp{\Re{\parb{(x_a/r)\cdot
\end{align*}
In turn these assertions may be verified  essentially  as in
Step III (more precisely  by commutation along the lines of  (<ref>))  by showing

\begin{align}\label{eq:good}
\begin{split}
&\tilde f_2(\brH_a) \chi_-\parb{\tfrac{\abs{x^a}}{\epsilon
p_a}}}/(4\epsilon_0)}^2\parb{A_{1\pm}^2-I}\tilde f_2(\brH_a)\\&\quad
\quad
\quad\quad
\quad
\quad
\vO(r^{-1/2})=\vO(r^{0_-}),
\end{split}
\end{align}

\begin{align}\label{eq:lessgood}\begin{split}
&\tilde f_2(\brH_a) \chi_-\parb{\tfrac{\abs{x^a}}{\epsilon
p_a}}}/(4\epsilon_0)}^2A^2_{1\pm}\tilde f_2(\brH_a)\\&\quad \quad \quad\quad \quad \quad = \vO(r^{-1/2})=\vO(r^{0_-}). \end{split}
\end{align}

Obviously the right-hand side of (<ref>), i.e.  $\mp\tfrac
{2\lambda_\alpha}\pi f^4_1(\lambda)\norm{{m_a}g}_{L^2(C_a)}^2$, is continuous in
$\lambda$. We conclude that the function

\begin{align*}
\lambda\to \norm{Q^\pm f_1(\brH_a)f_1(\brh_\alpha)\phi^\pm
\end{align*}

is  continuous,  as wanted.

V. We  mimic Step IV using  now

\begin{align*}
\end{align*}
$\Phi^\pm_4=\Phi^\pm_4(a)$ (replacing $\Phi^\pm_3(a)$)
and using
again Steps II and III,
yielding similarly  the strong continuity for $Q_5^\pm(a),
Q_6^\pm(a),Q_7^\pm(a)$ and $ Q_8^\pm(a)$. In fact this  case is simpler
in that one readily shows the following analogue of (<ref>),

\begin{align*}
\lim_{\rho\to \infty}2\Re\inp{\i
\end{align*}
which obviously is continuous in $\lambda$.

§.§ Conclusion of argument, the weak

The  expression to the right in (<ref>)  can (and should)  be written, cf. the proof of
Lemma <ref>,  as a sum of terms on
the form

\begin{align*}
\parbb{f^2_2(\lambda)&{\gamma_b^+(\lambda_\beta)}J^*_\beta
&\parbb{Q^+(b,l)f_1 (H)\delta(H-\lambda)f_1 (H)Q^-(a,k)^*}\parbb{B_-Q^-(a,k)J_\alpha{\gamma_a^-(\lambda_\alpha)}^*},
\end{align*}
where $B_+=B_+(b,l)$ and $B_-=B_-(a,k)$ are bounded
operators. The middle factor is weakly continuous by
Remark <ref> <ref>. The factor to the right is
strongly continuous by Lemma <ref>. The
adjoint of the   factor to the left is
strongly continuous by the same result. Consequently the
entire product is weakly continuous, and  Theorem
<ref> follows (cf. the
discussion given before the statement).

§ CHANNEL WAVE MATRICES AND SCATTERING AT FIXED

In this concuding section we derive  various consequences of our proof
of Theorem <ref>. Whence   we concretely
construct the open channel wave
matrices and the scattering matrix for  all energies in the   small
neighbourhood $I_0 \ni \lambda_0$ fixed  in Section
<ref>. Away
from a null set of
energies, more precisely at any stationary scattering
energy in $I_0$, the scattering matrix  is unitary and strongly
continuous and it is characterized by asymptotic properties of minimum
generalized eigenfunctions.  The discussion
relies heavily on Appendix <ref>.

For all $b\in \vA'$ we pick an arbitrary  increasing sequence
$(C_{b,k})_k $ of open reflection symmetric subsets of  $C_b'$ with closure
$\overline C_{b,k}\subset C_b'$ and $\cup_k C_{b,k}=C_b'$. We pick for each such set a
function $m_b=m_{b,k}$  as in  Section <ref> such
that $m_b(\hat \xi)=1$ for all $\hat \xi\in C_{b,k}$.   Let for any channel  $\beta
=(b,\lambda^\beta, u^\beta)$

\begin{align*}
\chi_{\beta,k}&=\chi_{\beta,k}(p_b)=F_\beta^{-1}
1_{C_{b,k}}(\hat \xi_b)F_\beta,\\
\tilde\chi_{\beta,k}&=\tilde\chi_{\beta,k}(p_b)=F_\beta^{-1}(4\lambda_\beta)^{-1}
1_{C_{b,k}}(\hat \xi_b)F_\beta.
\end{align*}

By combining (<ref>), (<ref>)  and
(<ref>) we then obtain  (by an elementary approximation argument)

\begin{align}\label{eq:Wfinal2}
\begin{split}
=\widetilde W^+_\beta
&=\int \,\tfrac \pi {2\lambda_\beta} (f_2g)(\lambda) \parb{f_1(H)\delta(H-\lambda) T^+_{\beta,k} J_\beta
\gamma_{b}^+(\lambda_\beta)^*}1_{C_{b,k}}\gamma_{b,0}(\lambda_\beta)\varphi
\,\d \lambda;\\
&\quad \quad\quad\text{ for  any  }\varphi\in L^2_s(\mathbf  X_b) \text{ with   }s>1/2.
\end{split}
\end{align}
Note that indeed $
T^+_{\beta}=T^+_{\beta,k}$ depends on $k$.

We are lead to introduce the approximate  channel wave matrices

\begin{align}\label{eq:AppExa}
\Gamma^+_{\beta,k}(\lambda)^*=\tfrac \pi {2\lambda_\beta}  f_1(H){\delta(H-\lambda) T^+_{\beta,k} J_\beta
\gamma_{b}^+(\lambda_\beta)^*}1_{C_{b,k}};\quad k\in
\N,\,\lambda \in  I_0\,(\ni \lambda_0).
\end{align}

§.§ Channel wave matrices

By (<ref>) the right-hand side of (<ref>)
is independent of  details of the
construction of $T^+_{\beta,k}$. Considered as an $\vL(L^2(C_b), L_{-s}^2(\bX))$-valued function of   $\lambda\in  I_0$
any $s>1/2$, it is strongly continuous. Clearly

\begin{align*}
1_{C_{b,k}}\Gamma^+_{\beta,k+1}(\lambda)=\Gamma^+_{\beta,k}(\lambda)\in \vL( L_{s}^2(\bX),L^2(C_b)).
\end{align*}
We can then  for any  $\lambda\in  I_0$  and $\psi\in
L_{s}^2(\bX)$ with  $s>1/2$  introduce   a measurable function
$\Gamma^+_{\beta}(\lambda)\psi$  on $C_b$ by

\begin{align}\label{eq:GAM}
k\in \N.
\end{align}

We argue that $\Gamma^+_{\beta}(\lambda)\psi\in L^2(C_b)$ as follows.
With (<ref>) it follows from Stone's formula that

\begin{align*}
\int_\Delta\,\norm{\Gamma^+_{\beta,k}(\lambda)\psi}^2\,\d \lambda\leq
\int_\Delta\,\inp{\delta(H-\lambda)}_\psi\,\d \lambda\quad \text{for
any   interval }
\Delta\subseteq I_0,
\end{align*}
to the
conclusion  that

\begin{align}\label{eq:semiBND}
\norm{\Gamma^+_{\beta,k}(\lambda)\psi}^2\leq
\inp{\delta(H-\lambda)}_\psi \text{ for a.e. } \lambda \in I_0.
\end{align}

Since $\Gamma^+_{\beta,k}(\cdot)\psi\in L^2(C_b)$ is
weakly continuous and the functional $\norm{\cdot}^2$  is weakly lower semi-continuous we
conclude that

\begin{align*}
\norm{\Gamma^+_{\beta,k}(\lambda)\psi}^2 \leq
\inp{\delta(H-\lambda)}_\psi  \text{ for all } \lambda \in I_0.
\end{align*}
In particular also  $\norm{\Gamma^+_{\beta}(\lambda)\psi}^2=\lim _{k\to
\infty}\norm{\Gamma^+_{\beta,k}(\lambda)\psi}^2  \leq
\inp{\delta(H-\lambda)}_\psi $, showing that  indeed $\Gamma^+_{\beta}(\lambda)\psi\in L^2(C_b)$.

We can now record,  using  Lemma <ref> and
(<ref>),  that for any $ s>1/2$

\begin{align*}
\Gamma^+_{\beta}(\lambda)\in \vL( L_{s}^2(\bX),L^2(C_b))
\text{
with a weakly continuous  dependence  of }  \lambda \in I_0,
\end{align*}
and in fact  that the  outgoing channel wave matrix

\begin{align*}
\Gamma^+_{\beta}(\lambda)^*\in \vL(L^2(C_b),  L_{-s}^2(\bX))
\text{
depends strongly continuously of }  \lambda \in I_0.
\end{align*}

By taking $k\to \infty$  in (<ref>) we conclude that  for any $\psi\in L_{s}^2(\bX)$,  $s>1/2$,

\begin{align*}
\parb{F_\beta
(W^+_\beta)^*\psi}(\lambda)=\Gamma_\beta^+(\lambda)\psi\quad\text{for a.e.
\lambda\in I_0.
\end{align*}

We can argue similarly for $W^-_\beta$, cf. (<ref>),  and
introduce the incoming  channel wave
matrix $\Gamma^-_{\beta}(\lambda)^*$, concluding by  the same
reasoning that

\begin{align*}
&\Gamma^-_{\beta}(\lambda)\in \vL( L_{s}^2(\bX),L^2(C_b))\text{
depends weakly continuously of }  \lambda \in I_0,\\
&\Gamma^-_{\beta}(\lambda)^*\in \vL(L^2(C_b),  L_{-s}^2(\bX))
\text{
depends strongly continuously of }  \lambda \in I_0,\\
&\forall \,\psi\in L_{s}^2(\bX), \, s>1/2:\quad \parb{F_\beta
(W^-_\beta)^*\psi}(\lambda)=\Gamma_\beta^-(\lambda)\psi\quad\text{for a.e.
\lambda\in I_0.
\end{align*}

We summarize as follows.
Let $\beta$ be a  given channel  $\beta
=(b,\lambda^\beta, u^\beta)$,     $f:I^\beta=(\lambda^\beta,\infty)\to \C$ be
bounded and
continuous, and let  and $s>1/2$. For any $\varphi\in
L^2_s(\mathbf  X_b)$

\begin{align}\label{eq:wav1}
(f1_{I_0})\paro{ k_\beta}\varphi=\int_{I_0} \,f(\lambda)
\Gamma^\pm_{\beta}(\lambda)^* \gamma_{b,0}(\lambda_\beta)\varphi
\,\d \lambda\in  L_{-s}^2(\bX),
\end{align}

More generally for any $\varphi
\in1_{I_0}(k_\beta) L^2(\bX_b)$

\begin{align}\label{eq:wav2}
W^\pm _\beta
f\paro{ k_\beta}\varphi=\int_{I_0} \,f(\lambda)
\Gamma^\pm_{\beta}(\lambda)^* \paro{F_\beta \varphi)(\lambda,
\cdot} \,\d \lambda \in L_{-s}^2(\bX),
\end{align}

In (<ref>) the integrand is a bounded and continuous
$L_{-s}^2(\bX)$-valued function. For   (<ref>) the integral
has  the weak interpretation of a    measurable
$L_{-s}^2(\bX)$-valued function.

For $f(\lambda)= \e^{-\i t\lambda}$,
$t\in\R$,  the formulas (<ref>) and (<ref>) represent
Schrödinger wave packets of energy-localized states outgoing to  or incoming  from
the channel $\beta$, cf. a discussion in Subsection <ref>.

§.§ Parseval identities and   construction of the scattering matrix

Let $Q=Q_0$ be given as in (<ref>) (for an arbitrary $\epsilon>0$).
By the orthogonality and completeness of channel wave operators we deduce the Parseval

\begin{align}\label{eq:parseval}
\begin{split}
&\int_\Delta\,\,\sum_{\lambda^\beta< \lambda_0}\,\norm{\Gamma^\pm_{\beta}(\lambda)Q_0^*\varphi}^2\,\d \lambda=
\int_\Delta\,\inp{\delta(H-\lambda)}_{Q_0^*\varphi }\,\d \lambda;\\&\quad\quad\quad\quad\text{for
all intervals }
\Delta\subseteq I_0 \mand \varphi\in \vH.
\end{split}
\end{align}
Moreover, arguing more or less as before, we can deduce
from (<ref>)  that

\begin{align}\label{eq:pointw1}
\lambda_0}\,\norm{\Gamma^\pm_{\beta}(\lambda)Q_0^*\varphi }^2\leq
\inp{\delta(H-\lambda)}_{Q_0^*\varphi } \quad\text{ for all } \lambda \in I_0,\\
\lambda_0}\,\norm{\Gamma^\pm_{\beta}(\lambda)Q_0^*\varphi }^2=
\inp{\delta(H-\lambda)}_{Q_0^*\varphi } \quad\text{  a.e.  in }
I_0,\text{  say for } \lambda \in I_0
\setminus \vN_0.   \label{eq:pointw2}
\end{align}

By first taking first $\varphi$ from any  dense countable
subset $\vC\subseteq \vH$ we then
obtain (<ref>) and (<ref>)
for all vectors  from  $\vC$
with the null set $\vN_0$ in   (<ref>) being independent of
$\varphi\in \vC$. Secondly we extend by continuity  and conclude that  (<ref>) holds with the same null
set $\vN_0$ for  all
$\varphi\in\vH$. In particular we obtain
that for some null subset $\vN_0\subseteq I_0$

\begin{align}\label{eq:Finalgood0}
&\forall \psi\in L^2_\infty:\,\,\sum_{\lambda^\beta<
\lambda_0}\,\norm{\Gamma^\pm_{\beta}(\lambda)\psi }^2\leq
\inp{\delta(H-\lambda)}_{\psi}\text{ for all }  \lambda
\in I_0,\\
&\forall \psi\in L^2_\infty:\,\,\sum_{\lambda^\beta<
\lambda_0}\,\norm{\Gamma^\pm_{\beta}(\lambda)\psi }^2=
\inp{\delta(H-\lambda)}_{\psi}\text{ for all }  \lambda
\in I_0\setminus \vN_0.  \label{eq:Finalgood}
\end{align}

We are lead to introduce the following concept.

A stationary scattering energy is  a point  $\lambda\in I_0$ for which

\begin{align}\label{eq:ScatEnergy}
\forall \psi\in  L^2_\infty:\,\, \sum_{\lambda^\beta<
\lambda_0}\,\norm{\Gamma^\pm_{\beta}(\lambda)\psi}^2=
\inp{\delta(H-\lambda)}_{\psi}.
\end{align}

We have shown that almost all energies in $ I_0$ are stationary scattering
energies. The property missing for possibly having this conclusion for all
energies in $ I_0$ is clearly the continuity in $\lambda$ of the sum to the left in (<ref>).

Recall that the above operator $Q_0$ is only one out of many
`$Q$-operators' from our procedure. More
precisely  for each
$k$ the expansions of $
T^\pm_{\beta}$ used in Sections <ref> and <ref> involved
several (but
finitely many)  `$Q$-operators' either on the above form
$Q_0$ or being one of those from  Lemma <ref>. So when varying $k$ we
obtain in total an infinite  family of `$Q$-operators',
say  denoted  by $\vQ$.

For any  $Q\in \vQ$ and $\varphi \in L^2_\infty$   we  record that
$\tilde f_2(H) Q^*\varphi \in  L^2_\infty$, cf. Remark
<ref> <ref>, and

\begin{align*}
\Gamma^\pm_{\beta}(\lambda)Q^*\varphi
=\Gamma^\pm_{\beta}(\lambda)\tilde  f_2(H)  Q^*\varphi =\lim_{k\to \infty}
\Gamma^\pm_{\beta,k}(\lambda)\tilde  f_2(H) Q^*\varphi \in L^2(C_b);\,\lambda\in I_0.
\end{align*}

However (more generally) for any finite sum $\psi=\sum_j Q_j^*\varphi_j$, where now
$\varphi_j$ is  arbitrary  from $\vH$, the quantity  $ \Gamma^\pm_{\beta,k}(\lambda)\psi=\Gamma^\pm_{\beta,k}(\lambda)\tilde
f_2(H) \psi$ is well-defined. Thanks to
(<ref>) it is  unambiguously given   as

\begin{align*}
\Gamma^\pm_{\beta,k}(\lambda)
\psi=\sum_j 1_{C_{b,k}}\Gamma^\pm_{\beta,k}(\lambda)
Q_j^*\varphi_j\in L^2(C_b),
\end{align*}
and as used  before
allowing us to compute

\begin{align*}
\Gamma^\pm_{\beta}(\lambda) \psi:&=\lim_{k\to \infty} \Gamma^\pm_{\beta,k}(\lambda) \psi\\&=\sum_j \lim_{k\to \infty} 1_{C_{b,k}}\Gamma^\pm_{\beta,k}(\lambda)
=\sum_j \lim_{k\to \infty} 1_{C_{b,k}}\Gamma^\pm_{\beta}(\lambda)
Q_j^*\varphi_j\\&=\sum_j \Gamma^\pm_{\beta}(\lambda)
Q_j^*\varphi_j \in L^2(C_b).
\end{align*}
We  used the following  analogue of (<ref>),

\begin{align}\label{eq:SE}
\norm{\Gamma^+_{\beta,k}(\lambda)Q_j^*\varphi_j}^2 \leq
\inp{\delta(H-\lambda)}_{f_1(H)Q_j^*\varphi_j}  \text{ for all } \lambda \in I_0.
\end{align}
This reasoning yields that

\begin{align*}
\forall Q\in \vQ:\,\Gamma^\pm_{\beta}(\lambda)Q^*\in\vL(
\vH,L^2(C_b)),
\end{align*}
in fact with a weakly continuous dependence of
$\lambda \in I_0$.

We are lead  to introduce the  subspace $\vV\subseteq \vH^{-2}$ of  vectors
$\psi$ as above, i.e.  vectors being  given as a finite
sum  of terms on  the form $Q^* \varphi$, where $Q\in \vQ$ and
$\varphi\in\vH$. Mimicking the proof of (<ref>)   we
conclude the following extension of (<ref>),

\begin{align}\label{eq:Finalgood02a}
\forall \psi\in \vV:\,\,\sum_{\lambda^\beta<
\lambda_0}\,\norm{\Gamma^\pm_{\beta}(\lambda)\psi }^2\leq
\inp{\delta(H-\lambda)}_{f_1(H)\psi}\text{ for all }  \lambda
\in I_0.
\end{align}
We are lead  to define

\begin{align*}
\Gamma^\pm(\lambda):\vV\to  \vG:=\Sigma^\oplus_{\lambda^\beta<
\lambda_0}L^2(C_b);\quad\Gamma^\pm(\lambda)=\parb{\Gamma^\pm_{\beta}(\lambda)}_{\lambda^\beta
\end{align*}
By (<ref>) and the above computation,
$\Gamma^\pm(\lambda)=\Gamma^\pm(\lambda)\tilde  f_2(H)$ are  well-defined linear maps.  Using the natural
Hilbert space structure on  $\vG$,
(<ref>) may be phrased as

\begin{align}\label{eq:Finalgood023}
\forall \psi\in \vV:\,\,\norm{\Gamma^\pm(\lambda)\psi }^2\leq
\inp{\delta(H-\lambda)}_{f_1(H)\psi}\text{ for all }  \lambda
\in I_0.
\end{align}

expression $\inp{\delta(H-\lambda)}_{f_1(H)\psi}$  appearing to the right in
(<ref>)   is continuous in $\lambda \in I_0$,
cf. Remark <ref> <ref>. The
quantities on the left-hand side might not have this property,
however we can  record  that $\Gamma^\pm(\cdot)\psi $
are  weakly continuous and note
the following version af (<ref>),

\begin{align}\label{eq:parseval3}
\forall \psi\in \vV,\forall \text{ intervals }
\Delta\subseteq I_0:\,\int_\Delta\,\norm{\Gamma^\pm(\lambda)\psi}^2\,\d \lambda=
\int_\Delta\,\inp{\delta(H-\lambda)}_{f_1(H)\psi}\,\d \lambda.
\end{align}

Since almost all energies in $ I_0$ are stationary scattering
energies the Parseval formulas (<ref>) may be seen as a
consequence of the following extension of (<ref>).

Let  $\lambda\in I_0$ be
any  stationary scattering energy. Then

\begin{align}
\forall \psi\in \vV:\,\,\norm{\Gamma^\pm(\lambda)\psi }^2=\sum_{\lambda^\beta<
\lambda_0}\,\norm{\Gamma^\pm_{\beta}(\lambda)\psi }^2=
\inp{\delta(H-\lambda)}_{f_1(H)\psi}.  \label{eq:Finalgood2}
\end{align}

Fix an arbitrary $\psi=\Sigma_j Q^*_j \varphi_j\in \vV$,  let
$\chi_m=\chi_-(r/m)$ for $m\in\N$ and then
$\psi_m=\Sigma_jQ^*_j \chi_m\varphi_j$. Since
$\chi_m\varphi_j\in  L^2_\infty$ and $\tilde  f_2(H) Q^*_j=\vO(r^0)$
(cf. Remark <ref> <ref>), also $\tilde  f_2(H) \psi_m\in
L^2_\infty$, and   for such  vector  we know  the identities (<ref>). By (<ref>) the sequences
$\Gamma^\pm(\lambda)\tilde  f_2(H) \psi_m =\Gamma^\pm(\lambda) \psi_m\to  \Gamma^\pm(\lambda)\psi$  in
$\vG$. Whence also $\norm{\Gamma^\pm(\lambda)\tilde  f_2(H) \psi_m}^2 \to
\norm{\Gamma^\pm(\lambda)\psi}^2$. Similarly $
\inp{\delta(H-\lambda)}_{f_1(H)\psi_m}\to
\inp{\delta(H-\lambda)}_{f_1(H)\psi}$. Since the identities in (<ref>)
hold  for all elements of two sequences we conclude  the limiting identities,
agreeing with (<ref>).

Let  $\lambda\in I_0$. Then the subspaces

\begin{align*}
\vG^\pm_\infty=\set{{\Gamma^\pm(\lambda)\psi}=\parb{\Gamma^\pm_{\beta}(\lambda)\psi}_{{\lambda^\beta<
\lambda_0}}\in
\vG\,|\,\psi   \in L^2_\infty}\text{ are  dense in  }\vG.
\end{align*}

It suffices to prove  density of the subspaces $\vG^\pm_\vV=\set{{\Gamma^\pm(\lambda)\psi}\in
\vG\,|\,\psi   \in \vV}$, cf. the proof of Lemma <ref>.
Let us only
consider the `minus case'.  For any `open channel'  $\alpha$,  $k\in \N$ and $g_a\in L^2(C_a)$  we consider

\begin{align}\label{eq:gPsi}
\psi_{\alpha}=\psi_{\alpha,k,g_a}=T^- _{\alpha,k}
J_\alpha{\gamma_a^- (\lambda_\alpha)}^*1_{C_{a,k}}(\hat \xi_a)g_a.
\end{align}

We can write $ \psi_{\alpha}$ as a finite sum of vectors on
the form $Q^*\varphi$ where $Q\in \vQ$ and $\varphi\in\vH$. Whence
$\psi_{\alpha}\in \vV$ and
$\Gamma^-_{\beta}(\lambda)\psi_{\alpha}$ is a
well-defined vector in $ L^2(C_b)$. We compute for any $l\in \N$
the vector
$1_{C_{b,l}}(\hat \xi_b)\Gamma^-_{\beta}(\lambda)\psi_{\alpha}$
using  Remark <ref>, concluding
that $1_{C_{b,l}}(\hat \xi_b)\Gamma^-_{\beta}(\lambda)\psi_{\alpha}=0$ if $\beta\neq
\alpha$, while for $\beta=
\alpha$

\begin{align*}
1_{C_{b,l}}(\hat \xi_b)\Gamma^-_{\beta}(\lambda)\psi_{\alpha}=1_{C_{a,l}}(\hat \xi_a)\Gamma^-_{\alpha,l}(\lambda)\psi_{\alpha}= -
\tfrac {2\lambda_\alpha}\pi 1_{C_{a,l}\cap {C_{a,k}}}g_a.
\end{align*}
By taking $l\to \infty$ it follows that

\begin{align}\label{eq:fundCOMP}
\vG^-_\vV\ni \Gamma^-(\lambda)\psi_{\alpha}=( \delta_{ \beta \alpha}
\tfrac {-2\lambda_\alpha}\pi 1_{{C_{a,k}}}g_a)_{{\lambda^\beta<
\lambda_0}}.
\end{align}
By varying over $\alpha$ (with $\lambda^\alpha<\lambda_0$), $k\in \N$ and $g_a\in L^2(C_a)$
corresponding vectors to the right in (<ref>) span  a
dense  subset  of $\vG$.

Let  $\lambda\in I_0$ be
any  stationary scattering energy. Then we let  $U(\lambda)\in \vL(\vG)$ be
the uniquely defined unitary
operator $U(\lambda)$ on $\vG$ such that

\begin{align*}
\quad \psi   \in \vV.
\end{align*}
(Note that Lemma <ref> is used.)

Let  $\lambda\in I_0$. The  scattering matrix  $S(\lambda)$ is  the unique bounded operator  on $\vG$  with each $\beta\alpha$-entry
given by  $S_{\beta\alpha}(\lambda)$ as specified in Theorem
<ref>. (Note that $\norm {S(\lambda)} \leq 1$
by  the orthogonality of channels and the weak continuity.)

Let $\lambda\in
I_0$ be   any  stationary scattering energy. Then  $S(\lambda)$ is unitary.
the $\vL(\vG)$-valued function $S(\cdot )$ is strongly
continuous at $\lambda$, for
any $\psi \in \vV$ the $\vG$-valued functions
\lambda_0}}$ are  norm-continuous
at $\lambda$, and

\begin{align}\label{eq:scaSS}
S(\lambda )\Gamma^-(\lambda)\psi=\Gamma^+(\lambda)\psi.
\end{align}

It suffices to check that
$S_{\beta\alpha}(\lambda)=U_{\beta\alpha}(\lambda)$. By
(<ref>),  (<ref>),
(<ref>), (<ref>)  and (<ref>)

\begin{align*}
&1_{C_{b,l}}(\hat \xi_b)U_{\beta\alpha}(\lambda)1_{{C_{a,k}}}g_a\\&=-\tfrac {\pi}{2\lambda_\alpha}\Gamma^+_{\beta,l}(\lambda)T^- _{\alpha,k}
J_\alpha{\gamma_a^- (\lambda_\alpha)}^*1_{C_{a,k}}(\hat\xi_a)g_a\\&
=-\tfrac {\pi}{2\lambda_\alpha}\tfrac \pi {2\lambda_\beta} 1_{C_{b,l}}(\hat \xi_b){\gamma_b^+(\lambda_\beta)}J^*_\beta
\parb{T^+_{\beta,l}}^* \delta(H-\lambda)T^- _{\alpha,k}
=\tfrac{(2\pi\i)^2}{16\lambda_\beta \lambda_\alpha} 1_{C_{b,l}}(\hat \xi_b){\gamma_b^+(\lambda_\beta)}J^*_\beta
\parb{T^+_{\beta,l}}^* \delta(H-\lambda)T^- _{\alpha,k}
=\parb{16\lambda_\beta \lambda_\alpha}^{-1} 1_{C_{b,l}}(\hat \xi_b) \widetilde
S_{\beta\alpha}(\lambda)1_{C_{a,k}}(\hat\xi_a)g_a\\& = 1_{C_{b,l}}(\hat \xi_b) S_{\beta\alpha}(\lambda)1_{C_{a,k}}(\hat\xi_a)g_a.
\end{align*}
We take $k,l\to \infty$ and conclude that indeed
$U_{\beta\alpha}(\lambda)g_a=S_{\beta\alpha}(\lambda)g_a$ for any  $g_a\in L^2(C_a)$.

The free energy $K:=\diag(k_\alpha)$  is diagonalised on the neighbourhood
$I_0\ni \lambda_0$ by

\begin{align*}
F_{I_0}: \sum_{\lambda^\alpha < \lambda_0} &\oplus 1_{I_0}(k_\alpha) L^2(\bX_a)\to  \sum_{\lambda^\alpha < \lambda_0} \oplus L^2(I_0,L^2(C_a))=\int_{I_0}^\oplus \vG \,\d \lambda;\\&F_{I_0}=\diag(F_\alpha).
\end{align*}
Recall from Theorem <ref> the
notation $S_{\beta\alpha}=\parb{W_{\beta}^+}^*W_{\alpha}^-$.
The scattering matrix  considered
as a map

\begin{align*}
I_0\ni \lambda\to S(\lambda)\in\vL \paro{\vG}\text{ is
weakly continuous.}
\end{align*}
Within the class of such maps it is uniquely determined by the identity

\begin{align}
F_{I_0}SF^{-1}_{I_0}&=F_{I_0}\paro{S_{\beta\alpha}}_{\lambda^\beta,\,\lambda^\alpha < \lambda_0}\,F^{-1}_{I_0}=\int^\oplus_{I_0}
S(\lambda)\,\d \lambda.
\end{align}
For almost all energies in $I_0$ the scattering matrix
is unitary and strongly continuous.

It is stated in  <cit.> that for short-range
systems the  scattering
matrix is strongly continuous away from thresholds and eigenvalues. The proof
appears  incorrect  to the author (however the   assertion  of weak continuity
is correctly derived).

§.§ Besov space setting and  minimum  generalized eigenfunctions

In Lemma <ref> we considered $\Gamma^\pm(\lambda)=\parb{\Gamma^\pm_{\beta}(\lambda)}_{\lambda^\beta
<\lambda}$  as mappings
$\Gamma^\pm(\lambda):\vV\to \vG$;
$\lambda\in I_0$. The extended space $\vV\supseteq
L^2_\infty$ was convenient  for our constructions, however there is
a different extension which  appears 'more natural',  namely the Besov
space $\vB$ in  which
$L^2_\infty$ is densely embedded. Using
(<ref>), (<ref>) and extension by continuity it follows that
$\Gamma^\pm(\lambda)\in \vL(\vB,\vG)$ with a uniform bound in
$\lambda\in I_0$.
Clearly $\Gamma^\pm(\cdot)\in\vL(\vB,\vG)$ are
strongly continuous at  any stationary scattering energy $\lambda\in I_0$ in
which case (<ref>) also holds for $\psi\in \vB$. Due to
Theorem  <ref> (stated below) the vector
$\Gamma^-(\lambda)\psi$ with
$\psi\in \vB$  (appearing
the left in  (<ref>)) in this case is  the generic form of vectors in $\vG$.

Introducing the space of  minimum  generalized eigenfunctions,

\begin{align*}
\vE_\lambda=\set{\phi\in \vB^*|\, (H-\lambda)\phi=0},
\end{align*}

we note  that $\delta(H-\lambda)\psi\in\vE_\lambda$ for
$\psi\in \vB$ (or possibly for $\psi\in f_1(H)\vV$). However if $\lambda$ is a
stationary scattering energy, in fact  any
$\phi\in\vE_\lambda$ has this form. This is a
of  the following more general
result. We remark that  there are   analogous assertions   for various one-body type Hamiltonians in the literature, see for
example [15].
Let $\lambda\in
I_0$ be   any  stationary scattering energy.
Then the  operators $\Gamma^\pm(\lambda): \vB\to \vG$,
$\Gamma^\pm(\lambda)^*: \vG\to \vE_\lambda$   and $\delta(H-\lambda)
: \vB \to \vE_\lambda$  all map
onto. Moreover the wave matrices
$\Gamma^\pm(\lambda)^*: \vG\to \vE_\lambda(\subseteq \vB^*)$ are bi-continuous linear isomorphisms.

The ranges of $\Gamma^\pm(\lambda)$
are dense due to Lemma
<ref>, and    clearly $\vE_\lambda$ is closed in
Thanks to   the open
mapping and closed range theorems  (see for example see [25])
it then suffices
to show that  $\Gamma^\pm(\lambda)^*$ map onto $\vE_\lambda$.  (Here
and below we use the identities
$\Gamma^\pm(\lambda)^*\Gamma^\pm(\lambda)=\delta(H-\lambda)$ as
quadratic forms on $\vB$.)

For convenience we only show that  $\Gamma^+(\lambda)^*$ maps
$\vE_\lambda$.  Let $B$ is given by (<ref>) (possibly rescaled),
and let  $h$ be the   function $h(e)=e\tilde f_1(e)$  ($h(e)$ is
denoted by $g(\lambda)$  in
Section <ref>) and note that  $h(H)-\lambda$ annihilates the spaces $\ran \Gamma^+(\lambda)^*\subseteq\vE_\lambda$. Due to <cit.>  there exists  $\epsilon>0$ such that

\begin{align}\label{eq:MicroB}
\chi_-(\pm B/\epsilon) R(\lambda\pm\i0)\psi\in \vB_0^*
\text{ for any }\psi\in \vB.
\end{align}
(The fact  that $\chi_-(\pm B/\epsilon)=\vO(r^0)$ implies that
$\chi_-(\pm B/\epsilon)\in \vL(\vB)$, cf.  <cit.>.)
For  any given $\phi\in \vE_\lambda$ we introduce for $\epsilon>0$
taken small

\begin{align*}
\phi_+=\chi_+(
B/\epsilon)\phi,\quad \phi_-=\phi-\phi_+, \quad g_\rho=\Gamma^+(\lambda)\chi_\rho \parb{h(H)-\lambda}\phi_+;
\end{align*}

$\chi_\rho=\chi_-(r/\rho)$, $\rho>1$.  By commuting the factor $h(H)-\lambda$ through the factor $\chi_\rho$
we deduce that $\sup_{\rho>1}\,\norm
{g_\rho}<\infty$ (note that only the  commutator contributes).
Let $g_\infty=\wlim_{n\to \infty}g_{\rho_n}\in \vG$, the limit
taken  along a  suitable
sequence $\rho_n\to \infty$, and let $\breve
\phi=\Gamma^+(\lambda)^*g_\infty$. We can  now mimic the proof
of <cit.>,  using the identity
$\Gamma^+(\lambda)^*\Gamma^+(\lambda)=\delta(H-\lambda)$, the
decomposition $\phi=\phi_++\phi_-$  and
the bounds (<ref>) in combination with (<ref>),  and conclude that $\phi=\breve
\phi$. In particular $\phi\in \ran \Gamma^+(\lambda)^*$ as wanted.

The following functions $w^{\pm}_{\lambda,\beta} [g]$ are identical
with the functions $v^{\pm}_{\lambda,\beta} [g]$ of Subsection
<ref> up to  trivial factors.

Let $\beta=(b,\lambda^\beta, u^\beta)$ be any
channel with $\lambda^\beta < \lambda_0$. For $\epsilon>0$ taken
small enough the following bounds hold for any $\lambda\in
I_0$ and any $g\in L^2(C_b)$.

\begin{align*}
\chi^2_-(\mp B/\epsilon)\Gamma^\pm_{\beta}(\lambda)^*g-J_\beta
w^{\pm}_{\lambda,\beta} [g]\in \vB_0^*,
\end{align*}

\begin{align*}
w^{\pm}_{\lambda,\beta} [g]( x_b)&= d_\beta^\pm \lambda_\beta^{-1/4} \chi_+(\abs{x_b}) \abs{x_b}^{(1-n_b)/2}
\e^{\pm \i
K_b(x_b,\lambda_\beta)}g(\hat x_b);\\&\quad\quad\quad\quad d_\beta^\pm=\mp\tfrac \i 2 \e^{\mp
\i(n_b-3)/4}\pi^{-1/2}.
\end{align*}

By (<ref>) and its `minus-analogue'

\begin{align}\label{eq:AppExa2}
\Gamma^\pm_{\beta,k}(\lambda)^*=\pm \tfrac \pi {2\lambda_\beta} f_1(H){\delta(H-\lambda) T^\pm_{\beta,k} J_\beta
\gamma_{b}^\pm(\lambda_\beta)^*}1_{C_{b,k}}.
\end{align}
The identity (<ref>)  may  formally be seen as
a  replacement
of the factor
$\pm 2\pi\delta(H-\lambda)
T^\pm_{\beta,k}$ from (<ref>)  by   ${\Phi_{\beta,k}^\pm+\i
R(\lambda\mp\i 0)T^\pm_{\beta,k}} $, and from
the derivation of
(<ref>) we see that this is legitimate considering the image
space of the operators as being
$L^2_{-1}(\bf  X)$.
In particular there exist  in $\vL(L^2(C_b),L^2_{-1}(\bf  X))$

\begin{align*}
f_1(H)R(\lambda\mp\i 0)T^\pm_{\beta,k}J_\beta
\gamma_{b}^\pm(\lambda_\beta)^*1_{C_{b,k}}:=\wlim_{\epsilon\to 0_+}f_1(H)R(\lambda\mp\i \epsilon)T^\pm_{\beta,k}J_\beta
\gamma_{b}^\pm(\lambda_\beta)^*1_{C_{b,k}} .
\end{align*}

By  combining  (<ref>) and Lemma
<ref> we obtain  that

\begin{align*}
\sup_{\epsilon>0}\,\norm{R(\lambda\mp\i \epsilon)T^\pm_{\beta,k}J_\beta
\gamma_{b}^\pm(\lambda_\beta)^*1_{C_{b,k}}}_{
\vL(L^2(C_b),\vB^*)}<\infty.
\end{align*}

Whence  in fact

\begin{align*}
S^\mp:=f_1(H)R(\lambda\mp\i 0)T^\pm_{\beta,k}J_\beta
\gamma_{b}^\pm(\lambda_\beta)^*1_{C_{b,k}}\in\vL(L^2(C_b),\vB^*).
\end{align*}

Next we expand  for each sign $T^\pm_{\beta,k}$ in terms of the $Q$-operators and
consider the approximation of $ S^\mp$ given by replacing   $R(\lambda\mp\i 0)f_1(H) Q^*_j$  by
$R(\lambda\mp\i 0)f_1(H) Q^*_j \chi_\rho$, say denoted by
$S^\mp_\rho$ (as in the proof of Theorem
<ref>, here
$\chi_\rho=\chi_-(r/\rho)$ with   $\rho$ considered large).  Now $S^\mp_\rho \to S^\mp$ strongly in
$\vL(L^2(C_b),\vB^*)$, and by (<ref>)
B/\epsilon)S^\mp_\rho\in\vL(L^2(C_b),\vB_0^*)$. Whence also $\chi^2_-(\mp

By density and continuity   we are left with showing

\begin{align*}
{ \tfrac 1 {4\lambda_\beta}
\chi^2_-(\mp
B/\epsilon)f_1(H)\Phi_{\beta,k}^\pm J_\beta
\gamma_{b}^\pm(\lambda_\beta)^*1_{C_{b,k}}g_k-J_\beta
w^{\pm}_{\lambda,\beta} [g_k]}\in \vB_0^*
\end{align*}
for any $g_k=1_{C_{b,k}}g\in  C^\infty
(C_b)$, $k\in \N$. In turn, thanks to the form of $\Phi_{\beta,k}^\pm
$ (and commutation), it remains to show that

\begin{align*}
{ \tfrac 1 {4\lambda_\beta}
f_1(H)\Phi_{\beta,k}^\pm J_\beta
\gamma_{b}^\pm(\lambda_\beta)^*1_{C_{b,k}}g_k-J_\beta
w^{\pm}_{\lambda,\beta} [g_k]}\in \vB_0^*
\end{align*}
for any such  $g_k$.

By using the  analysis of Steps IV and V  in
the proof of Lemma <ref> (not to be repeated
here) the first term
exactly as the subtracted  second term modulo a term in $\vB_0^*$.

We can now give a `geometric interpretation' of the operator
$\Gamma^-_{\alpha}(\lambda)^*$ needed for a wave packet description of
an incoming  $\alpha$-channel  experiment, energy-localized in
$I_0$, cf. a discussion in Subsection <ref>. More precisely we  characterize for any stationary scattering
energy  the space  of
generalized eigenfunctions  in  the range of
$\Gamma^-_{\alpha}(\lambda)^*$ in terms of asymptotics expressed by
the quasi-modes $J_\beta
w^{\pm}_{\lambda,\beta} [\cdot]$ of Lemma <ref>. The result
may also be vieved as a  characterization of the    incoming
$\alpha$-channel  part of the
scattering matrix.
We equip the space $\vB^*/\vB_0^*$ with the quotient-norm, thus
making it  a Banach space.
Let $\lambda\in
I_0$ be   any  stationary scattering energy and $\alpha=(a,\lambda^\alpha, u^\alpha)$ be any
channel with $\lambda^\alpha < \lambda_0$. Then the following
existence and uniqueness result holds for any  $
g\in L^2(C_a)$.

*  Let   $
u=\Gamma^-_{\alpha}(\lambda)^* g$,  and let
$(g_\beta)_{\lambda^\beta<\lambda_0}\in \vG$ be   given by
$g_\beta=S_{\beta\alpha}(\lambda) g$. Then,  as an
identity in  $\vB^*/\vB_0^*$,

\begin{align}\label{eq:as}
w^{-}_{\lambda,\alpha} [ g]+\sum_{\lambda^\beta<\lambda_0} J_\beta
w^{+}_{\lambda,\beta} [g_\beta].
\end{align}

*  Conversely, if  (<ref>)
is fulfilled for some  $ u\in\vE_\lambda$
and some $(g_\beta)_{\lambda^\beta<\lambda_0}\in \vG$,
then  $ u=\Gamma^-_{\alpha}(\lambda)^* g$ and
$g_\beta=S_{\beta\alpha}(\lambda) g$ for all

I. We note that with $F_\rho:=F(\set{x\in X|\,\abs{x}<\rho})$, $\rho>1$, the expression

\begin{align*}
\norm{u}_{\rm quo}=\limsup_{\rho\to \infty}\,\rho^{-1/2}\norm{F_\rho u}
\end{align*}
is a norm on $\vB^*/\vB_0^*$ equivalent with the
quotient-norm. Using this norm the Cauchy criterion assures that for
any  $(g_\beta)_{\lambda^\beta<\lambda_0}\in \vG$ the right-hand
side of  (<ref>) is well-defined. The cross-terms do not
contribute for the following reason: For given open different channels
$\beta=(b,\lambda^\beta, u^\beta)$ and $\beta'=(b',\lambda^{\beta'},
u^{\beta'})$ with $b=b'$,    Fubini's theorem and the fact that
$\inp{u^\beta, u^{\beta'}}_{\vH^b}=0$ clearly implies

\begin{align}\label{eq:ort}
\inp{F_\rho J_\beta
w^{+}_{\lambda,\beta} [g_\beta],F_\rho J_{\beta'}
w^{+}_{\lambda,\beta'} [g_{\beta'}]}_{\vH}=\int_{\bX} \,\overline{F_\rho u^\beta
w^{+}_{\lambda,\beta} [g_\beta]}\, F_\rho u^{\beta'}
w^{+}_{\lambda,\beta'} [g_{\beta'}]\,\d x=0.
\end{align}

If $b\neq b'$ we let $c\in \vA$ be given by $
\bX^c=\bX^b+\bX^{b'}$ and $n_c=\dim \bX_c$, and  note that in this
$n_c\leq (n_b+n_{b'}-1)/2$. By approximation, and by using  Fubini's
theorem and ,  we can assume that the functions
$g_\beta,g_{\beta'}, u^\beta$ and $u^{\beta'}$ are all  bounded with
compact support. With this assumption we claim that the  integral in (<ref>) is $\vO(\rho^{1/2})$ (and hence in particular
$o(\rho)$  as wanted). On the support of the integrand,
$\abs{x^c}\leq C_1(\abs{x^b}+\abs{x^{b'}}) \leq C_2$ and

\begin{align*}
\abs{x_b}^{(1-n_b)/2}\abs{x_{b'}}^{(1-n_{b'})/2}\leq C_3
\abs{x}^{1-(n_b+n_{b'})/2} \leq C_4 \abs{x}^{(1-n_c)} \abs{x}^{-1/2}.
\end{align*}
If $n_c=0$ it follows that $\abs{x}\leq C_2$,  and
consequently the  integral is  $
\vO(\rho^{0})=\vO(\rho^{1/2})$. If  $n_c\geq 1$ we estimate  $\abs{x}^{(1-n_c)}
\abs{x}^{-1/2}\leq \abs{x_c}^{(1-n_c)} \abs{x_c}^{-1/2}$ and use
spherical coordinates in $\bX_c$ and the fact that $\abs{x_c}\leq
\abs{x}< \rho$, indeed yielding the desired bound  $\vO(\rho^{1/2})$.

We conclude that

\begin{align}\label{eq:pars}
\norm[\big]{u-J_\alpha
w^{-}_{\lambda,\alpha} [ g]}_{\rm
quo}^2=\sum_{\lambda^\beta<\lambda_0} \parb{4\pi
\lambda_\beta^{1/2}}^{-1}\,\norm{g_\beta}^2.
\end{align}

II. For <ref> we decompose $u=\Gamma^-_{\alpha}(\lambda)^* g$ as

\begin{align*}
u=\chi^2_-( B/\epsilon)u+\chi^2_+( B/\epsilon)u.
\end{align*}
The first term corresponds to the first
term to the  right in  (<ref>) thanks to  Lemma
<ref>. For the second  term we substitute,  cf. (<ref>) and (<ref>),

\begin{align*}
\Gamma^-_{\alpha}(\lambda)^*g=\sum_{\lambda^\beta<\lambda}
\Gamma^+_{\beta}(\lambda)^*g_\beta;\quad g_\beta= S_{\beta\alpha}(\lambda)g.
\end{align*}
The series converges in $\vB^*$. By continuity we can
take the factor
$\chi^2_+( B/\epsilon)$ inside the summation. Then we  obtain
<ref> by  applying  Lemma
<ref> to the  terms of  the (convergent)  series $\sum_{\lambda^\beta<\lambda}
\chi^2_+( B/\epsilon)\Gamma^+_{\beta}(\lambda)^*g_\beta$.

III.  For <ref> we note that
$ u':=u-\Gamma^-_{\alpha}(\lambda)^* g$ is in $\vE_\lambda$ and

\begin{align*}
\chi_-( B/\epsilon) u'=0 \text{ in }\vB^*/\vB_0^*.
\end{align*}

Here we first use (<ref>) to $\Gamma^-_{\alpha}(\lambda)^* g $ to obtain a similar representation of
$u'\in \vB^*/\vB_0^*$, then  we multiply by  $\chi_-( B/\epsilon)$ using
the fact
this operator  is continuous in $\vB^*/\vB_0^*$  and finally we use Appendix
<ref> to the terms of the resulting series. (Note that
(<ref>) for any $g\in L^2(C_a)$  follows by approximation
since we know the assertion for $ g\in
C_\c^\infty (C_a')$.)  This means that $\chi_-( B/\epsilon) u'\in
\vB_0^*$ and then in turn,   thanks to <cit.>, that $ u'=0$. Finally
(<ref>) applied to the mentioned representation of  $u'$
implies that also $g_\beta-S_{\beta\alpha}(\lambda) g=0$, as wanted.

§ COMPLETION OF THE PROOF  OF LEMMA

We complete the proof of Lemma <ref>
<ref>.  For convenience we only show that $\check
P^+_{\lambda,\alpha} [g]=0$. By  <cit.> it suffices to show that for  some (small)

\begin{align}\label{eq:bminus}
\chi_-(B/\epsilon) u^\alpha\otimes v^{+}_{\lambda,\alpha}
[g]\in \vB_0^*,
\end{align}
cf. (<ref>). Our proof of (<ref>)
does not rely on (<ref>).
We recall that

\begin{align*}
v^{+}_{\lambda,\alpha} [g]( x_a)= \chi_+(\abs{x_a})\abs{x_a}^{(1-n_a)/2}
\e^{ \i
K_a(x_a,\lambda_\alpha)}g(\hat x_a) ;\quad g\in
C_\c^\infty (C_a').
\end{align*}

Depending on the support of $g$ we can pick $\xi^+_a$ as in
(<ref>) such that

\begin{align*}
\parb{1-\xi^+_a}u^\alpha\otimes v^{+}_{\lambda,\alpha}
[g]\in \vB_0^*.
\end{align*}
Let $\brH_a$ be given by (<ref>) and $E<\min\sigma(\brH_a)$. By commutation, cf. Lemma <ref>, it suffices
to find $\epsilon>0$ such that

\begin{align}\label{eq:goal}
\begin{split}
&\rho^{-1}\norm{\varphi_\rho}^2\to 0\text{ for }\rho\to \infty;\\
&\varphi_\rho:=\chi_-(B/\epsilon) T_\rho u^\alpha\otimes
v^{+}_{\lambda,\alpha},\quad T_\rho=\chi_-(r/\rho)\xi^+_a(\brH_a-E)^{-1}.
\end{split}
\end{align}
Note at this point that

\begin{align}\label{eq:e-lam}
\begin{split}
v^{+}_{\lambda,\alpha}}\in \vB_0^*,\\
v^{+}_{\lambda,\alpha}}\in \vB_0^*,
\end{split}
\end{align}
cf. (<ref>). As we will below the introduced factor
$(\brH_a-E)^{-1}$ in $T_\rho$ facilitates  commutation in the following estimation:

We estimate (with $\inp{T}_\varphi:=\inp{\varphi,T\varphi}$),
commute repeatedly (using Lemma <ref>), apply
(<ref>) and (in the last step) (<ref>),

\begin{align*}
&\leq-\rho^{-1}\inp{B/\epsilon}_{\varphi_\rho}\\&=-(\rho \epsilon)^{-1}\inp{T^*_\rho
B T_\rho}_{\chi_-(B) \parb{u^\alpha\otimes
\\&=-(\rho \epsilon)^{-1}\Re\inp{T^*_\rho
r^{-1}\parb{2{x_a}\cdot p_a+{\mathop{\mathrm{grad}} (r^a)^2}\cdot p^a} T_\rho}_{\chi_-(B) \parb{u^\alpha\otimes
\\&=-(\rho \epsilon)^{-1}\inp{T^*_\rho
r^{-1}\parb{2\sqrt{\lambda_\alpha}\abs{x_a}} T_\rho}_{\chi_-(B) \parb{u^\alpha\otimes
\\&=-(\rho
\epsilon)^{-1}{2\sqrt{\lambda_\alpha}\inp[\big]{\inp{x}/r}}_{\varphi_\rho}+o(\rho^{0})
\\&=-(\rho \epsilon)^{-1}{2\sqrt\lambda_\alpha}\norm{\varphi_\rho}^2+o(\rho^{0}).
\end{align*}

Note  for the fourth step that we need to commute $r^{-1}{x_a}\cdot
p_a$ through a factor $\chi_-(B) $  (before we can replace $r^{-1}{x_a}\cdot
p_a\approx \sqrt\lambda_\alpha r^{-1}\abs{x_a}$ and commute back). After a
little commutation this may in turn be accomplished by showing

\begin{align*}
\comm[\big]{\brH_a-E)^{-1}r^{-1}{x_a}\cdot p_a(\brH_a-E)^{-1},\chi_-(B) }=\vO(r^{-1})
\end{align*}
in the sense of (<ref>). However  with the
two factors of  resolvents in place the result follows from the Helffer–Sjöstrand
formula and  Lemma <ref>, cf. Remark

Clearly the above estimation yields (<ref>) for any  positive
$\epsilon<{\sqrt\lambda_\alpha}$ (by a subtraction).

§ PROOF OF (<REF>)
(<REF>) AND (<REF>)

For convenience we consider only the assertions for $t\to
+\infty$. Our procedure relies on standard stationary phase analysis
on which we omit  details, see for
example [10, 12] for  elaborated accounts.

(<ref>):   We need to show that

\begin{align}\label{eq:simp1}
\begin{split}
&\forall \varphi \in L^2(\mathbf X_a):\quad \lim_{t\to +\infty}\norm{\parb{I-A_1 A_2^aA_3^aA_2^aA_1}\parb{u^\alpha\otimes  \varphi(t)}}=0;
\\&\quad \quad\quad\quad\varphi(t)=\e^{-\i S_a (p_a,t)}f_2(k_\alpha)\varphi.
\end{split}
\end{align}
We can assume that $\varphi$ is smooth in momentum
space. Then also $f_2(k_\alpha)\varphi$ is  smooth  in momentum space,
there it is
compactly supported
away from  origo.  Recalling  that $S_a (p_a,t)= t
\xi_a^2+\vO(t^{1-\mu})$ the stationary phase
method then yields the effective localization $\abs{x_a}\geq \epsilon'
t$ for some (computable) $\epsilon'>0$. On the other hand $x^a$ is a localized
variable due to the presence of the factor $u^\alpha$. This means more
precisely that  we may  freely insert to the left of the tensor product
in (<ref>) the   factor $\chi_-\parb{{
r^{-\delta'}}{\abs{x^a}}}$ for any  $\delta'>0$. We do this for a $\delta'<\delta$.

Noting  that we can also freely insert the  factor $\chi_+\parb{{\Re{\parb{(x_a/r)\cdot
p_a}}}/(4\epsilon_0)}$ to the left of the tensor product in
(<ref>)  (thanks to stationary phase analysis),  it follows essentially
from (<ref>) that $A_1^2=A_{1+}^2$
can be replaced by $I$. Similarly the factors $A_2^a$ and
$A_3^a=A_{3+}^a$ can one by one be replaced by
$I$. This is
due to the fact that
in $\set{\abs{x^a}<c r^\delta}$,
with $c>0$ given in  the property <ref> of Section <ref> (this is for $r$ replaced by $r^a$),
$r^a_\delta=r^\delta r^a(0)$ and $B_{\delta,\rho_1}^a=r^{\rho_1/2}B_\delta^a
r^{\rho_1/2}=0$. Clearly we
can then remove   $A_2^a$, and the
justification   of  removing $A_3^a$  is  provided by  (<ref>).  This completes the proof of (<ref>).

(<ref>) and (<ref>):
Due to
(<ref>) it suffices to show (recalling the notation
(<ref>))  that for all $\varphi  \in L^2(\mathbf X_a)$
with $\hat \varphi\in C^\infty_\c(\mathbf X'_a):$

\begin{align}
\label{eq:simp1bcde}
\begin{split}
\varphi(t)}-u^\alpha\otimes  \parb{{m_\alpha^+}\varphi(t)}}}=0,\\
+\infty}\norm{\parb{f_1(H)-f_1(\brH_a)}\parb{u^\alpha\otimes  \parb{\parb{m_\alpha^+}^2\varphi(t)}}}=0;
\\&\quad \quad\quad\quad\varphi(t)=\e^{-\i S_a (p_a,t)}f_2(k_\alpha)\varphi.
\end{split}
\end{align}

The first assertion of (<ref>) follows by combining
the properties <ref>
and <ref> from Section <ref> with arguments from our
proof  of  (<ref>) (including  in particular stationary
phase analysis). The second part follows easily from stationary
phase analysis.

§ PROOF  OF (<REF>) AND FORMULAS FOR CHANNEL WAVE MATRICES
derivation of the formula (<ref>) for $\widetilde
S_{\beta\alpha}(\lambda)$, interpretated correctly in
Subsection <ref>,  will be given using
smoothness   bounds from Section <ref>, arguments from Subsections <ref> and <ref>  and  finally bounds from [1]. We follow essentially the scheme of <cit.> (for a similar issue, see    for example the proof of <cit.>).

Formally (<ref>)  leads with <cit.> to the formula

\begin{align}\label{eq:SreP4}
\begin{split}
\widetilde
S_{\beta\alpha}(\lambda)&= -2\pi f^2_2(\lambda){\gamma_b^+(\lambda_\beta)}J^*_\beta\parb{\Phi_\beta^+}^*\parb{T^-_\alpha
-\i(T^+_\beta)^* R(\lambda+\i 0)T^-_\alpha}
(2\pi \i)^2f^2_2(\lambda){\gamma_b^+(\lambda_\beta)}J^*_\beta
\parb{T^+_\beta}^* \delta(H-\lambda)T^-_\alpha
\end{split}
\end{align}
Note that we have not given an independent justification
the above middle  term (with the factor $R(\lambda+\i 0)$), including its well-definedness. However
this will not be
needed, and the stated expression to the far right   does indeed have a clean
interpretation, cf. Remark <ref>
<ref> and Subsections <ref> and <ref>.

Let for $\epsilon>0$ and $\lambda\in \Lambda$

\begin{align*}
\delta_{\epsilon,\beta}(\lambda)= (2\i\pi)^{-1}\parb{(k_\beta -\lambda-\i \epsilon)^{-1}-(k_\beta -\lambda+\i \epsilon)^{-1}}.
\end{align*}
The outset for our analysis is the following
two formulas which may be derived exactly as in <cit.>. In the first formula $g$ is any complex continuous function
on $\R$ vanishing at infinity.

\begin{align}
\begin{split}
\label{eq:W}
&\widetilde W^+_\beta (g1_{\Lambda})(k_\beta)\varphi\\
&\quad=\lim_{\epsilon\to 0}\int_{\Lambda}\,g(\lambda)f_1(H)\parb{\Phi_\beta^+
+\i R(\lambda-\i \epsilon)T^+_\beta}J_\beta \breve w^+_b f_2(k_\beta)\delta_{\epsilon,\beta}(\lambda)
\varphi \,\d \lambda, \end{split}\\
\begin{split}
S_{\beta\alpha}\varphi_a}\\&\quad=-2\pi \lim_{\epsilon\to
\widetilde W^+_\beta }^*T^-_\alpha J_\alpha \breve w^-_a f_2(k_\alpha)\delta_{\epsilon,\alpha}(\lambda')
\varphi_a} \,\d \lambda'.\label{eq:S}
\end{split}
\end{align}
Note that (<ref>) is based on the fact that

\begin{align*}
\slim_{t\to +\infty}\e^{\i tH}\tilde{J}_\alpha^-\e^{-\i
\end{align*}
which in  turn is a consequence of (<ref>) and stationary phase analysis, cf. Appendix

Although (<ref>) is valid for any $\varphi\in L^2(\mathbf
X_b)$ we take below
$\varphi=\varphi_b\in L^2_s(\mathbf  X_b)$ for  $s>1/2$,  and compute by taking
$\epsilon\to 0$    using <cit.>, here done formally only,

\begin{align*}
\begin{split}
\widetilde W^+_\beta g(k_\beta)\varphi
&=2\pi\int_{\Lambda}\,g(\lambda) f_1(H)\delta(H-\lambda)T^+_\beta
J_\beta \breve w^+_b f_2(k_\beta)\delta_{0,\beta}(\lambda)
\varphi \,\d \lambda;\\
\delta_{0,\beta}(\lambda)&=(2\i\pi)^{-1}\parb{(k_\beta -\lambda-\i
0)^{-1}-(k_\beta -\lambda+\i
0)^{-1}}\\&=\gamma_{b,0}(\lambda_\beta)^* \gamma_{b,0}(\lambda_\beta).
\end{split}
\end{align*}
See  Section <ref> for
notation, and note that  this formula   has the following precise
meaning. We substitute, referring again to Section <ref>,

\begin{align*}
\breve w^+_b f_2(k_\beta)\delta_{0,\beta}(\lambda)
=f_2(\lambda) \gamma_{b}^+(\lambda_\beta)^*\gamma_{b,0}(\lambda_\beta).
\end{align*}

Then by combining the proof of
Lemma <ref>,
Remark <ref> <ref> and Lemma
<ref> we conclude that the integral to
the right  is well-defined,

\begin{align}\label{eq:Wfinal} \begin{split}
\widetilde W^+_\beta  g(k_\beta)\varphi\\&\quad
=2\pi\int \,(gf_2)(\lambda)  \parb{ f_1(H)\delta(H-\lambda) T^+_\beta J_\beta
\gamma_{b}^+(\lambda_\beta)^*}\gamma_{b,0}(\lambda_\beta)\varphi \,\d \lambda, \end{split}
\end{align}
where the correct interpretation of the product in
parentheses involves the `$Q$-operators' as in
Subsection <ref> (more precisely  obtained by expanding
$T^+_\beta$ into terms on the
form (<ref>) given below). We call (<ref>) a `channel
wave matrix' representation (examined closer in Section <ref>). Although the below proof of
(<ref>) reveals a different representation,  more
closely related to (<ref>), the formula
(<ref>) is more useful for most of our  purposes
(with exceptions  only in Subsection <ref>). The
alternative formula

\begin{align}\label{eq:AltForm}
\begin{split}
&\widetilde W^+_\beta  g(k_\beta)\varphi\\
&\quad =\int \,(gf_2)(\lambda)   f_1(H)\parb{\Phi_\beta^++\i R(\lambda-\i 0) T^+_\beta }J_\beta
\gamma_{b}^+(\lambda_\beta)^*\gamma_{b,0}(\lambda_\beta)\varphi \,\d \lambda.
\end{split}
\end{align}

Now there are two assertions
that need to be checked to justify  (<ref>)  by
taking the  `$\epsilon\to 0$'–limit  in
(<ref>),  in the precise meaning of taking  limit
in the weak topology of $L^2_{-1}(\mathbf

\begin{align}
\i \lim_{\epsilon\to 0}&\int_{\Lambda}\,g(\lambda) f_1(H)\parb{R(\lambda-\i \epsilon)-
R(\lambda+\i \epsilon)}T^+_\beta J_\beta \breve w^+_b f_2(k_\beta)\delta_{\epsilon,\beta}(\lambda)
\varphi \,\d \lambda \nonumber
\\&=\i \lim_{\epsilon\to 0} \int_{\Lambda}\,g(\lambda) f_1(H)\parb{R(\lambda-\i \epsilon)-
R(\lambda+\i \epsilon)}T^+_\beta J_\beta f_2(\lambda) \gamma_{b}^+(\lambda_\beta)^*\gamma_{b,0}(\lambda_\beta)
\varphi \,\d \lambda \nonumber\\&
=2\pi\int_{\Lambda}\,g(\lambda) f_1(H)\delta(H-\lambda)T^+_\beta
J_\beta f_2(\lambda) \gamma_{b}^+(\lambda_\beta)^*\gamma_{b,0}(\lambda_\beta)
\varphi \,\d \lambda,\label{eq:W3}
\end{align}

\begin{align}
\label{eq:W4}
\lim_{\epsilon\to 0}\int_{\Lambda}\,g(\lambda) f_1(H)\parb{\Phi_\beta^+
+\i R(\lambda+\i \epsilon)T^+_\beta}J_\beta \breve w^+_b f_2(k_\beta)\delta_{\epsilon,\beta}(\lambda)
\varphi \,\d \lambda=0.
\end{align}

For (<ref>)  we write with $\delta_{\epsilon,\lambda}(\lambda'):=\pi^{-1}\epsilon/\parb{(\lambda-\lambda')^2+\epsilon^2}$
for any  fixed  $\lambda\in \R$ and $\epsilon>0$,

\begin{align}\label{eq:delta}
\begin{split}
\breve w^+_b f_2(k_\beta)\delta_{\epsilon,\beta}(\lambda)
\varphi &=\breve w^+_b \int^\infty _{\lambda^\beta}
\gamma_{b,0}(\lambda'-\lambda^\beta)\varphi \,\d \lambda'\\
\gamma_{b,0}(\lambda'-\lambda^\beta)\varphi \,\d \lambda',
\end{split}
\end{align}
substitute  and replace by the limit (when taking  $\epsilon\to
0$). This is doable thanks to Remark <ref>
<ref>, Lemma <ref> and the continuity of $\gamma_{b,0}(\lambda'_\beta)\varphi$ justifying  the first equality, and the second is a
consequence of Remark <ref>
<ref> and Lemma <ref> too. Recall from Subsection <ref>  that $T^+_\beta$ is a finite sum of terms
expressed as

\begin{align}\label{eq:Ex1}
B_+\text{ bounded},
\end{align}
to be used below too.

For (<ref>) we will invoke <cit.>.  Indeed thanks to these results and the presence of the factors
$A_{1+}$ in $\Phi^+_\beta$ we can compute the left-hand side to be

\begin{align}\label{eq:EXP}
\int_{\Lambda}\,g(\lambda) f_1(H)\parb{\Phi_\beta^+ +\i R(\lambda+\i
0)T^+_\beta}J_\beta f_2(\lambda)
\gamma_{b}^+(\lambda_\beta)^*\gamma_{b,0}(\lambda_\beta) \varphi \,\d
\lambda.
\end{align}
To see this, considering only the `difficult term'

\begin{align*} \int_{\Lambda}\,g(\lambda) f_1(H)R(\lambda+\i
\epsilon)T^+_\beta J_\beta \breve w^+_b
f_2(k_\beta)\delta_{\epsilon,\beta}(\lambda) \varphi \,\d \lambda,
\end{align*}
we insert $I=\chi^2_+( 2B/\epsilon_0)+\chi^2_-(
2B/\epsilon_0)$ to the right of the factor $R(\lambda+\i
\epsilon)$. Using the factors of $A_{1+}$ and a little commutation
the contribution from the second term $\chi^2_-( 2B/\epsilon_0)$
allows the insertion of the weight $\inp{x}^{-s}$ for some  $s>1/2$  to the
right of $R(\lambda+\i \epsilon)$ and we can use (<ref>) and
(<ref>) to compute the `$\epsilon\to 0$'–limit for this
term (more precisely we use that $\inp{x}^{s}\chi^2_-(
2B/\epsilon_0)f_1(H)Q^+(b,l)^*B_+$ is bounded). As for the contribution from the first term $\chi^2_+(
2B/\epsilon_0)$ we write

\begin{align*} R(\lambda+\i \epsilon)\chi^2_+(
2B/\epsilon_0)=R(\lambda+\i \epsilon)\chi^2_+(
2B/\epsilon_0)\inp{x}^{-s}\inp{x}^{s};\quad s>1/2-\mu/2.
\end{align*}
Then by <cit.>

\begin{align*} \inp{x}^{-1}f_1(H)R(\lambda+\i \epsilon)\chi^2_+(
2B/\epsilon_0)\inp{x}^{-s}\text{ is uniformly bounded},
\end{align*}
and there exists

\begin{align*} \Lambda\ni \lambda\to\inp{x}^{-1}f_1(H)&R(\lambda+\i
0)\chi^2_+( 2B/\epsilon_0)\inp{x}^{-s}\\&=\lim_{\epsilon\to 0}
\,\inp{x}^{-1}f_1(H)R(\lambda+\i \epsilon)\chi^2_+(
\end{align*}
constituting a continuous $\vL(\vH)$-valued
function. Consequently, using again (<ref>), it suffices
to check the existence and continuity of the $\vL(L^2(C_b),\vH)$-valued

\begin{align*} \R\ni \lambda \to f_2(\lambda)\inp{x}^{s}T^+_\beta
J_\beta \gamma_{b}^+(\lambda_\beta)^*.
\end{align*}
Expanding $T^+_\beta$ as above,
we can use Lemma <ref> to combine for
each  term the factor $Q^+(b,l)$ with the factor $J_\beta
\gamma_{b}^+(\lambda_\beta)^*$ (as done above). Each  remaining factor of a `$Q$-operator'
(more precisely $Q^+(b,l)^*$ appearing to the left)   contributes by a factor
$\vO(r^{\rho_1/2-\delta/2}) $, cf. Remark <ref> <ref>. Whence we are left with checking that
$s+\rho_1/2-\delta/2\leq 0$ is possible. Due to (<ref>)
it suffices to produce an $s>1/2-\mu/2$ such that $s+ 1/2-2/(2+\mu)\leq 0$,
and therefore in turn to check that $ 1/2-\mu/2+ 1/2-2/(2+\mu)<0$.
But the latter condition is fulfilled for all $\mu>0$,
so the check is done. Consequently indeed the left-hand side of (<ref>) is given by the
integral (<ref>).

Next, using the above arguments and  <cit.>,  the expression
(<ref>) may be  computed   with  $\chi_\rho:=\chi_-(r/\rho)$ as

\begin{align*}
- R(\lambda+\i 0)(H-\lambda)\Phi^+_\beta}J_\beta f_2(\lambda) \gamma_{b}^+(\lambda_\beta)^*\gamma_{b,0}(\lambda_\beta)
\varphi \,\d \lambda\\
&=\lim_{\rho\to \infty}\int_{\Lambda}\,g(\lambda)f_1(H)\parb{\Phi_\beta^+
- R(\lambda+\i 0)\chi_\rho (H-\lambda)\Phi^+_\beta}J_\beta f_2(\lambda) \gamma_{b}^+(\lambda_\beta)^*\gamma_{b,0}(\lambda_\beta)
\varphi \,\d \lambda\\
&=\lim_{\rho\to \infty}\int_{\Lambda}\,g(\lambda) f_1(H)R(\lambda+\i
0)[H,\chi_\rho]\Phi^+_\beta J_\beta f_2(\lambda) \gamma_{b}^+(\lambda_\beta)^*\gamma_{b,0}(\lambda_\beta)
\varphi \,\d \lambda\\
\end{align*}
proving (<ref>).

Now to prove (<ref>),  we would like to substitute the adjoint expression  of (<ref>) with $g(\lambda)=\delta_{\epsilon,\lambda'}(\lambda)$ into
(<ref>) and then  interchange the order of the two
integrations. This requires of course some modification since the
meaning of the right-hand side of  (<ref>) is a vector in  $L^2_{-s}(\mathbf
X)$  for any $s>1/2$ (but unlikely any  smaller),  and there is no
reason to expect  that the quantity  $T^-_\alpha J_\alpha \breve w^-_a f_2(k_\alpha)\delta_{\epsilon,\alpha}(\lambda')
\varphi_a \in L^2_{s}(\mathbf
X)$ for any such $s$. However, writing    $T^-_\alpha$ as  a finite sum terms on the

\begin{align}\label{eq:Ex2}
B_-\text{ bounded},
\end{align}

we can introduce
the  modification, say denoted by
$ T^-_{\alpha,\rho}$, given by inserting for each such term  the above factor
$\chi_\rho$ to the right of $Q^-(a,k)^*$. Using  Fubini's theorem and
the computation

\begin{align*}
\int^\infty_{-\infty}\,\delta_{\epsilon,\lambda'}(\lambda)\delta_{\epsilon,\alpha}(\lambda')\,\d \lambda'=\delta_{2\epsilon,\alpha}(\lambda),
\end{align*}
we then obtain

\begin{align*}
&(2\pi \i)^{-2}\inp{\varphi_b,\widetilde
\infty}\\&\quad \int_{\Lambda}\,f_2(\lambda) \inp{\gamma_{b,0}(\lambda_\beta)\varphi_b,\gamma^+_{b}(\lambda_\beta)J^*_\beta (T^+_\beta)^*\delta(H-\lambda)T^-_{\alpha,\rho} J_\alpha \breve w^-_a f_2(k_\alpha)
\delta_{2\epsilon,\alpha}(\lambda)
\varphi_a} \,\d \lambda,
\end{align*}
valid  for any $\varphi_a\in L^2_s(\mathbf  X_a)$ and
$\varphi_b\in L^2_s(\mathbf  X_b)$ for

With an analogue of (<ref>) for $\alpha$ rather
than $\beta$ we compute

\begin{align*}
0}\int_{\Lambda}\,f_2(\lambda) \inp{\gamma_{b,0}(\lambda_\beta)\varphi_b,\gamma^+_{b}(\lambda_\beta)J^*_\beta (T^+_\beta)^*\delta(H-\lambda)T^-_{\alpha,\rho} J_\alpha \breve w^-_a f_2(k_\alpha)
\delta_{2\epsilon,\alpha}(\lambda)
\varphi_a} \,\d \lambda\\& =\int_{\Lambda}\,f_2(\lambda) \inp{\gamma_{b,0}(\lambda_\beta)\varphi_b,\gamma^+_{b}(\lambda_\beta)J^*_\beta (T^+_\beta)^*\delta(H-\lambda)T^-_{\alpha,\rho} J_\alpha f_2(\lambda)
\gamma_{a}^-(\lambda_\alpha)^*\gamma_{a,0}(\lambda_\alpha)
\varphi_a} \,\d \lambda,
\end{align*}
uniformly in $\rho>1$. Due to  these
features  we can interchange limits above and conclude that

\begin{align*}
&(2\pi \i)^{-2}\inp{\varphi_b,\widetilde
\infty}\\&\int_{\Lambda}\,f_2(\lambda) \inp{\gamma_{b,0}(\lambda_\beta)\varphi_b,\gamma^+_{b}(\lambda_\beta)J^*_\beta (T^+_\beta)^*\delta(H-\lambda)T^-_{\alpha,\rho} J_\alpha f_2(\lambda)
\gamma_{a}^-(\lambda_\alpha)^*\gamma_{a,0}(\lambda_\alpha)
\varphi_a} \,\d \lambda
\\&=\int_{\Lambda}\,f_2(\lambda) \inp{\gamma_{b,0}(\lambda_\beta)\varphi_b,\gamma^+_{b}(\lambda_\beta)J^*_\beta (T^+_\beta)^*\delta(H-\lambda)T^-_\alpha J_\alpha f_2(\lambda)
\gamma_{a}^-(\lambda_\alpha)^*\gamma_{a,0}(\lambda_\alpha)
\varphi_a} \,\d \lambda,
\end{align*}
showing  (<ref>). Note that indeed the right
interpretation is given by expansion into a (finite) sum,  substituting the expressions
(<ref>) and (<ref>), cf. Subsection <ref>.

Note the following analogues of
(<ref>) and (<ref>), cf. (<ref>) and (<ref>),

\begin{align}\label{eq:Wfinal2a}
\widetilde W^-_\beta g(k_\beta)\varphi
&=-2\pi\int \,(gf_2)(\lambda)  \parb{ f_1(H)\delta(H-\lambda) T^-_\beta J_\beta
\gamma_{b}^-(\lambda_\beta)^*}\gamma_{b,0}(\lambda_\beta)\varphi
\,\d \lambda,\\
\begin{split}
16\lambda_\beta \lambda_\alpha f^2_2(\lambda)&m_b(\pm\hat
\xi_b)^2m_b(\pm\hat \xi_a)^2\,\delta_{ \beta \alpha}\\&\quad =-(2\pi \i)^2f^2_2(\lambda){\gamma_b^\pm(\lambda_\beta)}J^*_\beta
\parb{T^\pm _\beta}^*\delta(H-\lambda)T^\pm _\alpha
J_\alpha{\gamma_a^\pm (\lambda_\alpha)}^*. \label{eq:Wfinal2b}
\end{split}
\end{align}
The  quantity to the  left in
(<ref>) is meant to be  an operator in
$\vL\parb{L^2(C_a),L^2(C_b)}$;  the use of the
Kronecker symbol  $\delta_{
\beta \alpha}$ specifies that it vanishes unless
$\beta =\alpha$. The operator  may be considered as   the fiber of $F_\beta \parb{\widetilde W_\beta^\pm}^*\widetilde W^\pm_\alpha
F_\alpha^{-1}$ at energy $\lambda$, invoking
the  orthogonality of channels.  The formula (<ref>)
by mimicking
the above procedure for showing (<ref>).

[1]
T. Adachi, K. Itakura, K. Ito, E. Skibsted,
New methods in spectral theory of $N$-body Schrödinger
operators,  Rev. Math. Phys. 33  (2021), 48 pp.

[2] S. Ashida, Scattering matrices and generalized
Fourier transforms in long-range $N$-body problems,
arXiv:1811.07527 (2018), 30 pp.

[3]
J. Dereziński, Asymptotic completeness for $N$-particle long-range quantum
systems, Ann. of Math.  38   (1993),

[4] J. Dereziński, E. Skibsted, Quantum
scattering at low
energies, J. Funct. Anal.,  257 (2009), 1828–1920.

[5] J. Dollard, Asymptotic convergence and Coulomb interaction, J. Math. Phys. 5 (1964), 729–738.

[6] Y. Gâtel and D. Yafaev, On the solutions of
the Schrödinger  equation with radiation conditions at infinity: the long-range case, Ann. Inst. Fourier (Grenoble)
49  (1999), 1581–1602.

[7] C. Gérard, H. Isozaki and E. Skibsted,
$N$-body resolvent estimates,
J. Math. Soc. Japan 48 no. 1 (1996), 135–160.

[8] G.M. Graf, Asymptotic completeness for
$N$-body short-range quantum systems: a new proof,
Commun. Math. Phys. 132 (1990), 73–101.

[9] I. Herbst, E. Skibsted, Free channel Fourier transform in the long-range $N$-body problem, J.
Anal. Math. 65 (1995),  297–332.

[10] L. Hörmander, The existence of wave
operators in scattering theory, Math. Z. 146 (1976), 68–91.

[11] L. Hörmander, The analysis of linear
partial differential operators. I–IV, Berlin, Springer

[12]
T. Ikebe, H. Isozaki, A stationary approach to the existence and
completeness of long-range operators, Integral equations and
operator theory 5
(1982), 18–49.
[13]
H. Isozaki, Eikonal equations and spectral representations for
long range  Schrödinger Hamiltonians,  J. Math. Kyoto Univ. 20
(1980), 243–261.

[14]
K. Ito, E. Skibsted,
Radiation condition bounds on manifolds with ends,
J. Funct. Anal. 278 (2020), 1–47.

[15] K. Ito, E. Skibsted, Stationary  scattering
theory on manifolds, Ann. Inst. Fourier, Grenoble (2021), 55 pp.

[16] K. Ito, E. Skibsted, Time-dependent  scattering
theory on manifolds, J. Funct. Anal. 277 (2019), 1423–1468.

[17] T. Kato, Smooth operators and commutators,
Studia Math. 31 (1968), 535–546.

[18] T. Kato, S.T.  Kuroda, The abstract theory of
scattering, Rocky Mountain journal of mathematics,  1
(1971), 127–171.

[19] E. Skibsted, Propagation estimates for $N$-body
Schrödinger operators, Commun. Math. Phys. 142 (1991),

[20]
E. Skibsted, Smoothness of $N$-body scattering amplitudes, Rev. Math. Phys.
4 no. 4
(1992), 619–658.

[21] D.R. Yafaev, Resolvent estimates and scattering
matrix for $N$-body Hamiltonians,
Intgr. Equat. Oper. Th. 21 (1995), 93–126.

[22] D.R. Yafaev, Scattering theory: some old and
new problems, Lecture Notes in Mathematics
1735, Springer, 2000.

[23] D.R. Yafaev, Radiation conditions and
scattering theory for $N$-particle Hamiltonians,
Commun. Math. Phys. 154  (1993), 523–554.

[24] D.R. Yafaev, Eigenfunctions of the continuous
spectrum for $N$-particle Schrödinger
operator, Spectral and scattering theory, Lecture notes in pure and
applied mathematics, Marcel Dekker, New York  1994, 259–286.

[25]
K. Yosida, Functional Analysis, Springer, Berlin, 1965.