Stationary Scattering Theory: The N-Body Long-Range Case

Abstract

Within the class of Dereziński–Enss pair-potentials which includes Coulomb potentials and for which asymptotic completeness is known (Dereziński in Ann Math 38:427–476, 1993), we show that all entries of the N-body quantum scattering matrix have a well-defined meaning at any given non-threshold energy. As a function of the energy parameter the scattering matrix is weakly continuous. This result generalizes a similar one obtained previously by Yafaev for systems of particles interacting by short-range potentials (Yafaev in Integr Equ Oper Theory 21:93–126, 1995). As for Yafaev’s paper we do not make any assumption on the decay of channel bound states. The main part of the proof consists in establishing a number of Kato-smoothness bounds needed for justifying a new formula for the scattering matrix. Similarly we construct and show strong continuity of channel wave matrices for all non-threshold energies. Away from a set of measure zero we show that the scattering and channel wave matrices constitute a well-defined ‘scattering theory’, in particular at such energies the scattering matrix is unitary, strongly continuous and characterized by asymptotics of generalized eigenfunctions of minimal growth.

Appendices

Appendix A: Proof of (3.2b), (3.7a) and (3.7b)

For convenience we consider only the assertions (3.2b), (3.7a) and (3.7b) for $t\rightarrow +\infty$. Our procedure of proof relies on standard stationary phase analysis on which we omit details, see for example [Hö1, Hö2, II] for elaborate accounts. In addition we will use Lemma 8.1 (proved in Sect. 8 independently of the mentioned assertions). Recall that Lemma 8.1 is stated in terms of the auxiliary operator $B_a=2\Re { \big ( (x_a/r)\cdot p_a\big )}$ introduced in (8.3).

(3.2b):    We need to show that

$$\begin{aligned} \mathop {\mathrm {s-lim}}\limits _{t\rightarrow +\infty } \big ( I-N^a_+ \big )J_\alpha \breve{w}_a^{+}{\mathrm e}^{-\textrm{i}tk_\alpha }{f_1}(k_\alpha )=0, \end{aligned}$$

which by the boundedness of $N^a_+ =A_1 A_2^a(A_3^a)^2A_2^aA_1$ (with $A_1=A_{1+}$ and $A^a_3=A^a_{3+}$) amounts to showing that

$$\begin{aligned} \begin{aligned}&\forall \varphi \in L^2({\textbf{X}}_a):\quad \lim _{t\rightarrow +\infty }\left\| \big ( I-A_1 A_2^a(A_3^a)^2A_2^aA_1\big ) \big ( u^\alpha \otimes \varphi (t)\big )\right\| =0; \\&\quad \quad \quad \quad \varphi (t)={\mathrm e}^{-\textrm{i}S_a (p_a,t)}\varphi _1,\quad \varphi _1={f_1}(k_\alpha )\varphi . \end{aligned} \end{aligned}$$

(A.1)

We can assume that $\varphi$ is smooth in momentum space, i.e. that its Fourier transform $\widehat{\varphi }\in C^\infty ({\textbf{X}}_a)$. Then $\widehat{\varphi _1}\in C^\infty _{\mathrm c}({\textbf{X}}_a{\setminus }\{0\})$, and

$$\begin{aligned} \varphi (t)(x_a)= (2\pi )^{-n_a/2} \int {\mathrm e}^{\textrm{i}x_a \cdot \xi _a}{\mathrm e}^{-\textrm{i}S_a (\xi _a,t)}\widehat{\varphi _1}(\xi _a)\,{\mathrm d}\xi _a. \end{aligned}$$

(A.2)

By assumption $B={\text {supp}}\widehat{\varphi _1}\subseteq {\textbf{X}}_a{\setminus }\{0\}$ is compact. Uniformly in $\xi \in B$ for any such B the function $S_a$ obeys the bounds (as $t\rightarrow \infty$)

$$\begin{aligned} \partial _\xi ^\gamma \partial _t^k \big ( S_a(\xi ,t)- t\xi ^2\big )= {\left\{ \begin{array}{ll} {\mathcal O}(t^{1-k-\mu })&{}\text { for }|\gamma |+k\le 2,\\ {\mathcal O}(t^{1-k})&{}\text { for }|\gamma |+k\ge 3. \end{array}\right. } \end{aligned}$$

(A.3)

The stationary phase method then applies, see for example [Hö2, Theorem 7.7.6]. We note that the part of (A.3) given by the cases $|\gamma |+k\le 1$ is a consequence of (2.8a)–(2.8c) (in turn appearing as [II, Lemma 6.1]). We give at the end of the “Appendix” a proof of (A.3) for the cases $|\gamma |+k\ge 2$. Taking for the moment the bounds for granted the stationary phase method yields the effective localization $|x_a|\ge \epsilon ' t$ for some $\epsilon '>0$, which means that we can freely insert the factor $\chi _+ \big ( {|x_a|/(\epsilon 't)}\big )$ for some $\epsilon '>0$ in front of $\varphi (t)$ in the tensor product in (A.1). (Recalling that $f_1$ is supported near $\lambda _0$ we can for example use this factor with $\epsilon '=\sqrt{\lambda _0-\lambda ^\alpha }$.)

Similarly we can record that

$$\begin{aligned} \left\| (B_a-2p_a^2) \big ( u^\alpha \otimes \varphi (t)\big )\right\| \rightarrow 0 \text { for }t\rightarrow \infty , \end{aligned}$$

(A.4)

which in addition to (A.2) and the stationary phase method is based on (5.2a) and the fact that $x^a$ is a localized variable due to the presence of the channel bound state $u^\alpha$. More precisely the localization can be implemented in terms of an insertion of a factor of $\chi _- \big ( {r^{-\delta '}}{|x^a|}\big )$ for any $\delta '\in (0,1)$ in front of the tensor product (which is harmless since effectively $|x_a|$ grows linearly in time). For the same reason we can freely insert the factor $\chi _- \big ( {r^{-\delta '}}{|x^a|}\big )$ in front of the tensor product in (A.1). It turns out to be useful to do this for some $\delta '\in (0,\delta )$ (where $\delta$ is fixed in (3.6)).

With these conventions it remains to argue that for the expression

$$\begin{aligned} A_1 A_2^a(A_3^a)^2A_2^aA_1\psi (t)\text { with } \psi (t)={\chi _- \big ( {r^{-\delta '}}{|x^a|}\big ) \big ( u^\alpha \otimes \big ( \chi _+ \big ( {|x_a|/(\epsilon 't)}\big )\varphi (t)\big )\big )}, \end{aligned}$$

(A.5)

all factors of $A_j$ (omitting for convenience the superscript a for $j=2,3$) can be replaced by I in the large time limit. This will be accomplished by showing that

$$\begin{aligned} \left\| (A_j-I)\psi (t)\right\| \rightarrow 0 \text { for }t\rightarrow \infty ;\quad j=1,2,3 . \end{aligned}$$

(A.6)

For the case of $j=1$ we can freely insert the factor $\chi _+({B_a}/(8\epsilon _0))$ in front of $\psi (t)={u^\alpha \otimes \varphi (t)}+o(t^0)$, which may be seen as follows, using in the second step (A.4) and a commutation,

$$\begin{aligned}&6^{-1}\left\| \chi _-({B_a}/(8\epsilon _0)) \big ( u^\alpha \otimes \varphi (t)\big )\right\| ^2\\&\quad \le \langle \chi _-({B_a}/(8\epsilon _0))^2 \big ( 2-{B_a}/(8\epsilon _0)\big )\rangle _{{u^\alpha \otimes \varphi (t)}}\\&\quad =-2\langle \big ( {p^2_a}/(8\epsilon _0)-1\big ){f_2}(k_\alpha )\rangle _{\chi _-({B_a}/(8\epsilon _0))({u^\alpha \otimes \varphi (t)})} +o(t^0)\\&\quad \le -2\langle \big ( {(\lambda _0-\lambda ^\alpha )}/(9\epsilon _0)-1\big ){f_2}(k_\alpha )\rangle _{\chi _-({B_a}/(8\epsilon _0))({u^\alpha \otimes \varphi (t)})} +o(t^0)\\&\quad \le o(t^0). \end{aligned}$$

Along with additional ‘free factors’ of $f_3({\breve{H}}_a)$ and $\chi _- \big ( |x^a|/(\epsilon r)\big )$ we then write, using in the last step Lemma 8.1,

$$\begin{aligned} (I-A^2_1)\psi (t)&=(I-A^2_1)\chi _+({B_a}/(8\epsilon _0))\psi (t)+o(t^0)\\ {}&=\chi _-({B/{\epsilon _0}})^2\chi _- \big ( |x^a|/(\epsilon r)\big )\chi _+({B_a}/(8\epsilon _0))f_3({\breve{H}}_a)\psi (t)+o(t^0)\\&=o(t^0). \end{aligned}$$

Since $A_1\ge 0$, the assertion (A.6) for $j=1$ follows.

To show (A.6) for $j=2,3$ we need the facts that in $\{|x^a|<c r^\delta \}$ with $c>0$ given in the property (ii) of Sect. 5 (with ii) used for $r^a$ rather than for r), $r^a_\delta =r^\delta r^a(0)$ and $B_{\delta ,\rho _1}^a=r^{\rho _1/2}B_\delta ^a r^{\rho _1/2}=0$. For $A_2=A_2^a$ we abbreviate $\chi _{\delta '}=\chi _- \big ( {r^{-\delta '}}{|x^a|}\big )$ and recall that we took it with $\delta '\in (0,\delta )$. Using then that for all large enough r,

$$\begin{aligned} A_2 \chi _{\delta '}=\chi _- \big ( r^{\rho _2-1+ \delta }r^a(0)\big )\chi _{\delta '} =\chi _{\delta '}, \end{aligned}$$

cf. (3.6), combined with the ‘free factor’ of $\chi _+ \big ( {|x_a|/(\epsilon 't)}\big )$, indeed (A.6) for $j=2$ follows. Similarly we can replace $A_3=\chi _-(B_{\delta ,\rho _1}^a)$ by arguing that for all large enough $\rho >1$,

$$\begin{aligned} (A_3-I)\chi _{\delta '}\chi _+(r/\rho )= \int _{{\mathbb {C}}} ( B_{\delta ,\rho _1}^a-z)^{-1}\, \big ( B_{\delta ,\rho _1}^a\chi _{\delta '}\chi _+(r/\rho )\big )\,z^{-1}\mathrm d\mu _{\chi _-}(z)=0, \end{aligned}$$

cf. (8.15). This proves (A.6) holds for $j=3$.

(3.7a) and (3.7b):    We take (3.7b), i.e.

$$\begin{aligned} F_\alpha m_\alpha ^+ F^{-1}_\alpha =\int ^\oplus _{I^\alpha } 2\lambda _\alpha ^{1/2 }m_a({\hat{\xi }}_a) \, {\mathrm d}\lambda = 2\lambda _\alpha ^{1/2 }m_a({\hat{\xi }}_a),\quad {\hat{\xi }}_a=\xi _a/|\xi _a|\in C_a, \end{aligned}$$

as a definition of $m_\alpha ^+$, and it remains (for (3.7a)) to show

$$\begin{aligned} \Big ( \mathop {\mathrm {s-lim}}\limits _{t\rightarrow \infty }\,{\mathrm e}^{\textrm{i}tH} \Phi _a^+ {\mathrm e}^{-\textrm{i}t{\breve{H}}_a}\Big )J_\alpha \breve{w}_a^{+}{f_1}(k_\alpha )&=\mathop {\mathrm {s-lim}}\limits _{t\rightarrow \infty }{{\mathrm e}^{\textrm{i}tH} {\mathrm e}^{-\textrm{i}t{\breve{H}}_a} J_\alpha \breve{w}_a^{\pm }}{f_1}(k_\alpha ) \big ( m_\alpha ^+\big )^2;\\ \Phi _a^+&={f_2}(H)M_a N^a_+ M_a {f_2}({\breve{H}}_a). \end{aligned}$$

Due to (A.1) it suffices to show (recalling the notation (4.2a)) that for all $\varphi \in L^2({\textbf{X}}_a)$ with ${\widehat{\varphi }}\in C^\infty _{\mathrm c}({\textbf{X}}'_a):$

$$\begin{aligned} \begin{aligned}&\lim _{t\rightarrow \infty }\left\| {M_a \big ( u^\alpha \otimes \varphi (t)\big )-u^\alpha \otimes \big ( {m_\alpha ^+}\varphi (t)\big )}\right\| =0,\\&\quad \lim _{t\rightarrow \infty }\left\| \big ( {f_2}(H)-{f_2}({\breve{H}}_a)\big ) \big ( u^\alpha \otimes \big ( \big ( m_\alpha ^+\big )^2\varphi (t)\big )\big )\right\| =0; \\&\quad \quad \quad \quad \varphi (t)={\mathrm e}^{-\textrm{i}S_a (p_a,t)}\varphi _1,\quad \varphi _1={f_1}(k_\alpha )\varphi . \end{aligned} \end{aligned}$$

(A.7)

As for the first assertion of (A.7) we can (as above) replace the tensor product $u^\alpha \otimes \varphi (t)$ with its modification $\psi (t)$ from (A.5). We can then use stationary phase analysis in combination with the properties (1) and (2) from Sect. 4 (here 2) is used with $b=a$). The second part also follows from standard stationary phase analysis noting that the support property of $\widehat{\varphi }$ yields ‘effective decay’ of $u^\alpha \otimes (m_\alpha ^+)^2\varphi (t)$ near the ‘collision planes’ ${\textbf{X}}_c$ with $c\not \le a$ (where the intercluster potential $I_a$ lacks decay), yielding the same conclusion for $I_a-\breve{I}_a$ when first representing

$$\begin{aligned} {f_2}(H)-{f_2}({\breve{H}}_a)=-\int _{{\mathbb {C}}} (H-z)^{-1}\bigl (I_a-\breve{I}_a\bigr ) ({\breve{H}}_a-z)^{-1}\,\mathrm d\mu _f(z), \end{aligned}$$

and then using a commutation. We skip the detailed arguments.

Remark A.1

To be used in “Appendix C” let us note the related result (with $\tilde{J}_\alpha ^-$ given by (3.1))

$$\begin{aligned} \mathop {\mathrm {s-lim}}\limits _{t\rightarrow +\infty }{\mathrm e}^{\textrm{i}tH}\tilde{J}_\alpha ^-{\mathrm e}^{-\textrm{i}tk_\alpha }{f_1}(k_\alpha )=0, \end{aligned}$$

(A.8)

which follows by the above arguments.

Proof of A.3for the cases $|\gamma |+k\ge 2$ Let $B\subseteq {\textbf{X}}_a\setminus \{0\}$ be compact. All bounds below are uniform in $\xi \in B$. Using the functions $x=x(\xi ,t)$ and $\lambda =\lambda (\xi ,t)$ of (2.8a) and (2.8b) for $\xi \in B$ and t large enough it follows (using also (2.8c) and representing as column vectors) that

$$\begin{aligned} \nabla _{(x.\lambda )} K_a=(\xi ^{{\text {tr}}}, t)^{{\text {tr}}}\text { and }\nabla _{(\xi .t)} S_a=(x^{{\text {tr}}}, \lambda )^{{\text {tr}}}. \end{aligned}$$

(A.9)

Motivated by (2.8b) we ‘renormalize’ the variables in terms of the ‘time’ $\tau :=\sqrt{t}$ as

$$\begin{aligned} {\bar{t}} =t/\tau \,\,(=\sqrt{t}=\tau ),\quad {\bar{\xi }}= \tau \xi ,\quad \bar{x}=x/\tau ,\quad {\bar{\lambda }} = \tau \lambda . \end{aligned}$$

We need to show the bounds

$$\begin{aligned} \partial ^\alpha _{({\bar{\xi }}, {\bar{t}})} ({\bar{x}}-2\bar{\xi })&={\mathcal O}(\tau ^{1-|\alpha |-2\mu });\quad |\alpha |\le 1, \end{aligned}$$

(A.10a)

$$\begin{aligned} \partial ^\alpha _{({\bar{\xi }}, {\bar{t}})} ({\bar{\lambda }}-|\bar{\xi }|^2/{\bar{t}})&={\mathcal O}(\tau ^{1-|\alpha |-2\mu });\quad |\alpha |\le 1, \end{aligned}$$

(A.10b)

$$\begin{aligned} \partial ^\alpha _{({\bar{\xi }}, {\bar{t}})} z&={\mathcal O}(\tau ^{1-|\alpha |});\quad z={\bar{x}},\quad z={\bar{\lambda }},\quad z={\bar{\xi }}\text { or } z={\bar{t}}. \end{aligned}$$

(A.10c)

Upon substituting the expressions for the dependent variables ${\bar{x}}$ and ${\bar{\lambda }}$, and using for the independent variables,

$$\begin{aligned} \partial _{ \xi }= {\bar{t}} \partial _{{\bar{\xi }}} \text { and }\partial _{t}= \big ( 2\bar{t}\big )^{-1} \Big ( \partial _{{\bar{t}}}+ \big ( {\bar{t}}\big )^{-1}{\bar{\xi }}\cdot \partial _{ {\bar{\xi }}}\Big ), \end{aligned}$$

it is an elementary check (here omitted) that (A.3) follows from (2.8a)–(2.8c) and (A.10a)–(A.10c).

For $|\alpha |=0$ the bounds (A.10a)–(A.10c) are already known by (2.8b). The proofs for $|\alpha |= 1$ will be based on the representation of the Hessian of

$$\begin{aligned} K_0=K_0(x,\lambda ):=\sqrt{\lambda }r, \quad r=|x|, \quad x=x_a, \end{aligned}$$

as the square block-matrix

$$\begin{aligned} \nabla ^2 K_0= \left( \begin{array}{cc} {\sqrt{\lambda }}{r^{-1}}P_{\perp }&{}(2\sqrt{\lambda })^{-1}{\hat{x}}\\ (2\sqrt{\lambda })^{-1}{\hat{x}}^{{\text {tr}}}&{}-4^{-1}\lambda ^{-3/2}r \end{array}\right) ; \quad {\hat{x}} =x/r,\quad P_{\perp }={I-| {\hat{x}}\rangle \langle {\hat{x}}|}. \end{aligned}$$

We introduce

$$\begin{aligned} A_0= \left( \begin{array}{cc} \tau ^2{r^{-1}}{\sqrt{\lambda }}P_{\perp }&{}(2\sqrt{\lambda })^{-1}{\hat{x}}\\ (2\sqrt{\lambda })^{-1}{\hat{x}}^{{\text {tr}}}&{}-\tau ^{-2}r4^{-1}\lambda ^{-3/2 } \end{array}\right) =R_\tau (\nabla ^2 K_0) R_\tau ;\quad R_\tau ={\textrm{diag}}(\tau , \tau ^{-1}), \end{aligned}$$

and compute

$$\begin{aligned} B_0:=A^{-1}_0= \left( \begin{array}{cc} \tau ^{-2}r\lambda ^{-1/2} I&{}2\sqrt{\lambda }\,{\hat{x}}\\ 2\sqrt{\lambda }\,{\hat{x}}^{{\text {tr}}}&{}0 \end{array}\right) . \end{aligned}$$

We record that the matrices

$$\begin{aligned} \begin{aligned}&A_0\text { and }A:=R_\tau (\nabla ^2 K_a) R_\tau =A_0+{\mathcal O}(\tau ^{-2\mu }),\\&B_0\text { and }B:=A^{-1}=R_\tau ^{-1}(\nabla ^2 K_a)^{-1} R_\tau ^{-1}=B_0+{\mathcal O}(\tau ^{-2\mu }) \end{aligned} \end{aligned}$$

(A.11)

are all bounded (with uniform bounds in $\xi \in B$). Note also the square matrix identity (obtained by differentiating (A.9)),

$$\begin{aligned} D_{({\bar{\xi }}, {\bar{t}})} (\xi ^{{\text {tr}}}, t)^{{\text {tr}}} =\nabla ^2 K_a(x,\lambda )D_{({\bar{\xi }}, {\bar{t}})} (x^{{\text {tr}}}, \lambda )^{{\text {tr}}}, \end{aligned}$$

(A.12)

where $D_zF$ is used as notation for the derivative of a vector field $F=F(z)$. Next we write (A.12) as

$$\begin{aligned}R_\tau ^{-1} C_1 =\nabla ^2 K_a(x,\lambda )R_\tau \big ( D_{({\bar{\xi }}, {\bar{t}})} ({\bar{x}}^{{\text {tr}}}, {\bar{\lambda }})^{{\text {tr}}}+C_2\big ), \end{aligned}$$

and therefore in turn as

$$\begin{aligned} C_1 =A \big ( D_{({\bar{\xi }}, {\bar{t}})} ({\bar{x}}^{{\text {tr}}}, {\bar{\lambda }})^{{\text {tr}}}+C_2\big ), \end{aligned}$$

(A.13)

where

$$\begin{aligned}C_1= \left( \begin{array}{cc} I&{} -{\bar{\xi }}/\tau \\ 0 &{}2 \end{array}\right) \text { and }C_2= \left( \begin{array}{cc} 0&{} {\bar{x}}/\tau \\ 0 &{}-{\bar{\lambda }}/\tau \end{array}\right) . \end{aligned}$$

I. We prove (A.10c) for $|\alpha |= 1$. The result follows from writing (A.13) as

$$\begin{aligned} D_{({\bar{\xi }}, {\bar{t}})} ({\bar{x}}^{{\text {tr}}}, {\bar{\lambda }})^{{\text {tr}}}= BC_1-C_2, \end{aligned}$$

(A.14)

since indeed (by the previous remarks) the right-hand side is bounded uniformly in $\xi \in B$ for all large $\tau$.

II. We prove (A.10a) and (A.10b) for $|\alpha |= 1$ (yielding (A.3) for the cases $|\gamma |+k\le 2$). Introducing the column vector

$$\begin{aligned} {\bar{\gamma }} ={\bar{\gamma }}(\tau ) = \big ( {\bar{x}}^{{\text {tr}}}- 2{{\bar{\xi }}}^{{\text {tr}}}, {\bar{\lambda }}- |{\bar{\xi }}|^2/{\bar{t}}\,\big )^{{\text {tr}}}, \end{aligned}$$

we record that ${\bar{\gamma }} ={\mathcal O}(\tau ^{1-2\mu })$ and compute using (A.14)

$$\begin{aligned} D_{({\bar{\xi }}, {\bar{t}})} {\bar{\gamma }} = BC_1-C_2-C_3;\quad C_3=\left( \begin{array}{cc} 2I&{} 0\\ 2{\bar{\xi }}^{{\text {tr}}}/\tau &{}- \big ( {\bar{\xi }}/\tau \big )^2 \end{array}\right) . \end{aligned}$$

We need to check that the right-hand side is ${\mathcal O}(\tau ^{-2\mu })$. Thanks to (A.11) it suffices to check that

$$\begin{aligned} B_0C_1-C_2-C_3={\mathcal O}(\tau ^{-2\mu }), \end{aligned}$$

but an elementary inspection yields the matrix bound

$$\begin{aligned} \left\| B_0C_1-C_2-C_3\right\| \le \text {Const. }\tau ^{-1}\,|\bar{\gamma }| \quad \text {for large } \tau , \end{aligned}$$

showing the desired bound ${\mathcal O}(\tau ^{-2\mu })$.

III. We prove (A.10c) for $|\alpha |\ge 2$ (yielding (A.3) for the cases $|\gamma |+k\ge 3$). Assuming by induction that the bounds hold for $|\alpha |\le m-1$ for some $m\ge 2$ we obtain, after $m-1$ differentiations of (A.13) (more precisely by applying $D^\beta _{({\bar{\xi }}, {\bar{t}})}$ with $|\beta |=m-1$ to (A.13)) computing by the chain rule, an expression for

$$\begin{aligned} A D^\beta _{({\bar{\xi }}, {\bar{t}})}D_{({\bar{\xi }}, {\bar{t}})} ({\bar{x}}^{{\text {tr}}}, {\bar{\lambda }})^{{\text {tr}}} \end{aligned}$$

in terms of derivatives of lower order. The resulting formula is an example of Faa di Bruno’s formula, see [Hö1, Lemma 3.6] for a similar problem. The induction scheme works thanks to proper control of derivatives of the ‘outer function’. More precisely we consider above the matrix $A=A(\tau , x, \lambda )$ as a function of $z:=({\bar{t}}, {\bar{x}},{\bar{\lambda }})$, introducing

$$\begin{aligned} {\bar{A}}( z)={\bar{A}} \big ( {\bar{t}}, {\bar{x}},{\bar{\lambda }}\big )=A \big ( {\bar{t}}, {\bar{t}} \,{\bar{x}},{\bar{\lambda }}/{\bar{t}}\,\big ). \end{aligned}$$

Then an elementary check shows that $\partial _z^\gamma {\bar{A}}( z)={\mathcal O}(\tau ^{-|\gamma |})$, and with these bounds the right-hand side of Faa di Bruno’s formula can be dealt with by the induction hypothesis, implementing the induction step. $\square$

Appendix B: Proof of (7.3c)

We shall prove (7.3c), i.e. the bound

$$\begin{aligned} \quad \sup _{\Im z\ne 0}\left\| Q^\pm (a,k){f_2} (H){R(z)}\right\| _{{\mathcal L}({\mathcal B},{\mathcal H})}< \infty . \end{aligned}$$

For that we invoke the following (standard) scheme of stationary scattering theory (see for example [Ya3, IS1]), proceeding partially abstractly and proving a more general version than needed for (7.3c) (useful in [Sk3] as well).

Pick any $f\in C^\infty _{\mathrm c}(\Lambda )$ such that ${f_2}\prec f \prec {f_3}$ (implying that the bound (5.6) of Theorem 5.1 is at disposal provided $\Re z$ is in some neighbourhood of ${\text {supp}}{f}$).

Lemma B.1

Suppose $\Psi =f_2(H)Pf_2(H)\in {\mathcal L}({\mathcal H})\cap {\mathcal L}({\mathcal B})$ is self-adjoint (on ${\mathcal H}$), and suppose

$$\begin{aligned} \textrm{i}[H, \Psi ]\ge f_2(H) \Big ( Q^*Q-T^*T\Big )f_2(H) \end{aligned}$$

(B.1)

for H-bounded operators Q and T. Then (with $f\in C^\infty _{\mathrm c}(\Lambda )$ given as above) the following estimates hold for all $z\in {\mathbb {C}}\setminus {\mathbb {R}}$ and all $\psi \in {\mathcal B}$.

$$\begin{aligned} \begin{aligned}&\left\| QR(z) f_2(H)\psi \right\| ^2\le \left\| TR(z) f_2(H)\psi \right\| ^2 \\&\quad +2 \big ( \left\| \Psi \right\| _{{\mathcal L}({\mathcal H})}+\left\| \Psi \right\| _{{\mathcal L}({\mathcal B})}\big ) \left\| R(z)f(H)\right\| _{{\mathcal L}({\mathcal B},{\mathcal B}^{*})}\left\| \psi \right\| ^2_{{\mathcal B}} . \end{aligned} \end{aligned}$$

(B.2)

In particular,

$$\begin{aligned} \text { if }\sup _{\Im z\ne 0}\left\| |T{f_2} (H)|{R(z)}\right\| _{{\mathcal L}({\mathcal B},{\mathcal H})}< \infty ,\text { then also }\sup _{\Im z\ne 0}\left\| |Q{f_2} (H)|{R(z)}\right\| _{{\mathcal L}({\mathcal B},{\mathcal H})}< \infty . \end{aligned}$$

Proof

Letting $\psi _2=f_2(H)\psi$ and $\phi =R(z) \psi _2$ the estimate (B.1) leads to

$$\begin{aligned} \left\| Q\phi \right\| ^2-\left\| T\phi \right\| ^2&\le 2 \big ( (\Im z)\langle R(z)\psi , \Psi R(z) \psi \rangle -\Im \langle \psi , \Psi R(z) \psi \rangle \big )\\&\le 2 \big ( \left\| \Psi \right\| _{{\mathcal L}({\mathcal H})} |\Im \langle \psi , R(z)f(H) \psi \rangle | +|\Im \langle \Psi \psi , R(z)f(H) \psi \rangle | \big ) \\&\le 2 \big ( \left\| \Psi \right\| _{{\mathcal L}({\mathcal H})} +\left\| \Psi \right\| _{{\mathcal L}({\mathcal B})}\big )\left\| R(z)f(H)\right\| _{{\mathcal L}({\mathcal B},{\mathcal B}^*)}\left\| \psi \right\| ^2_{{\mathcal B}} . \end{aligned}$$

$\square$

(7.3c): We repeatedly apply Lemma B.1 and the ‘propagation observables’ $\Psi _1,\dots , \Psi _4\in {\mathcal L}({\mathcal H})\cap {\mathcal L}({\mathcal B})$ of the proof of Lemma 7.1 (or alternatively the version of (8.10) with ${\breve{H}}_a$ replaced with H). Note for example that $Q_1=Q_1(a,j)=\xi ^+_j( x) G_{d_j}$ in Lemma 7.1 (1) may be treated by a single application of Lemma B.1, which is obvious from the proof of Lemma 7.1 (1).

Appendix C: Formulas for Scattering and Channel Wave Matrices

A main goal of this “Appendix” is to derive the formula (8.2), i.e.

$$\begin{aligned} \widetilde{S}_{\beta \alpha }(\lambda )=(2\pi \textrm{i})^2f^2_1(\lambda ){{\breve{\gamma }}_b^+(\lambda _\beta )}J^*_\beta \big ( T^+_b\big )^* \delta (H-\lambda )T^-_a J_\alpha {{\breve{\gamma }}_a^-(\lambda _\alpha )}^*,\quad \lambda _\alpha = \lambda -\lambda ^\alpha , \end{aligned}$$

which is interpreted correctly in Sect. 8.2. As in Sect. 3 here $\lambda \in \Lambda$, where $\Lambda$ is a small open interval containing a fixed $\lambda _0\notin {\mathcal F}_{\mathrm p}(H)$. The derivation will be given using smoothness bounds from Sect. 7, arguments from Sects. 8.1 and 8.2, (A.8) and finally bounds from Sect. 5.2. We follow essentially the scheme of [DS, Appendix A] (for a similar issue, see for example the proof of [Ya4, Proposition 7.2]).

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

$$\begin{aligned} \delta _{\epsilon ,\beta }(\lambda )&= (2\textrm{i}\pi )^{-1} \big ( (k_\beta -\lambda -\textrm{i}\epsilon )^{-1}-(k_\beta -\lambda +\textrm{i}\epsilon )^{-1}\big )\\ {}&=\tfrac{\epsilon }{\pi }(k_\beta -\lambda +\textrm{i}\epsilon )^{-1}(k_\beta -\lambda -\textrm{i}\epsilon )^{-1};\quad k_\beta =p_b^2+\lambda ^\beta . \end{aligned}$$

The outset for our analysis is the following two formulas which can be derived as in [DS, Appendix A]. In the first formula g is any complex continuous function on ${\mathbb {R}}$ vanishing at infinity.

$$\begin{aligned} \begin{aligned}&{\widetilde{W}}^+_\beta (g1_{\Lambda })(k_\beta )\varphi \\&\quad =\lim _{\epsilon \rightarrow 0_+}\int _{\Lambda }\,g(\lambda ){f_2}(H) \big ( \Phi _b^+ +\textrm{i}R(\lambda -\textrm{i}\epsilon )T^+_b\big )J_\beta \breve{w}^+_b {f_1}(k_\beta )\delta _{\epsilon ,\beta }(\lambda ) \varphi \,{\mathrm d}\lambda , \end{aligned} \end{aligned}$$

(C.1)

$$\begin{aligned} \begin{aligned}&\langle \varphi _b,\widetilde{S}_{\beta \alpha }\varphi _a\rangle \\ {}&\quad =-2\pi \lim _{\epsilon \rightarrow 0_+}\int ^\infty _{-\infty }\,\langle \varphi _b,\delta _{\epsilon ,\beta }(\lambda ') \big ( {\widetilde{W}}^+_\beta \big )^*T^-_a J_\alpha \breve{w}^-_a {f_1}(k_\alpha )\delta _{\epsilon ,\alpha }(\lambda ') \varphi _a\rangle \,{\mathrm d}\lambda '. \end{aligned} \end{aligned}$$

(C.2)

To make the exposition self-contained we prove these formulas, which only (compared to Sect. 3) require the properties ${f_1}\prec {f_2}$ for some ${f_1},{f_2}\in C^\infty _{\mathrm c}(\Lambda )$.

Proof of C.1 and C.2

Recall from Sect. 3 that

$$\begin{aligned} {\widetilde{W}}^\pm _\beta =\mathop {\mathrm {s-lim}}\limits _{t\rightarrow \pm \infty }{\mathrm e}^{\textrm{i}tH}\tilde{J}_\beta ^\pm {\mathrm e}^{-\textrm{i}tk_\beta }{f_1}(k_\beta )=\mathop {\mathrm {s-lim}}\limits _{t\rightarrow \pm \infty }{{\mathrm e}^{\textrm{i}tH} \Phi _b^\pm {\mathrm e}^{-\textrm{i}t{\breve{H}}_b} J_\beta \breve{w}_b^{\pm }}{f_1}(k_\beta ). \end{aligned}$$

By the intertwining property

$$\begin{aligned} {\widetilde{W}}^\pm _\beta = {\widetilde{W}}^\pm _\beta {f_2}(k_\beta ) = {f_2}(H) {\widetilde{W}}^\pm _\beta . \end{aligned}$$

(C.3)

For any interval I and any $\varphi \in L^2(\textbf{X}_b)$ we compute using the vector-valued Plancherel formula

$$\begin{aligned}&\lim _{t\rightarrow +\infty }{\mathrm e}^{\textrm{i}tH}\tilde{J}_\beta ^+{\mathrm e}^{-\textrm{i}tk_\beta }{f_1}(k_\beta )1_{I}(k_\beta )\varphi =\lim _{\epsilon \rightarrow 0_+}2\epsilon \int _0^\infty {\mathrm e}^{-2\epsilon t} {\mathrm e}^{\textrm{i}tH}\tilde{J}_\beta ^+{\mathrm e}^{-\textrm{i}tk_\beta }(1_{I}f_1)(k_\beta )\varphi \, {\mathrm d}t\\&\quad =\lim _{\epsilon \rightarrow 0_+} \, \tfrac{\epsilon }{\pi } \int _{\mathbb {R}}R(\lambda -\textrm{i}\epsilon )\tilde{J}_\beta ^+ (k_\beta -\lambda -\textrm{i}\epsilon )^{-1}(1_{I}f_1)(k_\beta )\varphi \,{\mathrm d}\lambda \\&\quad =\lim _{\epsilon \rightarrow 0_+} \, \tfrac{\epsilon }{\pi } \int _I R(\lambda -\textrm{i}\epsilon )\tilde{J}_\beta ^+ (k_\beta -\lambda -\textrm{i}\epsilon )^{-1}f_1(k_\beta )\varphi \,{\mathrm d}\lambda \\&\qquad -\lim _{\epsilon \rightarrow 0_+} \, \tfrac{\epsilon }{\pi } \int _I R(\lambda -\textrm{i}\epsilon )\tilde{J}_\beta ^+ (k_\beta -\lambda -\textrm{i}\epsilon )^{-1}(1_{I^{\mathrm c}}f_1)(k_\beta )\varphi \,{\mathrm d}\lambda \\&\qquad +\lim _{\epsilon \rightarrow 0_+} \, \tfrac{\epsilon }{\pi } \int _{I^{\mathrm c}} R(\lambda -\textrm{i}\epsilon )\tilde{J}_\beta ^+ (k_\beta -\lambda -\textrm{i}\epsilon )^{-1}(1_{I}f_1)(k_\beta )\varphi \,{\mathrm d}\lambda . \end{aligned}$$

The two last terms vanish thanks to the Cauchy–Schwarz inequality and the fact that $\tilde{J}_\beta ^+ f_2(k_\beta )$ is bounded, using as well the (general) features

$$\begin{aligned}&\left\| \tfrac{\epsilon }{\pi }\int _I R(\lambda -\textrm{i}\epsilon )R(\lambda +\textrm{i}\epsilon )\psi \,{\mathrm d}\lambda \right\| \le \Vert \psi \Vert ;\quad \psi \in {\mathcal H}, \end{aligned}$$

(C.4a)

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0_+}\, \tfrac{\epsilon }{\pi }\int _{I^{\mathrm c}} (k_\beta -\lambda +\textrm{i}\epsilon )^{-1}(k_\beta -\lambda -\textrm{i}\epsilon )^{-1}\tilde{ \varphi }\,{\mathrm d}\lambda = 1_{I^{\mathrm c}}(k_\beta )\tilde{ \varphi };\quad \tilde{ \varphi }\in L^2(\textbf{X}_b). \end{aligned}$$

(C.4b)

Note that in our application $1_{I^{\mathrm c}}(k_\beta )\tilde{ \varphi }=1_{I^{\mathrm c}}(k_\beta )(1_{I^{\mathrm c}}f_1)(k_\beta )\varphi =0$.

Any continuous function g vanishing at infinity can be uniformly approximated by $g_m$, finite linear combinations of characteristic functions of intervals, and for each such function $g_m$ we can record (using (C.3)) that

$$\begin{aligned} {\widetilde{W}}^+_\beta g_m(k_\beta )\varphi =\lim _{\epsilon \rightarrow 0_+} \, \tfrac{\epsilon }{\pi } \int g_m(\lambda ){f_2}(H) R(\lambda -\textrm{i}\epsilon )\tilde{J}_\beta ^+ (k_\beta -\lambda -\textrm{i}\epsilon )^{-1} f_1(k_\beta )\varphi \,{\mathrm d}\lambda . \end{aligned}$$

Thanks to (C.4a) and (C.4b) it follows that uniformly in $\epsilon >0$

$$\begin{aligned}&\tfrac{\epsilon }{\pi } \int g_m(\lambda ){f_2}(H) R(\lambda -\textrm{i}\epsilon )\tilde{J}_\beta ^+ (k_\beta -\lambda -\textrm{i}\epsilon )^{-1} f_1(k_\beta )\varphi \,{\mathrm d}\lambda \\&\quad \rightarrow \tfrac{\epsilon }{\pi } \int g(\lambda ){f_2}(H) R(\lambda -\textrm{i}\epsilon )\tilde{J}_\beta ^+ (k_\beta -\lambda -\textrm{i}\epsilon )^{-1} f_1(k_\beta )\varphi \,{\mathrm d}\lambda . \end{aligned}$$

Hence we can interchange limits (cf. [Ru1, Theorem 7.11]) obtaining that

$$\begin{aligned} {\widetilde{W}}^+_\beta g(k_\beta )\varphi =\lim _{\epsilon \rightarrow 0_+} \tfrac{\epsilon }{\pi } \int g(\lambda ){f_2}(H) R(\lambda -\textrm{i}\epsilon )\tilde{J}_\beta ^+f_2(k_\beta ) (k_\beta -\lambda -\textrm{i}\epsilon )^{-1} f_1(k_\beta )\varphi \,{\mathrm d}\lambda , \end{aligned}$$

from which we conclude (C.1) by substituting

$$\begin{aligned}&R(\lambda -\textrm{i}\epsilon )\tilde{J}_\beta ^+ f_2(k_\beta )\\&\quad =R(\lambda -\textrm{i}\epsilon )\Phi _b^+J_\beta \breve{w}^+_b\\&\quad = \big ( \Phi _b^+ +\textrm{i}R(\lambda -\textrm{i}\epsilon )T^+_b\big )J_\beta \breve{w}^+_b (k_\beta -\lambda +\textrm{i}\epsilon )^{-1} . \end{aligned}$$

As for (C.2) we compute using (A.8)

$$\begin{aligned}&{\widetilde{W}}^-_\alpha =\mathop {\mathrm {s-lim}}\limits _{t\rightarrow -\infty }{\mathrm e}^{\textrm{i}tH}\tilde{J}_\alpha ^-{\mathrm e}^{-\textrm{i}t k_\alpha }{f_1}(k_\alpha ) -\mathop {\mathrm {s-lim}}\limits _{t\rightarrow +\infty }{\mathrm e}^{\textrm{i}tH}\tilde{J}_\alpha ^-{\mathrm e}^{-\textrm{i}tk_\alpha }{f_1}(k_\alpha )\\&\quad =-\mathop {\mathrm {s-lim}}\limits _{t\rightarrow +\infty } \big ( {\mathrm e}^{\textrm{i}tH}\tilde{J}_\alpha ^-{\mathrm e}^{-\textrm{i}tk_\alpha }-{\mathrm e}^{-\textrm{i}tH}\tilde{J}_\alpha ^-{\mathrm e}^{\textrm{i}tk_\alpha }\big ){f_1}(k_\alpha )\\&\quad =-\mathop {\mathrm {s-lim}}\limits _{t\rightarrow +\infty }\int ^t_{-t}{{\mathrm e}^{\textrm{i}sH}T^-_aJ_\alpha \breve{w}^-_a {\mathrm e}^{-\textrm{i}sk_\alpha }{f_1}(k_\alpha )}\,{\mathrm d}s\\&\quad =-\mathop {\mathrm {s-lim}}\limits _{\epsilon \rightarrow 0_+} \int _0^\infty \epsilon {\mathrm e}^{-\epsilon t} \Big ( \int ^t_{-t}{{\mathrm e}^{\textrm{i}sH}T^-_aJ_\alpha \breve{w}^-_a {\mathrm e}^{-\textrm{i}sk_\alpha }{f_1}(k_\alpha )}{\mathrm d}s\Big )\,{\mathrm d}t\\&\quad =-\mathop {\mathrm {s-lim}}\limits _{\epsilon \rightarrow 0_+} \int _{\mathbb {R}}{\mathrm e}^{-\epsilon |s|}{{\mathrm e}^{\textrm{i}sH}T^-_aJ_\alpha \breve{w}^-_a {\mathrm e}^{-\textrm{i}sk_\alpha }{f_1}(k_\alpha )}\,{\mathrm d}s. \end{aligned}$$

Then we multiply by $\big ( {\widetilde{W}}^+_\beta \big )^*$ (in agreement with the definition of $\widetilde{S}_{\beta \alpha }$) and invoke again the intertwining property and the vector-valued Plancherel theorem, calculating for any $\varphi \in L^2(\textbf{X}_a)$

$$\begin{aligned} {\widetilde{S}}_{\beta \alpha }\varphi&=-\lim _{\epsilon \rightarrow 0_+} \int _{\mathbb {R}}{\mathrm e}^{-2\epsilon |s|}{{\mathrm e}^{\textrm{i}sk_\beta } \big ( {\widetilde{W}}^+_\beta \big )^*T^-_aJ_\alpha \breve{w}^-_a {\mathrm e}^{-\textrm{i}sk_\alpha }{f_1}(k_\alpha )\varphi }\,{\mathrm d}s \\&=-2\pi \lim _{\epsilon \rightarrow 0_+} \int ^\infty _{-\infty }\,\delta _{\epsilon ,\beta }(\lambda ) \big ( {\widetilde{W}}^+_\beta \big )^*T^-_a J_\alpha \breve{w}^-_a {f_1}(k_\alpha )\delta _{\epsilon ,\alpha }(\lambda ) \varphi \,{\mathrm d}\lambda . \end{aligned}$$

Whence (C.2) is proven. $\square$

1.1 Appendix C.1.: Taking $\epsilon \rightarrow 0$ in (C.1) and (C.2)

Although (C.1) is valid for any $\varphi \in L^2(\textbf{X}_b)$ we take below $\varphi =\varphi _b\in L^2_s({\textbf{X}}_b)$ for $s>1/2$, and compute by taking $\epsilon \rightarrow 0$ using Sect. 5.2, here first done formally,

$$\begin{aligned} \begin{aligned} {\widetilde{W}}^+_\beta g(k_\beta )\varphi&=2\pi \int _{\Lambda }\,g(\lambda ) {f_2}(H)\delta (H-\lambda )T^+_b J_\beta \breve{w}^+_b {f_1}(k_\beta )\delta _{0,\beta }(\lambda ) \varphi \,{\mathrm d}\lambda ;\\ \delta _{0,\beta }(\lambda )&=(2\textrm{i}\pi )^{-1} \big ( (k_\beta -\lambda -\textrm{i}0)^{-1}-(k_\beta -\lambda +\textrm{i}0)^{-1}\big )=\gamma _{b,0}(\lambda _\beta )^* \gamma _{b,0}(\lambda _\beta ). \end{aligned} \end{aligned}$$

(We prove in Step I below the two resulting formulas (C.5a) and (C.5b).) See Sect. 2.3 for notation, and note that the above formula has the following precise meaning. We substitute, referring again to Sect. 2.3,

$$\begin{aligned} \breve{w}^+_b {f_1}(k_\beta )\delta _{0,\beta }(\lambda ) ={f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ). \end{aligned}$$

Then by combining the proof of Lemma 7.3, Remark 7.2 (iii) and Lemma 8.3 we conclude that the integral to the right is well-defined,

$$\begin{aligned} \begin{aligned}&{\widetilde{W}}^+_\beta g(k_\beta )\varphi \\&\quad =2\pi \int \,(g{f_1})(\lambda ) \big ( {f_2}(H)\delta (H-\lambda ) T^+_b J_\beta {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\big )\gamma _{b,0}(\lambda _\beta )\varphi \,{\mathrm d}\lambda , \end{aligned} \end{aligned}$$

(C.5a)

where the correct interpretation of the product in parentheses involves the ‘Q-operators’ as in Sect. 8.2 (more precisely obtained by expanding $T^+_b$ into terms on the form (C.8) given below). We call (C.5a) a ‘channel wave matrix’ representation (examined closer in Sect. 9). The below proof of (C.5a) reveals a different representation, more closely related to (C.1), however the formula (C.5a) is more useful for most of our purposes (with exceptions in Sects. 9.3–9.5). The alternative formula reads

$$\begin{aligned} \begin{aligned}&{\widetilde{W}}^+_\beta g(k_\beta )\varphi \\&\quad =\int \,(g{f_1})(\lambda ) {f_2}(H) \big ( \Phi _b^++\textrm{i}R(\lambda -\textrm{i}0) T^+_b \big )J_\beta {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta )\varphi \,{\mathrm d}\lambda . \end{aligned} \end{aligned}$$

(C.5b)

Although it is not relevant for (8.2) we remark that the integrands in (C.5a) and (C.5b) take value in ${\mathcal B}({\textbf{X}})^*$. This is thanks to Remark 7.2 iv) and Lemma 8.3, see the proof of Lemma 9.12 for an elaboration. However the below procedure is somewhat softer. As the reader will see we shall consider the integrands in (C.5a) and (C.5b) as taking value in $L^2_{-1}(\textbf{X})$ and, writing $\delta (H-\lambda )=(2\textrm{i}\pi )^{-1} \big ( R(\lambda +\textrm{i}0)-R(\lambda -\textrm{i}0)\big )$, interpretating the integrands in agreement with taking limits in ${\mathcal L}({\mathcal G}_b,L^2_{-1}(\textbf{X}))$ as

$$\begin{aligned} {f_2}(H)R(\lambda \pm \textrm{i}0)T^+_{b}J_\beta {\breve{\gamma }}_{b}^+(\lambda _\beta )^*=\mathop {\mathrm {s- w-lim}}\limits _{\epsilon \rightarrow 0_+}{f_2}(H)R(\lambda \pm \textrm{i}\epsilon )T^+_{b}J_\beta {\breve{\gamma }}_{b}^+(\lambda _\beta )^*. \end{aligned}$$

With this interpretation the integrand in (C.5a) is better. Thanks to Remark 7.2 (iii) and Lemma 8.3 it obviously takes value in $L^2_{-s}(\textbf{X})$, not only for $s=1$, but for any $s>1/2$.

I. We prove (C.5a) and (C.5b). For (C.5a) we compute the ‘$\epsilon \rightarrow 0$’–limit in (C.1) in the precise meaning of taking limit in the weak topology of $L^2_{-1}(\textbf{X})$. There are two assertions that need to be checked:

$$\begin{aligned} \textrm{i}\lim _{\epsilon \rightarrow 0}&\int _{\Lambda }\,g(\lambda ) {f_2}(H) \big ( R(\lambda -\textrm{i}\epsilon )- R(\lambda +\textrm{i}\epsilon )\big )T^+_b J_\beta \breve{w}^+_b {f_1}(k_\beta )\delta _{\epsilon ,\beta }(\lambda ) \varphi \,{\mathrm d}\lambda \nonumber \\&=\textrm{i}\lim _{\epsilon \rightarrow 0} \int _{\Lambda }\,g(\lambda ) {f_2}(H) \big ( R(\lambda -\textrm{i}\epsilon )- R(\lambda +\textrm{i}\epsilon )\big )T^+_b J_\beta {f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ) \varphi \,{\mathrm d}\lambda \nonumber \\&=2\pi \int _{\Lambda }\,g(\lambda ) {f_2}(H)\delta (H-\lambda )T^+_b J_\beta {f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ) \varphi \,{\mathrm d}\lambda , \end{aligned}$$

(C.6a)

and

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}&\int _{\Lambda }\,g(\lambda ) {f_2}(H) \big ( \Phi _b^+ +\textrm{i}R(\lambda +\textrm{i}\epsilon )T^+_b\big )J_\beta \breve{w}^+_b {f_1}(k_\beta )\delta _{\epsilon ,\beta }(\lambda ) \varphi \,{\mathrm d}\lambda \nonumber \\&=\lim _{\epsilon \rightarrow 0}\int _{\Lambda }\,g(\lambda ) {f_2}(H) \big ( \Phi _b^+ +\textrm{i}R(\lambda +\textrm{i}\epsilon )T^+_b\big ) J_\beta {f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ) \varphi \,{\mathrm d}\lambda \nonumber \\&=\int _{\Lambda }\,g(\lambda ) {f_2}(H) \big ( \Phi _b^+ +\textrm{i}R(\lambda +\textrm{i}0)T^+_b\big ) J_\beta {f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ) \varphi \,{\mathrm d}\lambda \nonumber \\&=0. \end{aligned}$$

(C.6b)

For (C.6a) we write with $\delta _{\epsilon ,\lambda }(\lambda '):=\pi ^{-1}\epsilon / \big ( (\lambda -\lambda ')^2+\epsilon ^2\big )$ for any fixed $\lambda \in {\mathbb {R}}$ and $\epsilon >0$, and by using Lemma 2.3,

$$\begin{aligned} \breve{w}^+_b {f_1}(k_\beta )\delta _{\epsilon ,\beta }(\lambda ) \varphi =\int ^\infty _{\lambda ^\beta } {f_1}(\lambda ')\delta _{\epsilon ,\lambda }(\lambda '){\breve{\gamma }}^+_{b}(\lambda '-\lambda ^\beta )^* \gamma _{b,0}(\lambda '-\lambda ^\beta )\varphi \,{\mathrm d}\lambda ', \end{aligned}$$

(C.7)

substitute and replace by the limit (when taking $\epsilon \rightarrow 0$). This is doable thanks to Remark 7.2 (iii), Lemma 8.3 and the continuity of $\gamma _{b,0}(\lambda '_\beta )\varphi$ justifying the first equality, and the second equality is a consequence of Remark 7.2 (iii) and Lemma 8.3 too. In addition we used the fact that $T^+_b$ is a finite sum of terms expressed as

$$\begin{aligned} {f_2}(H)Q^+(b,l)^*B_+Q^+(b,l){f_2}({\breve{H}}_b)\text { with } B_+\text { bounded}, \end{aligned}$$

(C.8)

cf. Sect. 8.2. We shall use that $T^+_b$ is expanded this way again below.

For (C.6b) we will invoke Theorem 5.1 and Corollary 5.2. 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 equal

$$\begin{aligned} \begin{aligned}&\lim _{\epsilon \rightarrow 0}\int _{\Lambda }\,g(\lambda ) {f_2}(H) \big ( \Phi _b^+ +\textrm{i}R(\lambda +\textrm{i}\epsilon )T^+_b\big ) J_\beta {f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ) \varphi \,{\mathrm d}\lambda \\&\quad =\int _{\Lambda }\,g(\lambda ) {f_2}(H) \big ( \Phi _b^+ +\textrm{i}R(\lambda +\textrm{i}0)T^+_b\big )J_\beta {f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ) \varphi \,{\mathrm d}\lambda . \end{aligned} \end{aligned}$$

(C.9)

To see this, considering only the ‘difficult term’

$$\begin{aligned} \int _{\Lambda }\,g(\lambda ) {f_2}(H)R(\lambda +\textrm{i}\epsilon )T^+_b J_\beta \breve{w}^+_b {f_1}(k_\beta )\delta _{\epsilon ,\beta }(\lambda ) \varphi \,{\mathrm d}\lambda , \end{aligned}$$

we insert $I=\chi ^2_+( 2B/\epsilon _0)+\chi ^2_-( 2B/\epsilon _0)$ to the right of the factor $R(\lambda +\textrm{i}\epsilon )$. Using the factors of $A_{1+}$ and commutation the contribution from the second term $\chi ^2_-( 2B/\epsilon _0)$ allows the insertion of the weight $\langle x\rangle ^{-s}$ for some $s>1/2$ to the right of $R(\lambda +\textrm{i}\epsilon )$ and we can use (5.5a) and (C.7) to compute the ‘$\epsilon \rightarrow 0$’–limit for this term in agreement with (C.9). Note for this argument that $\langle x\rangle ^{s}\chi ^2_-( 2B/\epsilon _0){f_2}(H)Q^+(b,l)^*B_+$ is bounded. As for the contribution from the first term $\chi ^2_+( 2B/\epsilon _0)$ we write

$$\begin{aligned} R(\lambda +\textrm{i}\epsilon )\chi ^2_+( 2B/\epsilon _0)=R(\lambda +\textrm{i}\epsilon )\chi ^2_+( 2B/\epsilon _0)\langle x\rangle ^{-s}\langle x\rangle ^{s};\quad s>1/2-\mu /2. \end{aligned}$$

Then by Theorem 5.1 and Corollary 5.2

$$\begin{aligned} \langle x\rangle ^{-1}{f_2}(H)R(\lambda +\textrm{i}\epsilon )\chi ^2_+( 2B/\epsilon _0)\langle x\rangle ^{-s}\text { is uniformly bounded}, \end{aligned}$$

and there exists

$$\begin{aligned}&\Lambda \ni \lambda \rightarrow \langle x\rangle ^{-1}{f_2}(H)R(\lambda +\textrm{i}0)\chi ^2_+( 2B/\epsilon _0)\langle x\rangle ^{-s}\\&\quad =\lim _{\epsilon \rightarrow 0} \,\langle x\rangle ^{-1}{f_2}(H)R(\lambda +\textrm{i}\epsilon )\chi ^2_+( 2B/\epsilon _0)\langle x\rangle ^{-s}, \end{aligned}$$

constituting a continuous ${\mathcal L}({\mathcal H})$-valued function (the limit is taken uniformly in $\lambda$). Consequently, using again (C.7), it suffices to check the existence and continuity of the ${\mathcal L}({\mathcal G}_b,{\mathcal H})$-valued function

$$\begin{aligned} {\mathbb {R}}\ni \lambda \rightarrow {f_1}(\lambda )\langle x\rangle ^{s}T^+_b J_\beta {\breve{\gamma }}_{b}^+(\lambda _\beta )^*. \end{aligned}$$

Expanding $T^+_b$ as above, we can use Lemma 8.3 to combine for each term the factor $Q^+(b,l)$ with the factor $J_\beta {\breve{\gamma }}_{b}^+(\lambda _\beta )^*$ (as done above). Each remaining factor of a ‘Q-operator’ (more precisely $Q^+(b,l)^*$ appearing to the left) contributes by a factor ${\mathcal O}(r^{\rho _1/2-\delta /2})$, cf. Remark 7.2 (i). Whence we are left with checking that $s+\rho _1/2-\delta /2\le 0$ is possible. By (3.6) it suffices to produce an $s>1/2-\mu /2$ such that $s+ 1/2-2/(2+\mu )\le 0$, 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 (C.6b) is given by the expressions (C.9).

It remains to show that the right-hand side of (C.9) vanishes. For that we note that the formula remains valid upon replacing the factors of $T^+_b$ by $\chi _\rho T^+_b$, where $\chi _\rho =\chi _-(r/\rho )$ for $\rho >1$. By the above arguments one can easily check that the ‘$\epsilon \rightarrow 0$’–limit in the modified versions of (C.9) is taken uniformly in $\rho >1$. Whence the integrand to the right can be computed (in the weak topology of $L^2_{-1}(\textbf{X})$) as

$$\begin{aligned}&g(\lambda ){f_2}(H) \big ( \Phi _b^+ - R(\lambda +\textrm{i}0)(H-\lambda )\Phi ^+_b\big )J_\beta {f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ) \varphi \\&\quad =\lim _{\rho \rightarrow \infty } g(\lambda ){f_2}(H) \big ( \Phi _b^+ - R(\lambda +\textrm{i}0)\chi _\rho (H-\lambda )\Phi ^+_b\big )J_\beta {f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ) \varphi \\&\quad =\lim _{\rho \rightarrow \infty } g(\lambda ) {f_2}(H)R(\lambda +\textrm{i}0)[H,\chi _\rho ]\Phi ^+_b J_\beta {f_1}(\lambda ) {\breve{\gamma }}_{b}^+(\lambda _\beta )^*\gamma _{b,0}(\lambda _\beta ) \varphi \\&\quad =0\quad (\text {by the first part of Corollary 5.3}), \end{aligned}$$

proving (C.6b). We have shown (C.5a).

The second formula (C.5b) (interpretated as an identity in $L^2_{-1}(\textbf{X})$) is a consequence of the above proof of (C.5a).

II. We prove (8.2). It is tempting to substitute the adjoint expression of (C.5a) with $g(\lambda )=\delta _{\epsilon ,\lambda '}(\lambda )$ into (C.2) and then interchange the order of the two integrations. This is doable but requires of course some modification since the meaning of the right-hand side of (C.5a) is a vector in $L^2_{-s}(\textbf{X})$ for $s=1$, or in fact for any $s>1/2$ (but unlikely any smaller), and there is no obvious way of controlling the vector $T^-_a J_\alpha \breve{w}^-_a {f_1}(k_\alpha )\delta _{\epsilon ,\alpha }(\lambda ') \varphi _a$ uniformly in $\epsilon >0$ in $L^2_{s}(\textbf{X})$ for some $s>1/2$. However, writing $T^-_a$ as a finite sum of terms on the form

$$\begin{aligned} {f_2}(H)Q^-(a,k)^*B_-Q^-(a,k){f_2}({\breve{H}}_a)\text { with } B_-\text { bounded}, \end{aligned}$$

(C.10)

we can introduce the modification, say denoted by $T^-_{a,\rho }$, 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

$$\begin{aligned} \int ^\infty _{-\infty }\,\delta _{\epsilon ,\lambda '}(\lambda )\delta _{\epsilon ,\alpha }(\lambda ')\,{\mathrm d}\lambda '=\delta _{2\epsilon ,\alpha }(\lambda ), \end{aligned}$$

we then obtain

$$\begin{aligned}&(2\pi \textrm{i})^{-2}\langle \varphi _b,\widetilde{S}_{\beta \alpha }\varphi _a\rangle =\lim _{\epsilon \rightarrow 0}\,\lim _{\rho \rightarrow \infty }\\&\quad \int _{\Lambda }\,{f_1}(\lambda ) \langle \gamma _{b,0}(\lambda _\beta )\varphi _b,{\breve{\gamma }}^+_{b}(\lambda _\beta )J^*_\beta (T^+_b)^*\delta (H-\lambda )T^-_{a,\rho } J_\alpha \breve{w}^-_a {f_1}(k_\alpha ) \delta _{2\epsilon ,\alpha }(\lambda ) \varphi _a\rangle \,{\mathrm d}\lambda , \end{aligned}$$

valid for any $\varphi _a\in L^2_s({\textbf{X}}_a)$ and $\varphi _b\in L^2_s({\textbf{X}}_b)$ for $s>1/2$.

With an analogue of (C.7) (with $\alpha$ rather than $\beta$) we compute

$$\begin{aligned}&\lim _{\epsilon \rightarrow 0}\int _{\Lambda }\,{f_1}(\lambda ) \langle \gamma _{b,0}(\lambda _\beta )\varphi _b,{\breve{\gamma }}^+_{b}(\lambda _\beta )J^*_\beta (T^+_b)^*\delta (H-\lambda )T^-_{a,\rho } J_\alpha \breve{w}^-_a {f_1}(k_\alpha ) \delta _{2\epsilon ,\alpha }(\lambda ) \varphi _a\rangle \,{\mathrm d}\lambda \\&\quad =\int _{\Lambda }\,{f_1}(\lambda ) \langle \gamma _{b,0}(\lambda _\beta )\varphi _b,{\breve{\gamma }}^+_{b}(\lambda _\beta )J^*_\beta (T^+_b)^*\delta (H-\lambda )T^-_{\alpha ,\rho } J_\alpha {f_1}(\lambda ) {\breve{\gamma }}_{a}^-(\lambda _\alpha )^*\gamma _{a,0}(\lambda _\alpha ) \varphi _a\rangle \,{\mathrm d}\lambda . \end{aligned}$$

Here the limit is taken uniformly in $\rho >1$. Due to these features we can interchange limits above and conclude that

$$\begin{aligned}&(2\pi \textrm{i})^{-2}\langle \varphi _b,\widetilde{S}_{\beta \alpha }\varphi _a\rangle =\lim _{\rho \rightarrow \infty }\\&\qquad \int _{\Lambda }\,{f_1}(\lambda ) \langle \gamma _{b,0}(\lambda _\beta )\varphi _b,{\breve{\gamma }}^+_{b}(\lambda _\beta )J^*_\beta (T^+_b)^*\delta (H-\lambda )T^-_{a,\rho } J_\alpha {f_1}(\lambda ) {\breve{\gamma }}_{a}^-(\lambda _\alpha )^*\gamma _{a,0}(\lambda _\alpha ) \varphi _a\rangle \,{\mathrm d}\lambda \\&\quad =\int _{\Lambda }\,{f_1}(\lambda ) \langle \gamma _{b,0}(\lambda _\beta )\varphi _b,{\breve{\gamma }}^+_{b}(\lambda _\beta )J^*_\beta (T^+_b)^*\delta (H-\lambda )T^-_a J_\alpha {f_1}(\lambda ) {\breve{\gamma }}_{a}^-(\lambda _\alpha )^*\gamma _{a,0}(\lambda _\alpha ) \varphi _a\rangle \,{\mathrm d}\lambda , \end{aligned}$$

showing (8.2). Note that indeed the right interpretation of the formula requires the expansion into a (finite) sum of expressions arising upon substituting for $T^+_b$ and $T^-_a$ sums of operators on the form (C.8) and (C.10), respectively. $\square$

Remark C.1

Note the following analogues of (C.5a) and (8.2), cf. (3.8a) and (3.8b),

$$\begin{aligned} {\widetilde{W}}^-_\beta g(k_\beta )\varphi&=-2\pi \int \,(g{f_1})(\lambda ) \big ( {f_2}(H)\delta (H-\lambda ) T^-_b J_\beta {\breve{\gamma }}_{b}^-(\lambda _\beta )^*\big )\gamma _{b,0}(\lambda _\beta )\varphi \,{\mathrm d}\lambda , \end{aligned}$$

(C.11a)

$$\begin{aligned}&\begin{aligned} 16\lambda _\beta \lambda _\alpha f^2_1(\lambda )&m_b(\pm \hat{\xi }_b)^2m_b(\pm {\hat{\xi }}_a)^2\,\delta _{ \beta \alpha }\\ {}&\quad =-(2\pi \textrm{i})^2f^2_1(\lambda ){{\breve{\gamma }}_b^\pm (\lambda _\beta )}J^*_\beta \big ( T^\pm _b\big )^*\delta (H-\lambda )T^\pm _a J_\alpha {{\breve{\gamma }}_a^\pm (\lambda _\alpha )}^*. \end{aligned} \end{aligned}$$

(C.11b)

The quantity to the left in (c.11b) (for each sign) is meant to be an operator in ${\mathcal L} \big ( {\mathcal G}_a,{\mathcal G}_b\big )$; the use of the Kronecker symbol $\delta _{ \beta \alpha }$ specifies that it vanishes unless $\beta =\alpha$. The operator can be considered as the fiber of $F_\beta \big ( {\widetilde{W}}_\beta ^\pm \big )^*{\widetilde{W}}^\pm _\alpha F_\alpha ^{-1}$ at energy $\lambda$, invoking the orthogonality of channels. Whence the formula (c.11b) results by mimicking the above procedure for showing (8.2). As before its correct interpretation requires the expansion into a sum of expressions arising upon substituting for $T^\pm _b$ and $T^\pm _a$ sums of operators on the form (C.8) or (C.10).