# Stationary completeness: the $N$-body short-range case

E. Skibsted  Institut for Matematik
Aarhus Universitet
Ny Munkegade 8000 Aarhus C, Denmark<EMAIL_ADDRESS>
###### Abstract.

For a general class of $N$-body Schrödinger operators with short-range pair-
potentials the wave and scattering matrices as well as the restricted wave
operators are all defined at any non-threshold energy. In fact this holds
without imposing any a priori decay condition on channel eigenstates and even
for models including long-range potentials of Dereziński-Enss type. In this
paper we improve for short-range models on the known _weak continuity_
statements in that we show that all non-threshold energies are _stationary
complete_ , resolving in this case a conjecture from [Sk1]. A consequence is
that the above scattering quantities depend _strongly continuously_ on the
energy parameter at all non-threshold energies (whence not only almost
everywhere as previously demonstrated). Another consequence is that the
scattering matrix is unitary at any such energy. As a side result we give a
new and purely stationary proof of asymptotic completeness for $N$-body short-
range systems.

Supported by DFF grant nr. 8021-00084B

Keywords: $N$-body Schrödinger operators: asymptotic completeness; stationary
scattering theory; scattering and wave matrices; minimum generalized
eigenfunctions.

Mathematics Subject Classification 2010: 81Q10, 35A01, 35P05.

###### Contents

  1. 1 Introduction
    1. 1.1 The standard $N$-body short-range model, results
    2. 1.2 Comparison with the literature, content of the paper
  2. 2 Preliminaries
    1. 2.1 $N$-body Hamiltonians, assumptions and notation
      1. 2.1.1 $N$-body short-range Hamiltonians
      2. 2.1.2 General notation
    2. 2.2 Basic stationary $N$-body theory
  3. 3 Yafaev type constructions
    1. 3.1 Homogeneous Yafaev type functions for a general $a_{0}$
    2. 3.2 Geometric considerations for $a_{0}=a_{\max}$
    3. 3.3 Geometric considerations for $a_{0}\neq a_{\max}$
  4. 4 $N$-body short-range scattering theory
  5. 5 A partition of unity
  6. 6 Non-threshold analysis
  7. 7 A squeezing argument
  8. 8 Proof of stationary completeness
  9. A Strong resolvent bounds, proof of Lemma 5.2

##  1\. Introduction

We address a recent conjecture of [Sk1] for the stationary scattering theory
of $N$-body systems of quantum particles interacting with long-range
potentials of Dereziński-Enss type long-range pair-potentials. More precisely
we focus on the conjecture in the short-range case, i.e. for models defined by
short-range pair-potentials only. In this setting we will show that indeed
_all_ non-threshold energies are _stationary complete_ , resolving the
conjecture to the affirmative. The paper depends on several results of [Sk1].
Although we do give an account of the most relevant parts of [Sk1], the
present paper contains proofs for which the reader presumably would benefit
from independent parallel consultance of [Sk1].

###  1.1. The standard $N$-body short-range model, results

Consider a system of $N$ charged particles, $i=1,\dots,N$, of dimension $n\geq
1$ interacting by pair-potentials

$V_{ij}(x_{i}-x_{j})={\mathcal{O}}(\lvert
x_{i}-x_{j}\rvert^{-1-\epsilon}),\quad\epsilon>0.$

Here $x_{j}\in{\mathbb{R}}^{n}$ is the position of particle $j$, and denoting
$m_{j}$ the corresponding mass, the Hamiltonian reads

$H=-\sum_{j=1}^{N}\frac{1}{2m_{j}}\Delta_{x_{j}}+\sum_{1\leq i<j\leq
N}V_{ij}(x_{i}-x_{j}).$

This Hamiltonian $H$ is regarded as a self-adjoint operator on
$L^{2}({\mathbf{X}})$, where ${\mathbf{X}}$ is the $(N-1)n$ dimensional real
vector space $\\{x=(x_{1},\dots,x_{N})\mid\sum_{j=1}^{N}m_{j}x_{j}=0\\}$. Let
${\mathcal{A}}$ denote the set of all cluster decompositions of the system.
The notation $a_{\max}$ and $a_{\min}$ refers to the $1$-cluster and
$N$-cluster decompositions, respectively. Let for $a\in{\mathcal{A}}$ the
notation $\\#a$ denote the number of clusters in $a$. For
$i,j\in\\{1,\dots,N\\}$, $i<j$, we denote by $(ij)$ the $(N-1)$-cluster
decomposition given by letting $C=\\{i,j\\}$ form a cluster and the other
particles form singletons. We write $(ij)\leq a$ if $i$ and $j$ belong to the
same cluster in $a$. More general, we write $b\leq a$ if each cluster of $b$
is a subset of a cluster of $a$. If $a$ is a $k$-cluster decomposition,
$a=(C_{1},\dots,C_{k})$, we let

${\mathbf{X}}^{a}=\big{\\{}x\in{\mathbf{X}}\mid\sum_{l\in
C_{j}}m_{l}x_{l}=0,j=1,\dots,k\big{\\}}={\mathbf{X}}^{C_{1}}\oplus\cdots\oplus{\mathbf{X}}^{C_{k}},$

and

${\mathbf{X}}_{a}=\big{\\{}x\in{\mathbf{X}}\mid x_{i}=x_{j}\mbox{ if }i,j\in
C_{m}\mbox{ for some }m\in\\{1,\dots,k\\}\big{\\}}.$

Note that $b\leq a\Leftrightarrow{\mathbf{X}}^{b}\subseteq{\mathbf{X}}^{a}$,
and that the subspaces ${\mathbf{X}}^{a}$ and ${\mathbf{X}}_{a}$ define an
orthogonal decomposition of ${\mathbf{X}}$ equipped with the quadratic form
$q(x)=\Sigma_{j}\,2m_{j}|x_{j}|^{2},\,x\in{{\mathbf{X}}}$. Consequently any
$x\in{\mathbf{X}}$ decomposes orthogonally as $x=x^{a}+x_{a}$ with
$x^{a}=\pi^{a}x\in{\mathbf{X}}^{a}$ and $x_{a}=\pi_{a}x\in{\mathbf{X}}_{a}$.

With these notations the $N$-body Schrödinger operator takes the form
$H=H_{0}+V$, where $H_{0}=p^{2}$ is (minus) the Laplace-Beltrami operator on
the Euclidean space $({\mathbf{X}},q)$ and
$V=V(x)=\sum_{b=(ij)\in{\mathcal{A}}}V_{b}(x^{b})$ with
$V_{b}(x^{b})=V_{ij}(x_{i}-x_{j})$ for the $(N-1)$-cluster decomposition
$b=(ij)$. Note for example that

$x^{(12)}=\big{(}\tfrac{m_{2}}{m_{1}+m_{2}}(x_{1}-x_{2}),-\tfrac{m_{1}}{m_{1}+m_{2}}(x_{1}-x_{2}),0,\dots,0\big{)}.$

More generally for any cluster decomposition $a\in{\mathcal{A}}$ we introduce
a Hamiltonian $H^{a}$ as follows. For $a=a_{\min}$ we define $H^{a_{\min}}=0$
on $\mathcal{H}^{a_{\min}}:=\mathbb{C}.$ For $a\neq a_{\min}$ we introduce the
potential

$V^{a}(x^{a})=\sum_{b=(ij)\leq a}V_{b}(x^{b});\quad x^{a}\in{\mathbf{X}}^{a}.$

Then

$H^{a}=-\Delta_{x^{a}}+V^{a}(x^{a})=(p^{a})^{2}+V^{a}\ \ \text{on
}\mathcal{H}^{a}=L^{2}({\mathbf{X}}^{a}).$

A channel $\alpha$ is by definition given as
$\alpha=(a,\lambda^{\alpha},u^{\alpha})$, where
$a\in{\mathcal{A}}^{\prime}={\mathcal{A}}\setminus\\{a_{\max}\\}$ and
$u^{\alpha}\in\mathcal{H}^{a}$ obeys $\lVert u^{\alpha}\rVert=1$ and
$(H^{a}-\lambda^{\alpha})u^{\alpha}=0$ for a real number $\lambda^{\alpha}$,
named a threshold. The set of thresholds is denoted ${\mathcal{T}}(H)$, and
including the eigenvalues of $H$ we introduce
${\mathcal{T}}_{{\mathrm{p}}}(H)=\sigma_{{\mathrm{pp}}}(H)\cup{\mathcal{T}}(H)$.
For any $a\in{\mathcal{A}}^{\prime}$ the intercluster potential is by
definition

$\displaystyle I_{a}(x)=\sum_{b=(ij)\not\leq a}V_{b}(x^{b}).$

Next we recall the short-range channel wave operators

$W_{\alpha}^{\pm}=\operatorname*{s-lim}_{t\to\pm\infty}{\mathrm{e}}^{\mathrm{i}tH}\big{(}{u^{\alpha}}\otimes{\mathrm{e}}^{-\mathrm{i}tp_{a}^{2}}(\cdot)\big{)}.$
(1.1)

There exist by now many proofs of the existence of the channel wave operators
and in fact also of their completeness [GM, SS, Gr, En, De, Ta, HS, Ya2] (the
list is not complete), see also the monographs [DG, Is5]. Several of the known
proofs are manifestly time-dependent. The work [Ya2] appears in this regard as
an exception, although Yafaev’s approach is not completely stationary.

Let $I^{\alpha}=(\lambda^{\alpha},\infty)$ and
$k_{\alpha}=p_{a}^{2}+\lambda^{\alpha}$. By the intertwining property
$HW_{\alpha,{\rm atom}}^{\pm}\supseteq W_{\alpha,{\rm atom}}^{\pm}k_{\alpha}$
and the fact that $k_{\alpha}$ is diagonalized by the unitary map
$F_{\alpha}:L^{2}(\mathbf{X}_{a})\to L^{2}(I^{\alpha};{\mathcal{G}}_{a})$,
${\mathcal{G}}_{a}=L^{2}(\mathbf{S}_{a})$,
$\mathbf{S}_{a}=\mathbf{X}_{a}\cap{\mathbb{S}}^{d_{a}-1}$ with
$d_{a}=\dim\mathbf{X}_{a}$, given by

$\displaystyle(F_{\alpha}\varphi)(\lambda,\omega)=(2\pi)^{-d_{a}/2}2^{-1/2}\lambda_{\alpha}^{(d_{a}-2)/4}\int{\mathrm{e}}^{-\mathrm{i}\lambda^{1/2}_{\alpha}\omega\cdot
x_{a}}\varphi(x_{a})\,{\mathrm{d}}x_{a},\quad\lambda_{\alpha}=\lambda-\lambda^{\alpha},$

we can for any two given channels $\alpha$ and $\beta$ write

$\hat{S}_{\beta\alpha}:=F_{\beta}(W_{\beta}^{+})^{*}W_{\alpha}^{-}F_{\alpha}^{-1}=\int^{\oplus}_{I_{\beta\alpha}}S_{\beta\alpha}(\lambda)\,{\mathrm{d}}\lambda,\quad
I_{\beta\alpha}=I^{\beta}\cap I^{\alpha}.$

The fiber operator
$S_{\beta\alpha}(\lambda)\in{\mathcal{L}}({\mathcal{G}}_{a},{\mathcal{G}}_{b})$
is from an abstract point of view a priori defined only for a.e. $\lambda\in
I_{\beta\alpha}$, however it is known to exist away from the set of thresholds
(see Theorem 1.1 stated below). It is the $\beta\alpha$-entry of the
scattering matrix
$S(\lambda)=\big{(}S_{\beta\alpha}(\lambda)\big{)}_{\beta\alpha}$ (here the
‘dimension’ of the ‘matrix’ $S(\lambda)$ is $\lambda$-independent on any
interval not containing thresholds).

Introducing the standard notation for weighted spaces

$\displaystyle L_{s}^{2}(\mathbf{X})=\langle
x\rangle^{-s}L^{2}(\mathbf{X}),\quad s\in{\mathbb{R}},\quad\langle
x\rangle=\big{(}1+\lvert x\rvert^{2}\big{)}^{1/2},$

we recall the following result.

######  Theorem 1.1 ([Sk1]).

  1. 1)

Let $\alpha$ be a given channel $\alpha=(a,\lambda^{\alpha},u^{\alpha})$,
$f:I^{\alpha}\to{\mathbb{C}}$ be continuous and compactly supported away from
${\mathcal{T}}_{{\mathrm{p}}}(H)$, and let $s>1/2$. For any $\varphi\in
L^{2}(\mathbf{X}_{a})$

$\displaystyle
W^{\pm}_{\alpha}f(k_{\alpha})\varphi=\int_{I^{\alpha}\setminus{\mathcal{T}}_{\mathrm{p}}(H)}\,\,f(\lambda)W^{\pm}_{\alpha}(\lambda)(F_{\alpha}\varphi)(\lambda,\cdot)\,{\mathrm{d}}\lambda\in
L_{-s}^{2}({\mathbf{X}}),$ (1.2)

where the ‘wave matrices’
$W^{\pm}_{\alpha}(\lambda)\in{\mathcal{L}}\big{(}{\mathcal{G}}_{a},L_{-s}^{2}({\mathbf{X}})\big{)}$
with a strongly continuous dependence on $\lambda$. In particular for
$\varphi\in L_{s}^{2}(\mathbf{X}_{b})$ the integrand is a continuous compactly
supported $L_{-s}^{2}({\mathbf{X}})$-valued function. In general the integral
has the weak interpretation of an integral of a measurable
$L_{-s}^{2}({\mathbf{X}})$-valued function.

  2. 2)

The operator-valued function
$I_{\beta\alpha}\setminus{\mathcal{T}}_{{\mathrm{p}}}(H)\ni\lambda\to
S_{\beta\alpha}(\cdot)$ is weakly continuous.

Using the notation $L^{2}_{\infty}(\mathbf{X})=\cap_{s}L^{2}_{s}(\mathbf{X})$
the delta-function of $H$ at $\lambda$ is given by

$\delta(H-\lambda)=\pi^{-1}\operatorname{Im}{(H-\lambda-\mathrm{i}0)^{-1}}\text{
as a quadratic form on }L^{2}_{\infty}(\mathbf{X}).$

######  Definition 1.2.

An energy
$\lambda\in{\mathcal{E}}:=(\min{\mathcal{T}}(H),\infty)\setminus{\mathcal{T}}_{\mathrm{p}}(H)$
is stationary complete for $H$ if

$\forall\psi\in
L^{2}_{\infty}(\mathbf{X}):\,\,\sum_{\lambda^{\beta}<\lambda}\,\lVert
W^{\pm}_{\beta}(\lambda)^{*}\psi\rVert^{2}=\langle\psi,\delta(H-\lambda){\psi}\rangle.$
(1.3)

In this paper we take the existence of the channel wave operators for granted,
although this property is an easy consequence of our method (see Remark 4.6
ii). With this in place it is readily seen that their completeness is
equivalent to the property that Lebesgue almost all energies are stationary
complete (see [Sk1]). In particular we obtain asymptototic completeness as a
consequence of the following main result of the paper (stated here for the
standard $N$-body short-range model only).

######  Theorem 1.3.

All $\lambda\in{\mathcal{E}}$ are stationary complete for the Hamiltonian $H$.

Note that while (1.2) may be taken as a definition (although implicit) of the
wave matrices, (1.3) is a non-trivial derived property. The notion of
stationary complete energies was introduced and studied in [Sk1] in fact for a
model including long-range potentials of the type studied previously in [De,
En], in particular for the atomic $N$-body Hamiltonian. Even in the long-range
setting Lebesgue almost all non-threshold energies are stationary complete,
however it was left as an open problem to show stationary completeness at
fixed energy in that case. For the atomic $N$-body model the problem is still
open for $N\geq 4$, however recently solved for a natural class of long-range
potentials for $N=3$ [Sk2].

One can regard (1.3) as an ‘on-shell Parseval formula’. By integration it
yields asymptotic completeness, whence providing an alternative to other
proofs. Moreover there are immediate consequences for the discussed scattering
quantities considered as operator-valued functions on ${\mathcal{E}}$
(recalled for the generalized $N$-body short-range model in Section 4):

  1. I)

The scattering matrix $S(\cdot)$ is a strongly continuous unitary operator
determined uniquely by asymptotics of minimum generalized eigenfunctions (at
any given energy). The latter are taken from the ranges of the wave matrices
(at this energy).

  2. II)

The restricted wave operators $W^{\pm}_{\alpha}(\cdot)^{*}$ are strongly
continuous.

  3. III)

The scattering matrix links the incoming and outgoing wave matrices,

$W^{-}_{\alpha}(\lambda)=\sum_{\lambda^{\beta}<\lambda}W^{+}_{\beta}(\lambda)S_{\beta\alpha}(\lambda);\quad\lambda\in{\mathcal{E}},\,\lambda^{\alpha}<\lambda.$

Stationary scattering theory appears so far somewhat poorly developed (see a
discussion in Subsection 1.2). In particular, to our knowledge, the results
I–III are all new. The heart of the matter is (1.3), the proof of which relies
on a characterization of this property from [Sk1] (recalled in (4.7)). In
particular we derive the top-order asymptotics of any vector on the form
$(H-\lambda-\mathrm{i}0)^{-1}\psi$, $\psi\in L^{2}_{\infty}(\mathbf{X})$,
yielding (1.3). The asymptotics is also valid in the $N$-body setting of
[Sk1], however there only proven away from a Lebesgue null-set.

###  1.2. Comparison with the literature, content of the paper

For $3$-body systems there is a fairly big literature on stationary scattering
theory both on the mathematical side and the physics side. This is to a large
extent based on the Faddeev method or some modification of that, see for
example [GM, Ne]. The Faddeev method, as for example used in the
mathematically rigorous paper [GM], requires fall-off like
$V_{a}={\mathcal{O}}(\lvert x^{a}\rvert^{-2-\epsilon})$. Moreover there are
additional complications in that the threshold zero needs be be regular for
the two-body systems (i.e. zero-energy eigenvalues and resonances are
excluded) and ‘spurious poles’ cannot be ruled out (these poles would arise
from lack of solvability of a certain Lippmann–Schwinger type equation).

The work on the $3$-body stationary scattering theory [Is4] (see also its
partial precursor [Is3] or the recent book [Is5]) is different. In fact
Isozaki does not assume any regularity at zero energy for the two-body systems
and his theory does not have spurious poles. He overcomes these deficencies by
avoiding the otherwise prevailing Faddeev method. On the other hand [Is4]
still needs, in some comparison argument, a very detailed information on the
spectral theory of the (two-body) sub-Hamiltonians at zero energy, and this
requires the fall-off condition $V_{a}={\mathcal{O}}(\lvert
x^{a}\rvert^{-5-\epsilon})$ (as well as a restriction on the particle
dimension). The present paper does not involve such detailed information. In
fact in comparison with Isozaki’s papers a Borel calculus argument suffices.
Otherwise the overall spirit of the present paper and [Is4] is the same, in
particular the use of resolvent equations are kept at a minimum (solvability
of Lippmann–Schwinger type equations is not an issue) and both works employ
intensively Besov spaces not only in proofs but also in the formulation of
various results. In conclusion we recover all of Isozaki’s results by a
different method that works down to the critical exponent $1$ (i.e. for
general short-range potentials).

The literature on $N$-body stationary scattering theory for $N\geq 4$ is very
limited, see though [Ya1, Ya2, Ya3] which overlaps with [Sk1] regarding
Theorem 1.1.

We remark that the strong continuity assertion for the scattering matrix, cf.
I, cannot in general be replaced by norm continuity, see [Ya1, Subsection 7.6]
for a counterexample for a short-range potential. In this sense the stated
regularity in I of the scattering matrix is optimal.

We also remark that from a physics point of view well-definedness and
continuity of basic scattering quantities is only one out of several relevant
problems, for example like the structure of the singularities of the kernel of
the scattering matrix (possibly considered as function of the energy
parameter), see for example [Is2] for results on this topic. This and related
topics go beyond the scope of the present paper, although our formulas
potentially could reveal some insight.

The paper is organized as follows. In Section 2 we define a generalized
$N$-body short-range model and state various basic stationary theory. In
particular we introduce in Remarks 2.3 a notion of ‘$Q$-bounds’ that will be
vital for our procedure. Such bounds arise in our application of vector field
constructions, essentially due to Yafaev, as well as for other issues. The
Yafaev type constructions are studied in detail in the somewhat lengthy
Section 3. The reader may first skip this section and go directly to Section
4, where we give an account of various relevant results on scattering theory
from [Sk1]. In particular our strategy of proof of the pointwise stationary
completeness is explained in Section 4. In Section 5 we introduce a convenient
phase-space partition of unity, which then is used in Section 7 in combination
with the Borel calculus (trivializing threshold problems). We are left with
certain energy-localized states for which a ‘$Q$-bound’ of Section 6 will
apply in an iteration procedure, eventually to be explained and completed in
Section 8. Finally we devote Appendix A to an elaboration of some technical
details left out in Section 5.

##  2\. Preliminaries

We explain our abstract setting and give an account of some basic stationary
$N$-body theory.

###  2.1. $N$-body Hamiltonians, assumptions and notation

First we explain our model and then we introduce a number of useful general
notation.

####  2.1.1. $N$-body short-range Hamiltonians

Let ${\mathbf{X}}$ be a (nonzero) finite dimensional real inner product space,
equipped with a finite family $\\{{\mathbf{X}}_{a}\\}_{a\in{\mathcal{A}}}$ of
subspaces closed under intersection: For any $a,b\in\mathcal{A}$ there exists
$c\in\mathcal{A}$, denoted $c=a\vee b$, such that
${\mathbf{X}}_{a}\cap{\mathbf{X}}_{b}={\mathbf{X}}_{c}$. We order
${\mathcal{A}}$ by writing $a\leq b$ (or equivalently as $b\geq a$) if
${\mathbf{X}}_{a}\supseteq{\mathbf{X}}_{b}$. It is assumed that there exist
$a_{\min},a_{\max}\in{\mathcal{A}}$ such that
${\mathbf{X}}_{a_{\min}}={\mathbf{X}}$ and ${\mathbf{X}}_{a_{\max}}=\\{0\\}$.
The subspaces ${\mathbf{X}}_{a}$, $a\neq a_{\min}$, are called _collision
planes_. We will also use the notation $0=a_{\min}$ and
$d=d_{0}=\dim\mathbf{X}$. In general $d_{a}=\dim\mathbf{X}_{a}$. Let
${\mathcal{A}}^{\prime}={\mathcal{A}}\setminus\\{a_{\max}\\}$.

The $2$-body model (or more correctly named ‘the one-body model’) is based on
the structure ${\mathcal{A}}=\\{a_{\min},a_{\max}\\}$. The scattering theory
for such models is well-understood, in fact (here including long-range
potentials) there are several doable approaches, see for example the
monographs [DG, Is5] for accounts on time-dependent and stationary scattering
theories. In this paper we consider short-range potentials only, see the below
Condition 2.1, and the scattering theory for the $2$-body as well as for the
$N$-body model has a canonical meaning (to be discussed in Section 4).

Let ${\mathbf{X}}^{a}\subseteq{\mathbf{X}}$ be the orthogonal complement of
${\mathbf{X}}_{a}\subseteq{\mathbf{X}}$, and denote the associated orthogonal
decomposition of $x\in{\mathbf{X}}$ by

$x=x^{a}\oplus
x_{a}=\pi^{a}x\oplus\pi_{a}x\in{\mathbf{X}}^{a}\oplus{\mathbf{X}}_{a}.$

The vectors $x_{a}$ and $x^{a}$ may be called the _inter-cluster_ and
_internal components_ of $x$, respectively. The momentum operator
$p=-\mathrm{i}\nabla$ decomposes similarly, $p=p^{a}\oplus p_{a}$. For $a\neq
a_{\min}$ the unit sphere in $\mathbf{X}^{a}$ is denoted by $\mathbf{S}^{a}$.
For $a\neq a_{\max}$ the unit sphere in $\mathbf{X}_{a}$ is denoted by
$\mathbf{S}_{a}$.

A real-valued measurable function $V\colon{\mathbf{X}}\to\mathbb{R}$ is a
potential of many-body type if there exist real-valued measurable functions
$V_{a}\colon{\mathbf{X}}^{a}\to\mathbb{R}$ such that

$V(x)=\sum_{a\in\mathcal{A}}V_{a}(x^{a})\ \ \text{for }x\in\mathbf{X}.$

We take $V_{a_{\min}}=0$ and impose throughout the paper the following
condition for $a\neq a_{\min}$.

######  Condition 2.1.

There exists $\mu\in(0,1/2)$ such that for all
$a\in{\mathcal{A}}\setminus\\{a_{\min}\\}$ the potential $V_{a}=V_{a}(x^{a})$
fulfills:

  1. (1)

$V^{a}(-\Delta_{x^{a}}+1)^{-1}$ is compact.

  2. (2)

$\lvert x^{a}\rvert^{1+2\mu}V^{a}(-\Delta_{x^{a}}+1)^{-1}$ is bounded.

For any $a\in{\mathcal{A}}$ we introduce an associated Hamiltonian $H^{a}$ as
follows. For $a=a_{\min}$ we define $H^{a_{\min}}=0$ on
$\mathcal{H}^{a_{\min}}=L^{2}(\\{0\\})=\mathbb{C}.$ For $a\neq a_{\min}$ we
let

$V^{a}(x^{a})=\sum_{b\leq a}V_{b}(x^{b}),\quad x^{a}\in{\mathbf{X}}^{a},$

and introduce then

$H^{a}=-\Delta_{x^{a}}+V^{a}\ \ \text{on
}\mathcal{H}^{a}=L^{2}({\mathbf{X}}^{a})\text{ \,and\, }H_{a}=H^{a}\otimes
I+I\otimes p^{2}_{a}\text{ on }L^{2}({\mathbf{X}}).$

We abbreviate

$\displaystyle V^{a_{\max}}=V,\quad
H^{a_{\max}}=H,\quad\mathcal{H}^{a_{\max}}=\mathcal{H}=L^{2}({\mathbf{X}}).$

The operator $H$ (with domain $\mathcal{D}(H)=H^{2}(\mathbf{X})$) is the full
Hamiltonian of the $N$-body model, and the thresholds of $H$ are by definition
the eigenvalues of the sub-Hamiltonians $H^{a}$; $a\in{\mathcal{A}}^{\prime}$.
Equivalently stated the set of thresholds is

${\mathcal{T}}(H):=\bigcup_{a\in{\mathcal{A}}^{\prime}}\sigma_{{\mathrm{pp}}}(H^{a}).$

This set is closed and countable. Moreover the set of non-threshold
eigenvalues is discrete in ${\mathbb{R}}\setminus{\mathcal{T}}(H)$, and it can
only accumulate at points in ${\mathcal{T}}(H)$ from below. The essential
spectrum is given by the formula
$\sigma_{{\mathrm{ess}}}(H)=\bigl{[}\min{\mathcal{T}}(H),\infty\bigr{)}$. We
introduce the notation
${\mathcal{T}}_{{\mathrm{p}}}(H)=\sigma_{{\mathrm{pp}}}(H)\cup{\mathcal{T}}(H)$,
and more generally
${\mathcal{T}}_{{\mathrm{p}}}(H^{a})=\sigma_{{\mathrm{pp}}}(H^{a})\cup{\mathcal{T}}(H^{a})$.
Denote $R_{a}(z)=(H_{a}-z)^{-1}$ for $z\notin\sigma(H_{a})$ and
$R(z)=R_{a_{\max}}(z)$.

####  2.1.2. General notation

Any function $f\in C^{\infty}_{\mathrm{c}}({\mathbb{R}})$ taking values in
$[0,1]$ is referred to aa a _standard support function_ (or just a ‘support
function’). For any such functions $f_{1}$ and $f_{2}$ we write $f_{2}\succ
f_{1}$, if $f_{2}=1$ in a neighbourhood of $\operatorname{supp}f_{1}$.

If $T$ is a self-adjoint operator on a Hilbert space and $f$ is a support
function we can represent the operator $f(T)$ by the well-known
Helffer–Sjöstrand formula

$\displaystyle\begin{split}f(T)&=\int_{{\mathbb{C}}}(T-z)^{-1}\,\mathrm{d}\mu_{f}(z)\,\text{
with}\\\
&\mathrm{d}\mu_{f}(z)=\pi^{-1}(\bar{\partial}\tilde{f})(z)\,\mathrm{d}u\mathrm{d}v;\quad
z=u+\mathrm{i}v.\end{split}$ (2.1)

Here $\tilde{f}$ is an ‘almost analytic’ extension of $f$, which in this case
may be taken compactly supported. The formula (2.1) extends to more general
classes of functions and serves as a standard tool for commuting operators.
Since it will be used only tacitly in this paper the interested reader might
benefit from consulting [Sk1, Section 6] which is devoted to applications to
$N$-body Schrödinger operators, whence being equally relevant for the present
paper.

Consider and fix $\chi\in C^{\infty}(\mathbb{R})$ such that

$\displaystyle\chi(t)=\left\\{\begin{array}[]{ll}0&\mbox{ for }t\leq 4/3,\\\
1&\mbox{ for }t\geq 5/3,\end{array}\right.\quad\chi^{\prime}\geq 0,$

and such that the following properties are fulfilled:

$\displaystyle\sqrt{\chi},\sqrt{\chi^{\prime}},(1-\chi^{2})^{1/4},\sqrt{-\big{(}(1-\chi^{2})^{1/2}\big{)}^{\prime}}\in
C^{\infty}.$

We define correspondingly $\chi_{+}=\chi$ and $\chi_{-}=(1-\chi^{2})^{1/2}$
and record that

$\displaystyle\chi_{+}^{2}+\chi_{-}^{2}=1\text{ \,and\,
}\sqrt{\chi_{+}},\sqrt{\chi_{+}^{\prime}},\sqrt{\chi_{-}},\sqrt{-\chi_{-}^{\prime}}\in
C^{\infty}.$

We shall use the notation $\langle x\rangle=\big{(}1+\lvert
x\rvert^{2}\big{)}^{1/2}$ for $x\in\mathbf{X}$ (or more generally for any $x$
in a normed space). For $x\in\mathbf{X}\setminus\\{0\\}$ we write
$\hat{x}=x/\lvert x\rvert$. If $T$ is an operator on a Hilbert space
${\mathcal{G}}$ and $\varphi\in{\mathcal{G}}$ then $\langle
T\rangle_{\varphi}:=\langle\varphi,T\varphi\rangle$. We denote the space of
bounded operators from one (general) Banach space $X$ to another one $Y$ by
${\mathcal{L}}(X,Y)$ and abbreviate $\mathcal{L}(X)=\mathcal{L}(X,X)$. The
dual space of $X$ is denoted by $X^{*}$.

To define _Besov spaces associated with the multiplication operator $|x|$ on
${\mathcal{H}}$_ let

$\displaystyle F_{0}$
$\displaystyle=F\bigl{(}\bigl{\\{}x\in\mathbf{X}\,\big{|}\,\lvert
x\rvert<1\bigr{\\}}\bigr{)},$ $\displaystyle F_{m}$
$\displaystyle=F\bigl{(}\bigl{\\{}x\in\mathbf{X}\,\big{|}\,2^{m-1}\leq\lvert
x\rvert<2^{m}\bigr{\\}}\bigr{)}\quad\text{for }m=1,2,\dots,$

where $F(U)=F_{U}$ is the sharp characteristic function of any given subset
$U\subseteq{\mathbf{X}}$. The Besov spaces
$\mathcal{B}=\mathcal{B}(\mathbf{X})$,
$\mathcal{B}^{*}=\mathcal{B}(\mathbf{X})^{*}$ and
$\mathcal{B}^{*}_{0}=\mathcal{B}^{*}_{0}(\mathbf{X})$ are then given as

$\displaystyle\mathcal{B}$ $\displaystyle=\bigl{\\{}\psi\in
L^{2}_{\mathrm{loc}}(\mathbf{X})\,\big{|}\,\|\psi\|_{\mathcal{B}}<\infty\bigr{\\}},\quad\|\psi\|_{\mathcal{B}}=\sum_{m=0}^{\infty}2^{m/2}\|F_{m}\psi\|_{{\mathcal{H}}},$
$\displaystyle\mathcal{B}^{*}$ $\displaystyle=\bigl{\\{}\psi\in
L^{2}_{\mathrm{loc}}(\mathbf{X})\,\big{|}\,\|\psi\|_{\mathcal{B}^{*}}<\infty\bigr{\\}},\quad\|\psi\|_{\mathcal{B}^{*}}=\sup_{m\geq
0}2^{-m/2}\|F_{m}\psi\|_{{\mathcal{H}}},$ $\displaystyle\mathcal{B}^{*}_{0}$
$\displaystyle=\Bigl{\\{}\psi\in\mathcal{B}^{*}\,\Big{|}\,\lim_{m\to\infty}2^{-m/2}\|F_{m}\psi\|_{{\mathcal{H}}}=0\Bigr{\\}},$

respectively. Denote the standard _weighted $L^{2}$ spaces_ by

$L_{s}^{2}=L_{s}^{2}(\mathbf{X})=\langle x\rangle^{-s}L^{2}(\mathbf{X})\ \
\text{for }s\in\mathbb{R},\quad
L_{-\infty}^{2}=\bigcup_{s\in{\mathbb{R}}}L^{2}_{s},\quad
L^{2}_{\infty}=\bigcap_{s\in\mathbb{R}}L_{s}^{2}.$

Then for any $s>1/2$

$L^{2}_{s}\subsetneq\mathcal{B}\subsetneq
L^{2}_{1/2}\subsetneq\mathcal{H}\subsetneq
L^{2}_{-1/2}\subsetneq\mathcal{B}^{*}_{0}\subsetneq\mathcal{B}^{*}\subsetneq
L^{2}_{-s}.$

The abstract quotient-norm on the Banach space
${\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$ and

$\lVert\psi\rVert_{\rm
quo}:=\limsup_{n\to\infty}\,2^{-n/2}\Big{\lVert}\sum_{m=0}^{n}F_{m}\psi\Big{\rVert}_{{\mathcal{H}}},\quad\psi\in{\mathcal{B}}^{*},$

are equivalent norms.

Let $T$ be a linear operator on $\mathcal{H}=L^{2}(\mathbf{X})$ such that
$T,T^{*}:L^{2}_{\infty}\to L^{2}_{\infty}$, and let $t\in\mathbb{R}$. Then we
say that $T$ is an _operator of order $t$_, if for each $s\in\mathbb{R}$ the
restriction $T_{|L^{2}_{\infty}}$ extends to an operator
$T_{s}\in{\mathcal{L}}(L^{2}_{s},L^{2}_{s-t})$. Alternatively stated,

$\|\langle x\rangle^{s-t}T\langle x\rangle^{-s}\psi\|\leq C_{s}\|\psi\|\text{
for all }\psi\in L^{2}_{\infty}.$ (2.2a) If $T$ is of order $t$, we write
$T={\mathcal{O}}(\langle x\rangle^{t}).$ (2.2b)

For any given $a\in{\mathcal{A}}\setminus\\{a_{\min},a_{\max}\\}$, $\kappa>0$
(considered small) and $t\in\mathbb{R}$ and any given operator $T$ of order
$t$ we write

$T={\mathcal{O}}^{a}_{\kappa}(\langle x\rangle^{t}),$ (2.3a)

if for some ${\breve{\epsilon}}>0$ and some operator ${\breve{T}}$ also of
order $t$

$T-\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|){\breve{T}}\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|)={\mathcal{O}}(\langle
x\rangle^{t-{\breve{\epsilon}}}).$ (2.3b)

###  2.2. Basic stationary $N$-body theory

Under a rather weak condition (in particular weaker than Condition 2.1) it is
demonstrated in [AIIS] that the following limits exist locally uniformly in
$\lambda\not\in{\mathcal{T}}_{{\mathrm{p}}}(H)$:
$R(\lambda\pm\mathrm{i}0)=\lim_{\epsilon\to
0_{+}}\,R(\lambda\pm\mathrm{i}\epsilon)\in{\mathcal{L}}\big{(}L^{2}_{s},L^{2}_{-s}\big{)}\text{
for any }s>1/2.$ (2.4a) Furthermore (the strong weak∗-topology is explained in
Theorem A.1) $\displaystyle\begin{split}\sup_{0<\epsilon\leq 1}\lVert
R(\lambda\pm\mathrm{i}\epsilon)\rVert_{{\mathcal{L}}\big{(}{\mathcal{B}},{\mathcal{B}}^{*}\big{)}}&<\infty\text{
with locally uniform bounds in
}\lambda\not\in{\mathcal{T}}_{{\mathrm{p}}}(H),\\\ &\text{and there exist the
limits}\\\
R(\lambda\pm\mathrm{i}0)&=\operatorname*{s-w^{\star}-lim}_{\epsilon\to
0_{+}}\,R(\lambda\pm\mathrm{i}\epsilon)\in{\mathcal{L}}\big{(}{\mathcal{B}},{\mathcal{B}}^{*}\big{)};\quad\lambda\not\in{\mathcal{T}}_{{\mathrm{p}}}(H).\end{split}$
(2.4b) These results follow from a ‘Mourre estimate’ for the operator
$A=r^{1/2}Br^{1/2}$, where $B=2\operatorname{Re}(\omega\cdot
p)=-\mathrm{i}\sum_{j\leq
d}\big{(}\omega_{j}\partial_{x_{j}}+\partial_{x_{j}}\omega_{j}\big{)},\quad\omega=\mathop{\mathrm{grad}}r$
(2.5a) and $r$ is a certain smooth positive function on $\mathbf{X}$
fulfilling the following property, cf. [Sk1, Section 5]:

For all $k\in{\mathbb{N}}_{0}:={\mathbb{N}}\cup\\{0\\}$ and
$\beta\in{\mathbb{N}}^{d_{0}}_{0}$ there exists $C>0$ such that

$\big{\lvert}\partial^{\beta}(x\cdot\nabla)^{k}\big{(}r(x)-\langle
x\rangle\big{)}\big{\rvert}\leq C\langle x\rangle^{-1}.$ (2.5b)

Fixing $\lambda\not\in{\mathcal{T}}_{{\mathrm{p}}}(H)$ we can indeed find such
a function (possibly taken rescaled, meaning that we replace $r(x)$ by
$Rr(x/R)$ with $R\geq 1$ large) and an open neighbourhood $U$ of $\lambda$
such that

$\forall\text{ real }f\in
C_{\mathrm{c}}^{\infty}(U):\quad{f}(H)\mathrm{i}[H,r^{1/2}Br^{1/2}]{f}(H)\geq
c\,{f}(H)^{2};$ (2.5c)

here the constant $c>0$ depends on the number

$d(\lambda,H)=4\operatorname{dist}\big{(}\lambda,\\{\mu\in{\mathcal{T}}(H)\mid\mu<\lambda\\}\big{)}.$
(2.5d)

In fact $c$ can be taken arbitrarily smaller than the number $d(\lambda,H)$
(upon correspondingly adjusting the scaling parameter $R$ and the
neighbourhood $U$).

The Mourre estimate (2.5c) has consequences beyond (2.4a) and (2.4b). In
particular, cf. [Sk1, Subsection 5.2], for any
$\lambda\not\in{\mathcal{T}}_{{\mathrm{p}}}(H)$ and with
$2\epsilon_{0}=\sqrt{d(\lambda,H)}$

$\chi_{-}(\pm
B/2\epsilon_{0})R(\lambda\pm\mathrm{i}0)\psi\in{\mathcal{B}}_{0}^{*}\text{ for
all }\psi\in{\mathcal{B}}.$ (2.5e)

For completeness of presentation we note that $\chi_{-}(\pm
B/2\epsilon_{0})\in{\mathcal{L}}({\mathcal{B}})$, cf. [Hö, Theorem 14.1.4]. We
will need various partially extended versions of (2.5e) from [Sk1], which will
be stated in Appendix A. See also the related bound (2.11b) stated below.

Of course the above assertions for $H$ are also valid for the Hamiltonians
$H_{a}$, $a\in{\mathcal{A}}^{\prime}$.

We recall the following scheme of estimating the resolvent from [Sk1, Appendix
B].

######  Lemma 2.2.

Let $f_{1}$ and $f_{2}$ be standard support functions with $f_{2}\succ f_{1}$.
Suppose $P$ is symmetric and form-bounded relatively to $H$ such that the
self-adjoint operator $\Psi=f_{1}(H)Pf_{1}(H)$ is bounded when restricted to
${\mathcal{B}}$ and such that

$\mathrm{i}[H,\Psi]\geq f_{1}(H)\Big{(}Q^{*}Q-T^{*}T\Big{)}f_{1}(H)$ (2.6)

for $H$-bounded operators $Q$ and $T$ (possibly taking values in an abstract
Hilbert space). Then the following estimates hold for all
$z\in{\mathbb{C}}\setminus{\mathbb{R}}$ and all $\psi\in{\mathcal{B}}$.

$\displaystyle\begin{split}\lVert&QR(z)f_{1}(H)\psi\rVert^{2}\leq\lVert
TR(z)f_{1}(H)\psi\rVert^{2}\\\
&+2\big{(}\lVert\Psi\rVert_{{\mathcal{L}}({\mathcal{H}})}+\lVert\Psi\rVert_{{\mathcal{L}}({\mathcal{B}})}\big{)}\lVert
R(z)f_{2}(H)\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{B}}^{*})}\lVert\psi\rVert^{2}_{{\mathcal{B}}}.\end{split}$
(2.7a) In particular, if $\sup_{\operatorname{Im}z\neq 0}\lVert\lvert
T{f_{1}}(H)\rvert{R(z)}\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}})}<\infty\text{
\,and\, }\sup_{\operatorname{Im}z\neq 0}\lVert
R(z)f_{2}(H)\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{B}}^{*})},$ then
also $\sup_{\operatorname{Im}z\neq 0}\lVert\lvert
Q{f_{1}}(H)\rvert{R(z)}\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}})}<\infty.$
(2.7b)

###### Proof.

The estimate (2.6) leads for $\phi=R(z)f_{1}(H)\psi$ to

$\displaystyle\lVert Q\phi\rVert^{2}-\lVert T\phi\rVert^{2}$
$\displaystyle\leq 2\big{(}(\operatorname{Im}z)\langle R(z)\psi,\Psi
R(z)\psi\rangle-\operatorname{Im}\langle\psi,\Psi R(z)\psi\rangle\big{)}$
$\displaystyle\leq
2\big{(}\lVert\Psi\rVert_{{\mathcal{L}}({\mathcal{H}})}\lvert\operatorname{Im}\langle\psi,R(z)f_{2}(H)\psi\rangle\rvert+\lvert\operatorname{Im}\langle\Psi\psi,R(z)f_{2}(H)\psi\rangle\rvert\big{)}$
$\displaystyle\leq
2\big{(}\lVert\Psi\rVert_{{\mathcal{L}}({\mathcal{H}})}+\lVert\Psi\rVert_{{\mathcal{L}}({\mathcal{B}})}\big{)}\lVert
R(z)f_{2}(H)\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{B}}^{*})}\lVert\psi\rVert^{2}_{{\mathcal{B}}}.$

∎

######  Remarks 2.3.

  1. i)

The bounds (2.7a) and (2.7b) are useful in combination with the limiting
absorption principle expressed by (2.4a) and (2.4b) (and similarly when
applied to $H_{a}$ rather than to $H$ for any $a\in{\mathcal{A}}^{\prime}$).
Note for example that if $Q$ maps into ${\mathcal{H}}$,
$Q{f_{1}}(H)={\mathcal{O}}(\langle x\rangle^{0})$ (in the sense of (2.2b)) and
$\lambda\not\in{\mathcal{T}}_{{\mathrm{p}}}(H)$, then the bound (2.7b) leads
to the existence of the weak limits

${Q{f_{1}}(H)}R(\lambda\pm\mathrm{i}0)=\operatorname*{w-lim}_{\epsilon\to
0_{+}}Q{f_{1}}(H)R(\lambda\pm\mathrm{i}\epsilon)\in{\mathcal{L}}\big{(}{\mathcal{B}},{\mathcal{H}}\big{)}.$
(2.8)

  2. ii)

Bounds of the form (2.7b) are very useful for us (there will be several of
those, see v below). We shall use the generic notation $Q$ for $H$-bounded
operators obeying (2.7b), in all cases with $f_{1}$ being any narrowly
supported standard support function obeying $f_{1}=1$ on a neighbourhood of
any given $\lambda\not\in{\mathcal{T}}_{{\mathrm{p}}}(H)$. Such $Q=Q_{1}$ may
show up from a direct application of Lemma 2.2, or possibly the expression
$Q_{1}^{*}Q_{1}$ may be bounded from above by $Q_{2}^{*}Q_{2}$ where the bound
for $Q_{2}$ results from such direct application (obviously yielding the bound
for $Q_{1}$ also). In the latter case we will always have
$Q_{1}f_{1}(H)={\mathcal{O}}(\langle x\rangle^{-1/2})$ in the sense of (2.2b),
and in commutator calculations this product will generically arise as the
combination (for $a\in{\mathcal{A}}$)

$f_{1}(H_{a})Q_{1}^{*}B_{a}Q_{1}f_{1}(H);\quad\,B_{a}\text{ bounded}.$ (2.9)

Such expressions will be treated by using the bound (2.7b) with $Q=Q_{1}$ for
$H$ (as well as for $H$ replaced by $H_{a}$).

  3. iii)

The following basic example of a ‘$Q$-bound’ does not fit into the scheme of
Lemma 2.2:

$\displaystyle\begin{split}\sup_{\operatorname{Im}z\neq 0}\lVert
Q_{s}{f_{1}}&(H)R(z)\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}})}<\infty;\\\
&\quad Q_{s}=r^{-s},\,\,s\in(1/2,1).\end{split}$ (2.10)

Here $f_{1}$ is given as in ii, and of course the bound is a consequence of
(2.4b).

  4. iv)

Yet another example of a bound of the form (2.7b) is the following version of
(2.5e) from [Sk1]. We apply Lemma 2.2 with

$\Psi=\pm f_{1}(H)\chi_{+}(\pm B/\epsilon)f_{1}(H),$ (2.11a) where
$\epsilon\in(0,2\epsilon_{0}]$ and $f_{1}$ is given as in ii. Then (2.4a),
(2.4b) and (2.5c) lead to a version of (2.7b) from which we can deduce the
following concrete bounds (with ‘prime’ denoting the derivative), cf. ii:
$\displaystyle\begin{split}\sup_{\operatorname{Im}z\neq 0}\lVert
Q_{\epsilon\pm}&{f_{1}}(H)R(z)\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}})}<\infty;\\\
&\quad Q_{\epsilon\pm}=r^{-1/2}\sqrt{\chi_{+}^{\prime}}(\pm
B/\epsilon),\,\,0<\epsilon\leq 2\epsilon_{0}.\end{split}$ (2.11b)

  5. v)

A complete list of ‘$Q$-bounds’ to be applied in this paper to expressions of
the form (2.9) may be provided by adding the bounds (3.28c), (3.36d) and
(6.5b) to the bounds (2.10) and (2.11b) stated above.

##  3\. Yafaev type constructions

In [Ya2] various real functions $m_{a}\in C^{\infty}(\mathbf{X})$,
$a\in{\mathcal{A}}^{\prime}$, and $m_{a_{\max}}\in C^{\infty}(\mathbf{X})$ are
constructed. They are homogeneous of degree $1$ for $\lvert x\rvert>5/6$, and
in addition $m_{a_{\max}}$ is (locally) convex in that region. These functions
are constructed as depending of a small parameter $\epsilon>0$, and once
given, they are used in the constructions

$M_{a}=2\operatorname{Re}(w_{a}\cdot p)=-\mathrm{i}\sum_{j\leq
d}\big{(}(w_{a})_{j}\partial_{x_{j}}+\partial_{x_{j}}(w_{a})_{j}\big{)};\quad
w_{a}=\mathop{\mathrm{grad}}m_{a}.$ (3.1)

The operators $M_{a}$, $a\in{\mathcal{A}}^{\prime}$, may be considered as
‘channel localization operators’, while the operator $M_{a_{\max}}$ enters as
a technical quantity controlling commutators of the Hamiltonian and the
channel localization operators.

We are going to use and extend these constructions/ideas partly in generalized
settings.

A basic ingredient of [Ya2] are small conical neighbourhoods of the ‘collision
planes’. More precisely they are given for
$a\in{\mathcal{A}}^{\prime}={\mathcal{A}}\setminus\\{a_{\max}\\}$ in agreement
with the recipe

$\mathbf{X}_{a}(\varepsilon)=\\{x\in{\mathbf{X}}\mid\lvert
x_{a}\rvert>(1-\varepsilon)\lvert x\rvert\\};\quad\varepsilon\in(0,1).$ (3.2)

Note that indeed
$\mathbf{X}_{a}\setminus\\{0\\}\subseteq\mathbf{X}_{a}(\varepsilon)$ and that
in fact
$\mathbf{X}_{a}\setminus\\{0\\}=\cap_{\varepsilon\in(0,1)}\,\mathbf{X}_{a}(\varepsilon)$.
Note also that $\mathbf{X}_{0}(\varepsilon)=\mathbf{X}\setminus\\{0\\}$.

We will need the ‘width’ of the above cones to have a dependence of
$a\in{\mathcal{A}}^{\prime}$ roughly proportional to $\epsilon^{d_{a}}$ for a
small parameter $\epsilon>0$ (recall $d_{a}=\dim\mathbf{X}_{a}$) as determined
(to be elaborated on Subsection 3.1) by the following geometric property:
There exists $C>0$ such that for all $a,b\in{\mathcal{A}}^{\prime}$ and
$x\in\mathbf{X}$

$\lvert x^{c}\rvert\leq C\big{(}\lvert x^{a}\rvert+\lvert
x^{b}\rvert\big{)};\quad c=a\vee b.$ (3.3)

###  3.1. Homogeneous Yafaev type functions for a general $a_{0}$

In this subsection we fix any $a_{0}\in{\mathcal{A}}\setminus\\{a_{\min}\\}$
and let ${\mathcal{A}}_{a_{0}}=\\{a\leq a_{0}\\}$,
${\mathcal{A}}^{\prime}_{a_{0}}=\\{a\lneq a_{0}\\}$ and for
$a\in{\mathcal{A}}^{\prime}_{a_{0}}$

$\mathbf{X}^{a_{0}}_{a}(\varepsilon)=\\{x\in{\mathbf{X}}\mid\lvert
x^{a_{0}}_{a}\rvert>(1-\varepsilon)\lvert
x^{a_{0}}\rvert\\};\quad\varepsilon\in(0,1).$ (3.4)

Note that (3.4) coincdes with (3.2) if $a_{0}=a_{\max}$. We record

$\mathbf{X}^{a_{0}}_{a}(\varepsilon)\subseteq\\{\lvert x^{a}\rvert<\sqrt{2\
\varepsilon}\lvert x^{a_{0}}\rvert\\}.$ (3.5)

For small $\epsilon>0$ and any $a\in{\mathcal{A}}^{\prime}_{a_{0}}$ we define

$\varepsilon^{a}_{j}=j\epsilon^{d_{a}},\quad j=1,2,3,$

and view the numbers $\varepsilon_{a}$ in the interval
$(\varepsilon^{a}_{2},\varepsilon^{a}_{3})$ as ‘admissible’. Alternatively,
viewed as a free parameter, $\varepsilon_{a}$ is called admissible if

$\varepsilon^{a}_{2}=2\epsilon^{d_{a}}<\varepsilon_{a}<3\epsilon^{d_{a}}=\varepsilon^{a}_{3}.$
(3.6)

Using an arbitrary ordering of ${\mathcal{A}}^{\prime}_{a_{0}}$ we introduce
for these parameters the ‘admissible’ vector

$\bar{\varepsilon}=(\varepsilon_{a_{1}},\dots\varepsilon_{a_{n}}),\quad
n=n(a_{0})=\\#{\mathcal{A}}^{\prime}_{a_{0}},$

and denote by ${\mathrm{d}}\bar{\varepsilon}$ the corresponding Lebesgue
measure. The smallness of $\epsilon>0$ will be needed at various points below;
it will be determined by geometric considerations (primarily (3.3)) and in
particular taken independently of the vectors
$x\in\mathbf{X}\setminus\mathbf{X}_{a_{0}}$.

Let for $\varepsilon>0$ and $a\in{\mathcal{A}}^{\prime}_{a_{0}}$

$h_{a,\varepsilon}(x)=h^{a_{0}}_{a,\varepsilon}(x)=\big{(}(1+\varepsilon)^{2}\lvert
x_{a}^{a_{0}}\rvert^{2}+\lvert x_{a_{0}}^{2}\rvert\big{)}^{1/2};\quad
x\in\mathbf{X}\setminus\mathbf{X}_{a_{0}}.$

(We prefer for simplicity below to use the indicated short-hand notation
$h_{a,\varepsilon}(x)$.)

######  Lemma 3.1.

Let $a,b\in{\mathcal{A}}^{\prime}_{a_{0}}$, ${a\not\leq b}$ and
$x\in\mathbf{X}^{a_{0}}_{a}(\varepsilon^{a}_{1})$. Then (for all small
$\epsilon>0$)

$h_{b,\varepsilon^{b}_{3}}(x)<\max_{{\mathcal{A}}^{\prime}_{a_{0}}\ni c\geq
a}h_{c,\varepsilon^{c}_{2}}(x).$ (3.7)

###### Proof.

Note that while here $a,b,c\in{\mathcal{A}}^{\prime}_{a_{0}}$ it could be that
$\tilde{c}:=a\vee b=a_{0}$. By assumption $b\lneq\tilde{c}$. This leads us to
consider three cases, depending on whether $\tilde{c}=a_{0}$ or
$\tilde{c}\in{\mathcal{A}}^{\prime}_{a_{0}}$. We can assume that $\lvert
x^{a_{0}}\rvert=1$.

I. Suppose $\tilde{c}\in{\mathcal{A}}^{\prime}_{a_{0}}$ and
$x\in\mathbf{X}^{a_{0}}_{\tilde{c}}(\varepsilon^{\tilde{c}}_{1})$. We then
verify that

$h_{b,\varepsilon^{b}_{3}}(x)<h_{{\tilde{c}},\varepsilon^{\tilde{c}}_{2}}(x),$
(3.8)

proving (3.7). By assumptions

$\lvert x^{a_{0}}_{a}\rvert\leq 1\text{ \,and\, }\lvert
x^{a_{0}}_{\tilde{c}}\rvert>(1-\varepsilon^{\tilde{c}}_{1}).$

So it suffices to check that

$(1+\varepsilon^{b}_{3})\leq(1+\varepsilon^{\tilde{c}}_{2})(1-\varepsilon^{\tilde{c}}_{1}).$

Since $b\lneq\tilde{c}$, this is indeed valid for small $\epsilon>0$.

II. Suppose $\tilde{c}\in{\mathcal{A}}^{\prime}_{a_{0}}$ and
$x\in\mathbf{X}^{a_{0}}_{a}(\varepsilon^{a}_{1})\setminus\mathbf{X}^{a_{0}}_{\tilde{c}}(\varepsilon^{\tilde{c}}_{1})$.
We then verify that

$h_{b,\varepsilon^{b}_{3}}(x)<h_{{a},\varepsilon^{a}_{2}}(x),$ (3.9)

proving (3.7). Thanks to the bounds

$\lvert x^{a_{0}}_{a}\rvert>(1-\varepsilon^{a}_{1})\text{ \,and\, }\lvert
x^{a_{0}}_{\tilde{c}}\rvert\leq(1-\varepsilon^{\tilde{c}}_{1}),$

we deduce

$\lvert
x^{a}\rvert^{2}<1-(1-\varepsilon^{a}_{1})^{2}<\varepsilon^{a}_{2}\text{
\,and\, }\lvert x^{\tilde{c}}\rvert^{2}\geq
1-(1-\varepsilon^{\tilde{c}}_{1})^{2}\geq\varepsilon^{\tilde{c}}_{1}.$

Also it follows from our assumption on $x$ that $a\lneq\tilde{c}$. Due to
(3.3) and these bounds we can find $\kappa>0$ independent of small
$\epsilon>0$ (and $x$) such that

$\lvert x^{a_{0}}_{b}\rvert^{2}\leq 1-2\kappa\varepsilon^{\tilde{c}}_{1}.$

Then (3.9) is trivially fulfilled, since

$(1+\varepsilon^{b}_{3})(1-\kappa\varepsilon^{\tilde{c}}_{1})<1<(1+\varepsilon^{a}_{2})(1-\varepsilon^{a}_{1}).$

III. Suppose $\tilde{c}=a_{0}$ and
$x\in\mathbf{X}^{a_{0}}_{a}(\varepsilon^{a}_{1})$. Then using (3.3) as before
we find $\kappa>0$ independent of small $\epsilon>0$ such that

$\lvert x^{a_{0}}_{b}\rvert^{2}\leq 1-2\kappa.$

This leads to (3.9), noting that similarly

$(1+\varepsilon^{b}_{3})(1-\kappa)<(1+\varepsilon^{a}_{2})(1-\varepsilon^{a}_{1}),$

and then in turn to (3.7). ∎

Letting $\Theta=1_{[0,\infty)}$, we define for any
$a\in{\mathcal{A}}^{\prime}_{a_{0}}$ and any admissible vector
$\bar{\varepsilon}$

$m_{a}(x,\bar{\varepsilon})=m^{a_{0}}_{a}(x,\bar{\varepsilon})=h_{a,\varepsilon_{a}}(x)\Theta\big{(}h_{a,\varepsilon_{a}}(x)-\max_{{\mathcal{A}}^{\prime}_{a_{0}}\ni
c\neq a}h_{c,\varepsilon_{c}}(x)\big{)};\quad
x\in\mathbf{X}\setminus\mathbf{X}_{a_{0}}.$

Here the second factor can be replaced by the product

$\Theta(\cdot)=\Pi_{b\in{\mathcal{A}}^{\prime}_{a_{0}}}\,\Theta\big{(}h_{a,\varepsilon_{a}}(x)-h_{b,\varepsilon_{b}}(x)\big{)}.$

######  Lemma 3.2.

For all $a,b\in{\mathcal{A}}^{\prime}_{a_{0}}$,
$x\in\mathbf{X}^{a_{0}}_{b}(\varepsilon^{b}_{1})$ and admissible vectors
$\bar{\varepsilon}$:

  1. 1)

If ${b\not\leq a}$, then $m_{a}(x,\bar{\varepsilon})=0$.

  2. 2)

If ${b\leq a}$, then
$m_{a}(x,\bar{\varepsilon})=m_{a}(x_{b},\bar{\varepsilon})$ and

$m_{a}(x,\bar{\varepsilon})=h_{a,\varepsilon_{a}}(x)\Theta\big{(}h_{a,\varepsilon_{a}}(x)-\max_{{\mathcal{A}}^{\prime}_{a_{0}}\ni
c\geq b}h_{c,\varepsilon_{c}}(x)\big{)}.$

###### Proof.

Fix $x\in\mathbf{X}^{a_{0}}_{b}(\varepsilon^{b}_{1})$.

By (3.6) and Lemma 3.1, if ${b\not\leq a}$

$h_{a,\varepsilon_{a}}(x)\leq
h_{a,\varepsilon^{a}_{3}}(x)<\max_{{\mathcal{A}}^{\prime}_{a_{0}}\ni c\geq
b}h_{c,\varepsilon^{c}_{2}}(x)\leq\max_{{\mathcal{A}}^{\prime}_{a_{0}}\ni
c\geq b}h_{c,\varepsilon_{c}}(x),$ (3.10)

showing 1.

If on the other hand ${b\leq a}$, then it suffices for 2 to show that

$\max_{c\in{\mathcal{A}}^{\prime}_{a_{0}}}h_{c,\varepsilon_{c}}(x)=\max_{{\mathcal{A}}^{\prime}_{a_{0}}\ni
c\geq b}h_{c,\varepsilon_{c}}(x),$

noting that the right-hand side only depends on $x_{b}$. Obviously it suffices
to show that for any $\breve{c}\in{\mathcal{A}}^{\prime}_{a_{0}}$

$h_{\breve{c},\varepsilon_{\breve{c}}}(x)\leq\max_{{\mathcal{A}}^{\prime}_{a_{0}}\ni
c\geq b}h_{c,\varepsilon_{c}}(x),$

and we can assume that $b\not\leq\breve{c}$. The stated bound then follows
from (3.10) applied to $(a,b)=(\breve{c},b)$. ∎

Next we apply Lemma 3.2 2 to $b=a$.

######  Corollary 3.3.

For any $x\in\mathbf{X}^{a_{0}}_{a}(\varepsilon^{a}_{1})$,
$a\in{\mathcal{A}}^{\prime}_{a_{0}}$, and any admissible vector
$\bar{\varepsilon}$

$m_{a}(x,\bar{\varepsilon})=h_{a,\varepsilon_{a}}(x)\Theta\big{(}h_{a,\varepsilon_{a}}(x)-\max_{c\in{\mathcal{A}}^{\prime}_{a_{0}},\,c\geq
a}h_{c,\varepsilon_{c}}(x)\big{)}.$

To average the functions $m_{a}(x,\bar{\varepsilon})$ over $\bar{\varepsilon}$
we introduce for any $c\in{\mathcal{A}}^{\prime}_{a_{0}}$ a non-negative
function $\varphi_{c}\in C^{\infty}_{\mathrm{c}}({\mathbb{R}})$ with

$\operatorname{supp}\varphi_{c}\subseteq(\varepsilon^{c}_{2},\varepsilon^{c}_{3})\text{
\,and\, }\int\varphi_{c}(\varepsilon)\,{\mathrm{d}}\varepsilon=1.$

We then define for each $a\in{\mathcal{A}}^{\prime}_{a_{0}}$

$m_{a}(x)=m^{a_{0}}_{a}(x)=\int_{{\mathbb{R}}^{n}}m^{a_{0}}_{a}(x,\bar{\varepsilon})\,\Pi_{c\in{\mathcal{A}}^{\prime}_{a_{0}}}\,\varphi_{c}(\varepsilon_{c})\,{\mathrm{d}}\bar{\varepsilon};\quad
x\in\mathbf{X}\setminus\mathbf{X}_{a_{0}}.$ (3.11)

Recall the notation $a_{\min}=0$.

######  Lemma 3.4.

Suppose $a\in{\mathcal{A}}^{\prime}_{a_{0}}\setminus\\{0\\}$,
$x\not\in\mathbf{X}^{a_{0}}_{a}(\varepsilon^{a}_{3})$ and $x^{a_{0}}\neq 0$.
Then for any admissible vector $\bar{\varepsilon}$

$m_{a}(x)=m_{a}(x,\bar{\varepsilon})=0.$

###### Proof.

We show that $m_{a}(x,\bar{\varepsilon})=0$. With our assumptions we can
assume that $\lvert x^{a_{0}}\rvert=1$ and it suffices to show that

$h_{a,\varepsilon_{a}}(x)-h_{0,\varepsilon_{0}}(x)<0.$

This bound follows from estimating

$(1+\varepsilon_{a})\lvert
x_{a}^{a_{0}}\rvert\leq(1+\varepsilon^{a}_{3})\lvert
x_{a}^{a_{0}}\rvert\leq(1+\varepsilon^{a}_{3})(1-\varepsilon^{a}_{3})<(1+\varepsilon_{0}).$

∎

######  Lemma 3.5.

Let $a\in{\mathcal{A}}^{\prime}_{a_{0}}$ and suppose
$x\in\mathbf{X}^{a_{0}}_{a}(\varepsilon^{a}_{1})$ obeys that for all
$b\in{\mathcal{A}}^{\prime}_{a_{0}}$ with $b\gneq a$ the vector
$x\not\in\mathbf{X}^{a_{0}}_{b}(\varepsilon^{b}_{3})$. Then

$m_{a}(x)=\int_{\mathbb{R}}h_{a,\varepsilon}(x)\,\varphi_{a}(\varepsilon)\,{\mathrm{d}}\varepsilon.$

###### Proof.

We check that $m_{a}(x,\bar{\varepsilon})=h_{a,\varepsilon_{a}}(x)$, assuming
that $\lvert x^{a_{0}}\rvert=1$. Using Corollary 3.3 it suffices to show that

$h_{a,\varepsilon_{a}}(x)\geq\max_{{\mathcal{A}}^{\prime}_{a_{0}}\ni c\geq
a}h_{c,\varepsilon_{c}}(x).$

With any $b$ as in the lemma in turn it suffices to show that

$h_{a,\varepsilon_{a}}(x)\geq h_{b,\varepsilon_{b}}(x).$

This bound follows from

$(1+\varepsilon_{a})\lvert
x_{a}^{a_{0}}\rvert>(1+\varepsilon^{a}_{2})(1-\varepsilon^{a}_{1})\geq(1+\varepsilon^{b}_{3})(1-\varepsilon^{b}_{3})\geq(1+\varepsilon_{b})\lvert
x^{a_{0}}_{b}\rvert.$

∎

######  Lemma 3.6.

For any $a\in{\mathcal{A}}^{\prime}_{a_{0}}$ the function $m_{a}\in
C^{\infty}({\mathbf{X}}\setminus{{\mathbf{X}}_{a_{0}}})$.

###### Proof.

By homogeneity it suffices to consider $x$ obeying $\lvert x^{a_{0}}\rvert>2$.
By Lemma 3.4 we can then also assume that $\lvert x^{a_{0}}_{a}\rvert>1$. We
represent

$\displaystyle\begin{split}m_{a}(x)=\int\,{\mathrm{d}}\varepsilon_{a}h_{a,\varepsilon_{a}}(x)&\,\varphi_{a}(\varepsilon_{a})\,\\\
&\prod_{{\mathcal{A}}^{\prime}_{a_{0}}\ni b\neq
a}\,\Big{(}\int\,\Theta\big{(}h_{a,\varepsilon_{a}}(x)-h_{b,\varepsilon_{b}}(x)\big{)}\varphi_{b}(\varepsilon_{b})\,{\mathrm{d}}\varepsilon_{b}\Big{)}.\end{split}$
(3.12)

Clearly

$\Theta\big{(}h_{a,\varepsilon_{a}}(x)-h_{b,\varepsilon_{b}}(x)\big{)}=\Theta\big{(}(1+\varepsilon_{a})\lvert
x_{a}^{a_{0}}\rvert-(1+\varepsilon_{b})\lvert x_{b}^{a_{0}}\rvert\big{)},$

so introducing
$\varPhi_{b}(t)=\int_{0}^{t}\,\varphi_{b}(\varepsilon)\,{\mathrm{d}}\varepsilon$,

$\displaystyle m_{a}(x)$
$\displaystyle=\int\,{\mathrm{d}}\varepsilon_{a}h_{a,\varepsilon_{a}}(x)\,\varphi_{a}(\varepsilon_{a})\,\prod_{b\neq
a}\,\Big{(}\int\,\Theta\big{(}(1+\varepsilon_{a})\lvert
x_{a}^{a_{0}}\rvert/\lvert
x_{b}^{a_{0}}\rvert-1-\varepsilon_{b})\big{)}\varphi_{b}(\varepsilon_{b})\,{\mathrm{d}}\varepsilon_{b}\Big{)}$
$\displaystyle=\int\,{\mathrm{d}}\varepsilon_{a}h_{a,\varepsilon_{a}}(x)\,\varphi_{a}(\varepsilon_{a})\,\Pi_{b\neq
a}\,\big{(}\varPhi_{b}((1+\varepsilon_{a})\lvert x_{a}^{a_{0}}\rvert/\lvert
x_{b}^{a_{0}}\rvert-1)\big{)}.$

This is a smooth in $x\in\\{\lvert x^{a_{0}}_{a}\rvert>1\\}$. ∎

We summarize most of the proven properties as follows.

######  Lemma 3.7.

For any $a\in{\mathcal{A}}^{\prime}_{a_{0}}$ the function
$m_{a}:\mathbf{X}\setminus\mathbf{X}_{a_{0}}\to{\mathbb{R}}$ fulfills the
following properties for any sufficiently small $\epsilon>0$ and any
$b\in{\mathcal{A}}^{\prime}_{a_{0}}$:

  1. 1)

$m_{a}$ is homogeneous of degree $1$.

  2. 2)

$m_{a}\in C^{\infty}({\mathbf{X}}\setminus{{\mathbf{X}}_{a_{0}}})$.

  3. 3)

If $b\leq a$ and $x\in\mathbf{X}^{a_{0}}_{b}(\varepsilon^{b}_{1})$, then
$m_{a}(x)=m_{a}(x_{b})$.

  4. 4)

If ${b\not\leq a}$ and $x\in\mathbf{X}^{a_{0}}_{b}(\varepsilon^{b}_{1})$, then
$m_{a}(x)=0$.

  5. 5)

If $a\neq 0$, $x\not\in{\mathbf{X}}_{a_{0}}$ and $\lvert
x^{a}\rvert\geq\sqrt{2\ \varepsilon^{a}_{3}}\lvert x^{a_{0}}\rvert$, then
$m_{a}(x)=0$.

###### Proof.

The property 1 is obvious, and 2 coincides with Lemma 3.6. The properties 3
and 4 follow from applying Lemma 3.2 to the defining expression (3.11). The
property 5 follows from Lemma 3.4 and (3.5). ∎

Following [Ya2] we introduce for $x\in\mathbf{X}\setminus\mathbf{X}_{a_{0}}$
and admissible vectors $\bar{\varepsilon}$

$\displaystyle m_{a_{0}}(x,\bar{\varepsilon})$
$\displaystyle=m_{a_{0}}^{a_{0}}(x,\bar{\varepsilon})=\max_{a\in{\mathcal{A}}^{\prime}_{a_{0}}}h_{a,\varepsilon_{a}}(x),$
$\displaystyle m_{a_{0}}(x)$
$\displaystyle=m_{a_{0}}^{a_{0}}(x)=\int_{{\mathbb{R}}^{n}}m_{a_{0}}^{a_{0}}(x,\bar{\varepsilon})\,\Pi_{c\in{\mathcal{A}}^{\prime}_{a_{0}}}\,\varphi_{c}(\varepsilon_{c})\,{\mathrm{d}}\bar{\varepsilon}.$

Compared to the previously constructed functions
$m_{a}(\cdot,\bar{\varepsilon})$ and $m_{a}(\cdot)$ the functions
$m_{a_{0}}(\cdot,\bar{\varepsilon})$ and $m_{a_{0}}(\cdot)$ are convex (away
from $\mathbf{X}\setminus\mathbf{X}_{a_{0}}$).

######  Lemma 3.8.

For all $x\in\mathbf{X}\setminus\mathbf{X}_{a_{0}}$

$m_{a_{0}}(x)=\Sigma_{a\in{\mathcal{A}}^{\prime}_{a_{0}}}\,m_{a}(x).$ (3.13)

###### Proof.

To check (3.13) it suffices by continuity and density to consider

$x\in\cap_{a\in{\mathcal{A}}^{\prime}_{a_{0}}}\\{{x^{a_{0}}_{a}}\neq
0\\}\subseteq\mathbf{X}\setminus\mathbf{X}_{a_{0}}.$

For any such $x$ there is a Lebesgue null-set of admissible vectors such that
away from the null-set

$m_{a_{0}}(x,\bar{\varepsilon})=\Sigma_{a\in{\mathcal{A}}^{\prime}_{a_{0}}}\,m_{a}(x,\bar{\varepsilon}).$
(3.14)

Indeed for $a,b\in{\mathcal{A}}^{\prime}_{a_{0}}$, $a\neq b$, the relationship
$h_{a,\varepsilon_{a}}(x)=h_{b,\varepsilon_{b}}(x)$ is only possible at a
null-set in $(\varepsilon_{a},\varepsilon_{b})$. Consequently we can for
$\bar{\varepsilon}$ away from a null-set find
$a=a(x,\bar{\varepsilon})\in{\mathcal{A}}^{\prime}_{a_{0}}$ such that

$h_{a,\varepsilon_{a}}(x)>\max_{{\mathcal{A}}^{\prime}_{a_{0}}\ni b\neq
a}h_{b,\varepsilon_{b}}(x).$

Then $m_{b}(x,\bar{\varepsilon})=0$ for $b\neq a$ and
$m_{a_{0}}(x,\bar{\varepsilon})=m_{a}(x,\bar{\varepsilon})$, showing (3.14).
Clearly (3.13) follows from (3.14) by integration. ∎

We summarize the known properties as follows.

######  Lemma 3.9.

The function $m_{a_{0}}:\mathbf{X}\setminus\mathbf{X}_{a_{0}}\to{\mathbb{R}}$
fulfills the following properties for any sufficiently small $\epsilon>0$:

  1. i)

$m_{a_{0}}$ is convex and homogeneous of degree $1$.

  2. ii)

$m_{a_{0}}\in C^{\infty}({\mathbf{X}}\setminus{{\mathbf{X}}_{a_{0}}})$.

  3. iii)

If $b\in{\mathcal{A}}^{\prime}_{a_{0}}$ and
$x\in\mathbf{X}^{a_{0}}_{b}(\varepsilon^{b}_{1})$, then
$m_{a_{0}}(x)=m_{a_{0}}(x_{b})$.

  4. iv)

$m_{a_{0}}=\Sigma_{a\in{\mathcal{A}}^{\prime}_{a_{0}}}\,m_{a}$.

  5. v)

Let $a\in{\mathcal{A}}^{\prime}_{a_{0}}$ and suppose
$x\in\mathbf{X}^{a_{0}}_{a}(\varepsilon^{a}_{1})$ obeys that for all
$b\in{\mathcal{A}}^{\prime}_{a_{0}}$ with $b\gneq a$ the vector
$x\not\in\mathbf{X}^{a_{0}}_{b}(\varepsilon^{b}_{3})$. Then

$m_{a_{0}}(x)=m_{a}(x)=\int_{\mathbb{R}}h_{a,\varepsilon}(x)\,\varphi_{a}(\varepsilon)\,{\mathrm{d}}\varepsilon.$
(3.15)

  6. vi)

The derivative of $m_{a_{0}}\in
C^{\infty}({\mathbf{X}}\setminus{{\mathbf{X}}_{a_{0}}})$ is given by the
formula

$\displaystyle\begin{split}\quad\nabla m_{a_{0}}(x)\quad\quad&\\\
=\sum_{a\in{\mathcal{A}}^{\prime}_{a_{0}}}\,\int&\,{\mathrm{d}}\varepsilon_{a}\big{(}\nabla
h_{a,\varepsilon_{a}}(x)\big{)}\,\varphi_{a}(\varepsilon_{a})\,\\\
&\prod_{{\mathcal{A}}^{\prime}_{a_{0}}\ni b\neq
a}\,\Big{(}\int\,\Theta\big{(}h_{a,\varepsilon_{a}}(x)-h_{b,\varepsilon_{b}}(x)\big{)}\varphi_{b}(\varepsilon_{b})\,{\mathrm{d}}\varepsilon_{b}\Big{)}.\end{split}$
(3.16)

  7. vii)

The derivative of $m_{a_{0}}\in
C^{\infty}({\mathbf{X}}\setminus{{\mathbf{X}}_{a_{0}}})$ obeys

$\displaystyle\begin{split}\quad\quad\nabla\big{(}m_{a_{0}}(x)-\lvert
x\rvert\big{)}&\\\
=\sum_{a\in{\mathcal{A}}^{\prime}_{a_{0}}}\,\int\,{\mathrm{d}}\varepsilon_{a}&\nabla\big{(}{h_{a,\varepsilon_{a}}(x)}-\lvert
x\rvert\big{)}\,\varphi_{a}(\varepsilon_{a})\\\
&\prod_{{\mathcal{A}}^{\prime}_{a_{0}}\ni b\neq
a}\,\Big{(}\int\,\Theta\big{(}h_{a,\varepsilon_{a}}(x)-h_{b,\varepsilon_{b}}(x)\big{)}\varphi_{b}(\varepsilon_{b})\,{\mathrm{d}}\varepsilon_{b}\Big{)}.\end{split}$
(3.17)

In particular there exists $C>0$ such that for all
$x\in{\mathbf{X}}\setminus{{\mathbf{X}}_{a_{0}}}$ and all small parameter
values $\epsilon>0$

$\lvert\nabla\big{(}m_{a_{0}}(x)-\lvert x\rvert\big{)}\rvert\leq
C\epsilon^{3/2}.$ (3.18)

###### Proof.

The property iv coincides with (3.13). By the construction $m$ is convex, and
apart from this i and ii follow from iv and Lemma 3.7 1 and 2.

By Lemma 3.2 and iv it follows under the conditions of iii that

$m_{a_{0}}(x)=\Sigma_{{\mathcal{A}}^{\prime}_{a_{0}}\ni a\geq
b}\,m_{a}(x_{b})=\Sigma_{{\mathcal{A}}^{\prime}_{a_{0}}\ni
a}\,m_{a}(x_{b})=m_{a_{0}}(x_{b}).$

Here we used the trivial fact that since
$x\in\mathbf{X}^{a_{0}}_{b}(\varepsilon^{b}_{1})$, also
$x_{b}\in\mathbf{X}^{a_{0}}_{b}(\varepsilon^{b}_{1})$. We have verified iii.

As for v we use Lemma 3.5. It suffices to show that $m_{a_{0}}(x)=m_{a}(x)$
under the given conditions. Usng again iv we need to show that $m_{b}(x)=0$
for any $a\neq b$. If ${b\not\geq a}$, then $m_{b}(x)=0$ thanks to Lemma 3.7
4. If $b\gneq a$, we can apply Lemma 3.4, and consequently we conclude again
that $m_{b}(x)=0$.

As for vi obviously (3.16) is related to iv and (3.12) (it may be considered
as a variant of [GT, Lemma 7.6]). First we note that the proof of the
smoothness of the right-hand side expression of (3.12) shows that also the
right-hand side of (3.16) is smooth. Whence by continuity and density it
suffices to verify (3.16) as a formula for the distributional derivative of
$m$ as a function on the set

$\cap_{a,b\in{\mathcal{A}}^{\prime}_{a_{0}},\,a\neq
b}\\{x\mid{x^{a_{0}}_{a}}\text{ \,and\, }x^{a_{0}}_{b}\text{ span a two-
dimensional subspace}\\}.$

So let us consider a small open neighbourhood $U$ of any point $y$ in this
set. Pick a basis $e_{1},\dots,e_{d}$ of ${\mathbf{X}}$ such that for all
$j=1,\dots,d$ and all different $a,b\in{\mathcal{A}}^{\prime}_{a_{0}}$ the
vectors

$(e_{j}\cdot y^{a_{0}}_{a},e_{j}\cdot y^{a_{0}}_{b}),\,(\lvert
y^{a_{0}}_{a}\rvert^{2},\lvert
y^{a_{0}}_{b}\rvert^{2})\in{\mathbb{R}}^{2}\,\text{ are linear independent.}$
(3.19)

To see this is doable it suffices to show the linear independence for a single
vector $e_{1}$ (since then we can supplement with vectors $e_{2},\dots e_{d}$
close to $e_{1}$). In turn (by the same reasoning) it suffices to show that
(3.19) can be realized for $j=1$ and a fixed pair $(a,b)$. The latter is
obviously doable for a suitable vector in
$\mathrm{span}(y^{a_{0}}_{a},y^{a_{0}}_{b})$. Upon possibly shrinking $U$
(3.19) holds with $y$ replaced by an arbitrary $x\in U$, implying that for all
different $a,b\in{\mathcal{A}}^{\prime}_{a_{0}}$ and all admissible vectors
$\bar{\varepsilon}$

$U\cap\\{h_{a,\varepsilon_{a}}(x)=h_{b,\varepsilon_{b}}(x)\\}\text{ is a
regular hypersurface.}$

In fact for any $j=1,\dots,d$, $a\neq b$ and $\bar{\varepsilon}$

$x\in
U\cap\\{h_{a,\varepsilon_{a}}(x)=h_{b,\varepsilon_{b}}(x)\\}\Rightarrow\partial_{j}h_{a,\varepsilon_{a}}(x)\neq\partial_{j}h_{b,\varepsilon_{b}}(x);\quad\partial_{j}:=e_{j}\cdot\nabla.$
(3.20)

Here and below we represent the variable $x$ in terms of the constructed
basis, make the corresponding linear change of variables and note that in the
new coordinates $\nabla$ is represented as
$(\partial_{1},\dots,\partial_{d})$. With (3.20) we can easily for each
$j=1,\dots,d$ do the associated one-dimensional integration by parts using the
Fubini theorem (varying in the direction of $e_{j}$ while fixing the other
variables, including $\bar{\varepsilon}$). The obtained formulas amount to
(3.16).

As for vii the assertion (3.17) follows from a similar reasoning as given
above for (3.16), and (3.18) follows from an easy direct computation and
estimation of (3.17), cf. Lemma 3.7 5. ∎

###  3.2. Geometric considerations for $a_{0}=a_{\max}$

Throughout this subsection we consider the case $a_{0}=a_{\max}$ only. For
this example the functions $m_{a}$ from Lemma 3.7 and $m_{a_{0}}$ from Lemma
3.9 are smooth on ${\mathbf{X}}\setminus{0}$. We can introduce smooth
modifications by multiplying them by a suitable factor, say specifically by
the factor $\chi_{+}(2|x|)$. We adapt these modifications and will use
(slightly abusively) the same notation $m_{a}$ and $m_{a_{0}}$ for the
smoothed out versions. In either case we may then consider the first order
operators $M_{a}$ from (3.1) realized as self-adjoint operators. To see how
they can be used we need first to consider various conical subsets of
$\mathbf{X}\setminus{0}$ (recalling the basic example (3.2)).

Let for $a\in{\mathcal{A}}^{\prime}$ and $\varepsilon,\delta\in(0,1)$

$\displaystyle\begin{split}\mathbf{X}^{\prime}_{a}&={\mathbf{X}_{a}}\setminus\cup_{{b\gneq
a,\,b\in{\mathcal{A}}}}\,\mathbf{X}_{b}={\mathbf{X}_{a}}\setminus\cup_{{b\not\leq
a,\,b\in{\mathcal{A}}}\,}\mathbf{X}_{b},\\\
\mathbf{X}_{a}(\varepsilon)&=\\{x\in{\mathbf{X}}\mid\lvert
x_{a}\rvert>(1-\varepsilon)\lvert x\rvert\\},\\\
\mathbf{\Gamma}_{a}(\varepsilon)&=\big{(}\mathbf{X}\setminus\\{0\\}\big{)}\setminus\cup_{{b\not\leq
a,\,b\in{\mathcal{A}}^{\prime}}}\,\mathbf{X}_{b}(\varepsilon),\\\
\mathbf{Y}_{a}(\delta)&=\mathbf{X}_{a}(\delta)\setminus\cup_{b\gneq
a,\,b\in{\mathcal{A}}^{\prime}}\,\overline{\mathbf{X}_{b}(3\delta^{1/d_{a}})}.\end{split}$
(3.21)

Here and henceforth the overline means topological closure in $\mathbf{X}$.
The structure of the sets $\mathbf{X}_{a}(\varepsilon)$,
$\mathbf{\Gamma}_{a}(\varepsilon)$ and $\mathbf{Y}_{a}(\varepsilon)$ is
${{\mathbb{R}}_{+}V}$, where $V$ is a subset of the unit sphere
$\mathbf{S}^{a_{\max}}={\mathbb{S}}^{d-1}$ in $\mathbf{X}$. For
$\mathbf{X}_{a}(\varepsilon)$ and $\mathbf{Y}_{a}(\varepsilon)$ the set $V$ is
relatively open, while for $\mathbf{\Gamma}_{a}(\varepsilon)$ the set is
compact.

######  Lemma 3.10.

For any $a\in{\mathcal{A}}^{\prime}$ and $\varepsilon\in(0,1)$

$\mathbf{\Gamma}_{a}(\varepsilon)\subseteq\cup_{b\leq
a}\,\mathbf{X}^{\prime}_{b}.$ (3.22)

###### Proof.

For any $x\in\mathbf{\Gamma}_{a}(\varepsilon)$ we introduce $b(x)\leq a$ by
$\mathbf{X}_{b(x)}=\cap_{b\leq a,\,x\in\mathbf{X}_{b}}\,\mathbf{X}_{b}$ and
then check that $x\in\mathbf{X}^{\prime}_{b(x)}$: If not,
$x\in\mathbf{X}_{b}\cap\mathbf{X}_{b(x)}$ for some $b$ with $b(x)\lneq b\leq
a$. Consequently
$\mathbf{X}_{b}=\mathbf{X}_{b}\cap\mathbf{X}_{b(x)}=\mathbf{X}_{b(x)}$,
contradicting that $b\neq b(x)$. ∎

Thanks to (3.22) we can for any $a\in{\mathcal{A}}^{\prime}$ and any
$\varepsilon,\delta_{0}\in(0,1)$ write

$\mathbf{\Gamma}_{a}(\varepsilon)\subseteq\cup_{b\leq
a}\cup_{\delta\in(0,\delta_{0}]}\,\mathbf{Y}_{b}(\delta).$ (3.23a) By
compactness (3.23a) leads for any fixed $\varepsilon,\delta_{0}\in(0,1)$ to
the existence of a finite covering
$\mathbf{\Gamma}_{a}(\varepsilon)\subseteq\cup_{j\leq
J}\,\,\mathbf{Y}_{b_{j}}(\delta_{j}),$ (3.23b) where
$\delta_{1},\dots,\delta_{J}\in(0,\delta_{0}]$ and $b_{1},\dots,b_{J}\leq a$.
Here and in the following we suppress the dependence of quantities on the
given $a\in{\mathcal{A}}^{\prime}$.

To make contact with Subsection 3.1 we recall that the functions $m_{a}$,
$a\in{\mathcal{A}}^{\prime}$, depend on a positive parameter $\epsilon$ and by
Lemma 3.7 4 fulfill

$\operatorname{supp}m_{a}\subseteq\mathbf{\Gamma}_{a}(\epsilon^{d}).$ (3.24)

Having Lemmas 3.7 and 3.9 at our disposal we also recall that the (small)
positive parameter $\epsilon$ appears independently in the two lemmas. First
we choose and fix the same small $\epsilon$ for the lemmas. Then we need
applications of Lemma 3.9 for $\epsilon_{1},\dots,\epsilon_{J}\leq\epsilon$
fixed as follows. The latter devices will be used to control commutators, cf.
Remark 2.3 ii.

For each $a\in{\mathcal{A}}^{\prime}$ we apply (3.23b) with
$\varepsilon=\delta_{0}=\epsilon^{d}$ (cf. (3.24)). Since
$\delta_{j}\leq\delta_{0}$, we can introduce positive
$\epsilon_{1},\dots,\epsilon_{J}\leq\epsilon$ by the requirement
$\epsilon_{j}^{d_{b_{j}}}=\delta_{j}$. The inputs $\epsilon=\epsilon_{j}$ in
Lemma 3.9 yield corresponding functions, say denoted $m_{j}$. In particular in
the region ${\mathbf{Y}}_{b_{j}}(\delta_{j})$ the function $m_{a}$ from Lemma
3.7 only depends on $x_{b_{j}}$ (thanks to Lemma 3.7 3 and the property
${\mathbf{X}}_{b_{j}}(\delta_{j})\subseteq{\mathbf{X}}_{b_{j}}(\epsilon^{d_{b_{j}}})$),
while (thanks to Lemma 3.9 v and recalling $a_{0}=a_{\max}$)

$m_{j}(x)=C_{j}\lvert x_{b_{j}}\rvert,\quad
C_{j}=\int_{\mathbb{R}}(1+\varepsilon)\,\varphi_{b_{j}}(\varepsilon)\,{\mathrm{d}}\varepsilon.$
(3.25a) Obviously $\lvert y^{\prime}\rvert$ is non-degenerately convex in
$y^{\prime}\in{\mathbf{X}}_{b_{j}}\setminus\\{0\\}$, meaning that the
restricted Hessian $\big{(}\nabla_{y^{\prime}}^{2}\lvert
y^{\prime}\rvert\big{)}_{|{\mathbf{X}}_{b_{j}}\cap\\{y\\}^{\perp}}\text{ is
positive definite at any }y\in{\mathbf{X}}_{b_{j}}\setminus\\{0\\}.$ (3.25b)

These properties can be applied as follows using the vector-valued first order
operators

$G_{a}={\mathcal{H}}_{a}(x_{a})\cdot p_{a},\text{ where
}{\mathcal{H}}_{a}(x_{a})=\chi_{+}(2\lvert x_{a}\rvert)\lvert
x_{a}\rvert^{-1/2}\big{(}I-\lvert x_{a}\rvert^{-2}\lvert x_{a}\rangle\langle
x_{a}\rvert\big{)}.$ (3.26)

We choose for the considered $a\in{\mathcal{A}}^{\prime}$ a quadratic
partition $\xi_{1},\dots,\xi_{J}\in C^{\infty}({\mathbb{S}}^{d-1})$ (viz
$\Sigma_{j}\,\xi_{j}^{2}=1$) subordinate to the covering (3.23b) (recalling
the discussion before Lemma 3.10). Then we can write, using the support
properties (3.24) and (3.23b),

$\displaystyle m_{a}(x)=\Sigma_{j\leq J}\,\,m_{a,j}(x);\quad
m_{a,j}(x)=\xi_{j}(\hat{x})^{2}m_{a}(x),\quad\hat{x}=x/\lvert x\rvert,$

and from the previous discussion it follows that

$\displaystyle\chi_{+}(\lvert
x\rvert)^{2}m_{a,j}(x)=\xi_{j}(\hat{x})^{2}\chi_{+}(\lvert
x\rvert)^{2}m_{a}(x_{b_{j}}),$

as well as

$\displaystyle\begin{split}&p\cdot\big{(}\chi_{+}(\lvert
x\rvert)^{2}\nabla^{2}m_{a}(x)\big{)}p=\Sigma_{j\leq
J}\,\,p\cdot\big{(}\xi_{j}(\hat{x})^{2}\chi_{+}(\lvert
x\rvert)^{2}\nabla^{2}m_{a}(x_{b_{j}})\big{)}p,\\\ &=\Sigma_{j\leq
J}\,\,G^{*}_{b_{j}}\big{(}\xi_{j}(\hat{x})^{2}\chi_{+}(\lvert
x\rvert)^{2}{\mathcal{G}}_{j}\big{)}G_{b_{j}};\quad{\mathcal{G}}_{j}={\mathcal{G}}_{j}(x_{b_{j}})\text{
bounded}.\end{split}$ (3.27a) In turn using the convexity property of $m_{j}$
and the previous discussion (cf. (3.25a) and (3.25b)) we conclude that
$\displaystyle G^{*}_{b_{j}}\xi_{j}(\hat{x})^{2}\chi_{+}(\lvert
x\rvert)^{2}G_{bj}\leq 2p\cdot\big{(}\chi_{+}(\lvert
x\rvert)^{2}\nabla^{2}m_{j}(x)\big{)}p.$ (3.27b)

Here the right-hand side is the ‘leading term’ of the commutator
$\tfrac{1}{2}\mathrm{i}[p^{2},M_{j}]$, where $M_{j}$ is given by (3.1) for the
modification of $m_{j}$ given by the function $\chi_{+}(2|x|)m_{j}(x)$. More
precisely, in the sense of (2.2b), for any real $f\in
C_{\mathrm{c}}^{\infty}({\mathbb{R}})$

$\displaystyle\begin{split}f(H)\Big{(}2p\cdot\big{(}\chi_{+}(\lvert
x\rvert)^{2}\nabla^{2}m_{j}(x)\big{)}p-\tfrac{1}{2}\mathrm{i}[H,&\chi_{+}(\lvert
x\rvert)M_{j}\chi_{+}(\lvert x\rvert)]\Big{)}f(H)\\\ &={\mathcal{O}}(\langle
x\rangle^{-1-2\mu}).\end{split}$ (3.27c)

We note that a basic application of (3.27a)–(3.27c) comes from combining these
features with Lemma 2.2 applied concretely with

$\Psi=\Psi_{j}=\tfrac{1}{2}f_{1}(H)\chi_{+}(\lvert
x\rvert)M_{j}\chi_{+}(\lvert x\rvert)f_{1}(H),$ (3.28a) where $f_{1}$ is any
narrowly supported standard support function obeying $f_{1}=1$ on a
neighbourhood of a given $\lambda\not\in{\mathcal{T}}_{{\mathrm{p}}}(H)$. This
leads in fact to (2.7b) with ${\lvert
Qf_{1}(H)\rvert}^{2}=2f_{1}(H){p\cdot\big{(}\chi_{+}(\lvert
x\rvert)^{2}\nabla^{2}m_{j}(x)\big{)}p}f_{1}(H),$ (3.28b) and therefore in
particular to the ‘$Q$-bounds’
$\displaystyle\begin{split}\sup_{\operatorname{Im}z\neq 0}\lVert
Q_{j}{f_{1}}&(H){R(z)}\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}}^{d})}<\infty;\\\
&\quad Q_{j}=\xi_{j}(\hat{x})\chi_{+}(\lvert x\rvert)G_{b_{j}},\,\,j\leq
J.\end{split}$ (3.28c)

Finally we introduce for $a\in{\mathcal{A}}^{\prime}$ functions $\xi^{+}_{a}$
and $\tilde{\xi}^{+}_{a}$ as follows. First choose any $\xi_{a}\in
C^{\infty}(\mathbf{S}^{a_{0}})$ (recall $a_{0}=a_{\max}$) such that
$\xi_{a}=1$ in $\mathbf{S}^{a_{0}}\cap\mathbf{\Gamma}_{a}(\varepsilon)$ and
$\xi_{a}=0$ on
$\mathbf{S}^{a_{0}}\setminus\mathbf{\Gamma}_{a}(\varepsilon/2)$. Choose then
any $\tilde{\xi}_{a}\in C^{\infty}(\mathbf{S}^{a_{0}})$ using this recipe with
$\varepsilon$ replaced by $\varepsilon/2$. Finally let
$\xi^{+}_{a}(x)=\xi_{a}(\hat{x})\chi_{+}(4\lvert x\rvert)$ and
$\tilde{\xi}^{+}_{a}(x)=\tilde{\xi}_{a}(\hat{x})\chi_{+}(8\lvert x\rvert)$,
and note that $\tilde{\xi}^{+}_{a}\xi^{+}_{a}=\xi^{+}_{a}$. Applied to
$\varepsilon=\epsilon^{d}$ it follows from Lemma 3.7 4 that the corresponding
channel localization operators $M_{a}$ fulfill

$\displaystyle
M_{a}=M_{a}\xi^{+}_{a}=\xi^{+}_{a}M_{a}=M_{a}\tilde{\xi}^{+}_{a}=\tilde{\xi}^{+}_{a}M_{a},$
(3.29)

which in applications provides ‘free factors’ of $\xi^{+}_{a}$ and
$\tilde{\xi}^{+}_{a}$ where convenient. In particular it is useful for
replacing $H$ by $H_{a}$ (or vice versa) in the presence of a factor $M_{a}$.

###  3.3. Geometric considerations for $a_{0}\neq a_{\max}$

For $a_{0}\neq a_{\max}$ the functions $m^{a_{0}}_{a}$,
$a\in{\mathcal{A}}^{\prime}_{a_{0}}$, from Lemma 3.7 and $m_{a_{0}}^{a_{0}}$
from Lemma 3.9 are smooth on ${\mathbf{X}}\setminus{{\mathbf{X}}_{a_{0}}}$
however not on ${\mathbf{X}}\setminus\\{0\\}$. We can introduce smooth
modifications by multiplying them by a factor of $\chi_{+}(2|x|)$ (as before)
as well as by a factor of $\chi_{+}(2|x^{a_{0}}|/\kappa|x|)$ for a small
parameter $\kappa>0$. Whence

$m_{a,\kappa}(x)=m_{a,\kappa}^{a_{0}}(x)=\chi_{+}(2|x|)\chi_{+}(2|x^{a_{0}}|/\kappa|x|)m^{a_{0}}_{a}(x);\quad
a\in{\mathcal{A}}_{a_{0}}.$ (3.30a) Note that (3.30a) applies to
$a\in{\mathcal{A}}^{\prime}_{a_{0}}$ as well as to $a=a_{0}$. In either case
we may then consider the correponding first order operators (3.1), say denoted
by $M_{a,\kappa}=M^{a_{0}}_{a,\kappa},$ (3.30b)

again realized as self-adjoint operators. They will be used in a similar way
as for the case $a_{0}=a_{\max}$ discussed in Subsection 3.2, however now in
combination with an additional input from Section 6.

We mimic the procedure of Subsection 3.2 and consider the following conical
subsets of $\mathbf{X}\setminus{{\mathbf{X}}_{a_{0}}}$.

Let for $a\in{\mathcal{A}}_{a_{0}}^{\prime}$ and $\varepsilon,\delta\in(0,1)$

$\displaystyle\begin{split}\mathbf{X}^{{}^{\prime}a_{0}}_{a}&={\mathbf{X}_{a}}\setminus\cup_{{b\gneq
a,\,b\in{\mathcal{A}}_{a_{0}}}}\,\mathbf{X}_{b}={\mathbf{X}_{a}}\setminus\cup_{{b\not\leq
a,\,b\in{\mathcal{A}}_{a_{0}}}\,}\mathbf{X}_{b},\\\
\mathbf{X}^{a_{0}}_{a}(\varepsilon)&=\\{x\in{\mathbf{X}}\mid\lvert
x^{a_{0}}_{a}\rvert>(1-\varepsilon)\lvert x^{a_{0}}\rvert\\},\\\
\mathbf{\Gamma}^{a_{0}}_{a}(\varepsilon)&=\big{(}\mathbf{X}\setminus{{\mathbf{X}}_{a_{0}}}\big{)}\setminus\cup_{{b\not\leq
a,\,b\in{\mathcal{A}}_{a_{0}}^{\prime}}}\,\mathbf{X}^{a_{0}}_{b}(\varepsilon),\\\
\mathbf{Y}^{a_{0}}_{a}(\delta)&=\mathbf{X}^{a_{0}}_{a}(\delta)\setminus\cup_{b\gneq
a,\,b\in{\mathcal{A}}_{a_{0}}^{\prime}}\,\overline{\mathbf{X}^{a_{0}}_{b}(3\delta^{1/d_{a}})}.\end{split}$
(3.31)

The structure of the sets $\mathbf{X}^{a_{0}}_{a}(\varepsilon)$,
$\mathbf{\Gamma}^{a_{0}}_{a}(\varepsilon)$ and
$\mathbf{Y}^{a_{0}}_{a}(\varepsilon)$ is
$\big{(}{\mathbb{R}}_{+}V^{a_{0}}\big{)}\oplus\mathbf{X}_{a_{0}}$, where
$V^{a_{0}}$ is a subset of the unit sphere $\mathbf{S}^{a_{0}}$ in
$\mathbf{X}^{a_{0}}$. For $\mathbf{X}^{a_{0}}_{a}(\varepsilon)$ and
$\mathbf{Y}^{a_{0}}_{a}(\varepsilon)$ the set
$V^{a_{0}}\subseteq\mathbf{S}^{a_{0}}$ is (relatively) open, while for
$\mathbf{\Gamma}^{a_{0}}_{a}(\varepsilon)$ the set $V^{a_{0}}$ is compact.

In particular we obtain from Lemma 3.10 (now applied to $a_{0}$ rather than to
$a_{\max}$)

$\text{For any }a\in{\mathcal{A}}_{a_{0}}^{\prime}\text{ \,and\,
}\varepsilon\in(0,1):\quad\mathbf{\Gamma}^{a_{0}}_{a}(\varepsilon)\subseteq\cup_{b\leq
a}\,\mathbf{X}^{{}^{\prime}a_{0}}_{b}.$ (3.32)

Thanks to (3.32) we can for any $a\in{\mathcal{A}}_{a_{0}}^{\prime}$ and any
$\varepsilon,\delta_{0}\in(0,1)$ write

$\mathbf{\Gamma}^{a_{0}}_{a}(\varepsilon)\subseteq\cup_{b\leq
a}\cup_{\delta\in(0,\delta_{0}]}\,\mathbf{Y}^{a_{0}}_{b}(\delta),$ (3.33a) and
then compactness leads to the existence of a finite covering
$\mathbf{\Gamma}^{a_{0}}_{a}(\varepsilon)\subseteq\cup_{j\leq
J}\,\,\mathbf{Y}^{a_{0}}_{b_{j}}(\delta_{j}),$ (3.33b) where
$\delta_{1},\dots,\delta_{J}\in(0,\delta_{0}]$ and $b_{1},\dots,b_{J}\leq a$.

Again, to make contact with Subsection 3.1, we recall that the functions
$m^{a_{0}}_{a}$, $a\in{\mathcal{A}}^{\prime}_{a_{0}}$, depend on a positive
parameter $\epsilon$ and by Lemma 3.7 4 fulfill

$\operatorname{supp}m^{a_{0}}_{a}\subseteq\mathbf{\Gamma}^{a_{0}}_{a}(\epsilon^{d}).$
(3.34)

Having Lemmas 3.7 and 3.9 at our disposal we also recall that the (small)
positive parameter $\epsilon$ appear independently in the two lemmas. As
before we first choose and fix the same small $\epsilon$ for the lemmas. Then
we need applications of Lemma 3.9 along the pattern of that in Section 3.2.
Whence for each $a\in{\mathcal{A}}_{a_{0}}^{\prime}$ we apply (3.33b) with
$\varepsilon=\delta_{0}=\epsilon^{d}$ (cf. (3.34)). Since
$\delta_{j}\leq\delta_{0}$, we can introduce positive
$\epsilon_{1},\dots,\epsilon_{J}\leq\epsilon$ by the requirement
$\epsilon_{j}^{d_{b_{j}}}=\delta_{j}$. The inputs $\epsilon=\epsilon_{j}$ in
Lemma 3.9 yield corresponding functions, say denoted $m_{j}$. In particular in
the region ${\mathbf{Y}}_{b_{j}}(\delta_{j})$ the function $m_{a}$ from Lemma
3.7 only depends on $x_{b_{j}}$ (thanks to Lemma 3.7 3 and the property
${\mathbf{X}}_{b_{j}}(\delta_{j})\subseteq{\mathbf{X}}_{b_{j}}(\epsilon^{d_{b_{j}}})$),
while (thanks to Lemma 3.9 v)

$m_{j}(x)=\int_{\mathbb{R}}h_{b_{j},\varepsilon}(x)\,\varphi_{b_{j}}(\varepsilon)\,{\mathrm{d}}\varepsilon.$
(3.35a) It is an important (elementary) property that
$h_{b_{j},\varepsilon}(x)=h_{b_{j},\varepsilon}(x_{b_{j}})=h_{b_{j},\varepsilon}(y^{\prime})$
is non-degenerately convex in
$y^{\prime}\in{\mathbf{X}}_{b_{j}}\setminus\\{0\\}$, meaning that the
restricted Hessian
$\big{(}\nabla_{y^{\prime}}^{2}h_{b_{j},\varepsilon}(y^{\prime})\big{)}_{|{\mathbf{X}}_{b_{j}}\cap\\{y\\}^{\perp}}\text{
is positive definite at any }y\in{\mathbf{X}}_{b_{j}}\setminus\\{0\\}.$
(3.35b) Note that the corresponding eigenvalues are close to one (and hence
positive) thanks to the smallness of $\epsilon$. Obviously the non-degeneracy
holds as well for the expression $m_{j}(x_{b_{j}})$ given in (3.35a).

Choose a quadratic partition $\xi_{1},\dots,\xi_{J}\in
C^{\infty}(\mathbf{S}^{a_{0}})$ (viz $\Sigma_{j}\,\xi_{j}^{2}=1$) subordinate
to the covering (3.33b) (recalling the discussion before (3.32). Then we can
write, using the support properties (3.34) and (3.33b) and recalling the
definition of $m_{a,\kappa}^{a_{0}}$ and $m_{a_{0},\kappa}^{a_{0}}$ from
(3.30a) as well as the notation $\hat{x}=x/\lvert x\rvert$,

$m_{a,\kappa}(x)=m^{a_{0}}_{a,\kappa}(x)=\Sigma_{j\leq
J}\,\,m_{a,j,\kappa}(x);\quad
m_{a,j,\kappa}(x)=m^{a_{0}}_{a,j,\kappa}(x)=\xi_{j}(\widehat{x^{a_{0}}})^{2}m^{a_{0}}_{a,\kappa}(x).$

We note that

$\displaystyle\chi_{\kappa+}(x)^{2}m_{a,j,\kappa}(x)=$
$\displaystyle\xi_{j}(\widehat{x^{a_{0}}})^{2}\chi_{\kappa+}(x)^{2}m_{a,\kappa}(x_{b_{j}}),$
$\displaystyle\chi_{\kappa+}(x):=\chi_{+}(\lvert
x\rvert)\chi_{+}(|x^{a_{0}}|/\kappa|x|),$

as well as (recalling the definition (3.26))

$\displaystyle\begin{split}&p\cdot\big{(}\chi_{\kappa+}(x)^{2}\nabla^{2}m_{a,\kappa}(x)\big{)}p=\Sigma_{j\leq
J}\,\,p\cdot\big{(}\xi_{j}(\widehat{x^{a_{0}}})^{2}\chi_{\kappa+}(x)^{2}\nabla^{2}m_{a,\kappa}(x_{b_{j}})\big{)}p,\\\
&=\Sigma_{j\leq
J}\,\,G^{*}_{b_{j}}\big{(}\xi_{j}(\widehat{x^{a_{0}}})^{2}\chi_{\kappa+}(x)^{2}{\mathcal{G}}_{j,\kappa}\big{)}G_{b_{j}};\quad{\mathcal{G}}_{j,\kappa}={\mathcal{G}}_{j,\kappa}(x_{b_{j}})\text{
bounded}.\end{split}$ (3.36a) In turn using the convexity property of $m_{j}$
and the previous discussion (cf. (3.35a) and (3.35b)) we conclude that
$G^{*}_{b_{j}}\xi_{j}(\widehat{x^{a_{0}}})^{2}\chi_{\kappa+}(x)^{2}G_{b_{j}}\leq
2p\cdot\big{(}\chi_{\kappa+}(x)^{2}\nabla^{2}m_{j}\big{)}(x)p.$ (3.36b)

Here the right-hand side is the ‘leading term’ of the commutator
$\tfrac{1}{2}\mathrm{i}[p^{2},M_{j,\kappa}]$, where $M_{j,\kappa}$ is given by
(3.1) for the modification of $m_{j}$ given by the function

$\chi_{+}(2|x|)\chi_{+}(2|x^{a_{0}}|/\kappa|x|)m_{j}(x).$

More precisely, in the sense of (2.2b) and (2.3a), for any real $f\in
C_{\mathrm{c}}^{\infty}({\mathbb{R}})$

$\displaystyle\begin{split}f(H_{a_{0}})\Big{(}2p\cdot\big{(}\chi_{\kappa+}(x)^{2}&\nabla^{2}m_{j}\big{)}(x)p-\tfrac{1}{2}\mathrm{i}[H_{a_{0}},\chi_{\kappa+}(x)M_{j,\kappa}\chi_{\kappa+}(x)]\Big{)}f(H_{a_{0}})\\\
&={\mathcal{O}}(\langle x\rangle^{-1-2\mu})+{\mathcal{O}}^{a}_{\kappa}(\langle
x\rangle^{-1}).\end{split}$ (3.36c)

Due to the appearance of the second term to the right we can not directly use
Lemma 2.2 as we derived (3.28c) from (3.27b) and (3.27c). We shall in Section
8 remedy this point for $H_{a_{0}}$,
$a_{0}\in{\mathcal{A}}\setminus\\{a_{\min},a_{\max}\\}$, using the ‘$Q$-bound’
(6.5b) from Section 6. With this bound and a required energy localization for
$H^{a_{0}}$ the scheme of proof of Lemma 2.2 applies again, leading to
‘$Q$-bounds’ on the form

$\displaystyle\begin{split}\sup_{\operatorname{Im}z\neq 0}\lVert
Q_{j,\kappa}{f_{1}}&(H_{a_{0}}){R_{a_{0}}(z)}\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}}^{d})}<\infty;\\\
&\quad
Q_{j,\kappa}=\xi_{j}(\widehat{x^{a_{0}}})\chi_{\kappa+}(x)G_{b_{j}}f(H^{a_{0}}),\,\,j\leq
J.\end{split}$ (3.36d)

Here $f_{1}$ is given as in (3.28c) and $f$ is any standard support function
supported in ${\mathbb{R}}\setminus{\mathcal{T}}_{\mathrm{p}}(H^{a_{0}})$.
Once $f$ is fixed (we explain in Section 7 how to choose it for our purposes),
also $\kappa>0$ may be fixed (sufficiently small). More precisely (3.36d) then
follows from applying Lemma 2.2 with

$\Psi=\Psi_{j,\kappa}=\tfrac{1}{2}f_{1}(H_{a_{0}})f(H^{a_{0}})\chi_{\kappa+}(x)M_{j,\kappa}\chi_{\kappa+}(x)f(H^{a_{0}})f_{1}(H_{a_{0}}),$
(3.36e)

Finally we introduce for $a\in{\mathcal{A}}^{\prime}$ functions $\xi^{+}_{a}$
and $\tilde{\xi}^{+}_{a}$ as follows. First choose any $\xi_{a}\in
C^{\infty}(\mathbf{S}^{a_{0}})$ such that $\xi_{a}=1$ in
$\mathbf{S}^{a_{0}}\cap\mathbf{\Gamma}^{a_{0}}_{a}(\varepsilon)$ and
$\xi_{a}=0$ in
$\mathbf{S}^{a_{0}}\setminus\mathbf{\Gamma}^{a_{0}}_{a}(\varepsilon/2)$.
Choose then any $\tilde{\xi}_{a}\in C^{\infty}(\mathbf{S}^{a_{0}})$ using this
recipe with $\varepsilon$ replaced by $\varepsilon/2$. Finally let
$\xi^{+}_{a,\kappa}(x)=\xi_{a}(\widehat{x^{a_{0}}})\chi_{+}(4\lvert
x\rvert)\chi_{+}(4|x^{a_{0}}|/\kappa|x|)$ and
$\tilde{\xi}^{+}_{a,\kappa}(x)=\tilde{\xi}_{a}(\widehat{x^{a_{0}}})\chi_{+}(8\lvert
x\rvert)\chi_{+}(8|x^{a_{0}}|/\kappa|x|)$, and note that
$\tilde{\xi}^{+}_{a}\xi^{+}_{a,\kappa}=\xi^{+}_{a,\kappa}$. Applied to
$\varepsilon=\epsilon^{d}$ it follows from Lemma 3.7 that the corresponding
channel localization operators $M_{a,\kappa}$ from (3.30b) fulfill

$\displaystyle
M_{a,\kappa}=M_{a,\kappa}\xi^{+}_{a,\kappa}=\xi^{+}_{a,\kappa}M_{a,\kappa}=M_{a,\kappa}\tilde{\xi}^{+}_{a,\kappa}=\tilde{\xi}^{+}_{a,\kappa}M_{a,\kappa},$
(3.37)

which in applications will provide ‘free factors’ of $\xi^{+}_{a,\kappa}$ and
$\tilde{\xi}^{+}_{a,\kappa}$ where convenient.

##  4\. $N$-body short-range scattering theory

For any $a\in{\mathcal{A}}^{\prime}$ we introduce

$K_{a}(x_{a},\lambda)=\sqrt{\lambda}\lvert x_{a}\rvert\text{ \,and\,
}S_{a}(\xi_{a},t)=t\xi^{2};\quad(x_{a},\lambda),(\xi_{a},t)\in{\mathbf{X}_{a}}\times{\mathbb{R}}_{+}.$

For any _channel_ $\alpha=(a,\lambda^{\alpha},u^{\alpha})$, i.e.
$a\in{\mathcal{A}}^{\prime}$, $\lambda^{\alpha}\in{\mathbb{R}}$ and
$(H^{a}-\lambda^{\alpha})u^{\alpha}=0$ for a normalized
$u^{\alpha}\in\mathcal{H}^{a}$, we introduce the corresponding _channel wave
operators_

$W_{\alpha}^{\pm}=\operatorname*{s-lim}_{t\to\pm\infty}{\mathrm{e}}^{\mathrm{i}tH}J_{\alpha}{\mathrm{e}}^{-\mathrm{i}(S_{a}^{\pm}(p_{a},t)+\lambda^{\alpha}t)},\quad
J_{\alpha}\varphi=u^{\alpha}\otimes\varphi,$ (4.1)

where $p_{a}=-\mathrm{i}\nabla_{x_{a}}$ and $S_{a}^{\pm}(\xi_{a},\pm|t|)=\pm
S_{a}(\pm\xi_{a},|t|)=t\xi_{a}^{2}$. (With long-range potentials included, not
considered in this paper, the functions $K_{a}$ and $S_{a}$ need to be chosen
slightly different, see [Is1, II, Sk1].) We outline a proof of the existence
of the limits (4.1) in Remark 4.6 ii following [Sk1, Ya2]. It is a general
fact that the existence of the channel wave operators implies their
orthogonality, see for example [RS, Theorem XI.36].

Let us for the channel $\alpha=(a,\lambda^{\alpha},u^{\alpha})$ introduce the
notation

$k_{\alpha}=p_{a}^{2}+\lambda^{\alpha},\quad
I^{\alpha}=(\lambda^{\alpha},\infty)\quad\text{and}\quad{\mathcal{E}}^{\alpha}=I^{\alpha}\setminus{{\mathcal{T}}_{{\mathrm{p}}}(H)}.$

Note the intertwining property $HW_{\alpha}^{\pm}\supseteq
W_{\alpha}^{\pm}k_{\alpha}$ and the fact that $k_{\alpha}$ is diagonalized by
the unitary map $F_{\alpha}:L^{2}(\mathbf{X}_{a})\to
L^{2}(I^{\alpha};{\mathcal{G}}_{a})$,
${\mathcal{G}}_{a}=L^{2}(\mathbf{S}_{a})$,
$\mathbf{S}_{a}=\mathbf{X}_{a}\cap{\mathbb{S}}^{d_{a}-1}$ with
$d_{a}=\dim\mathbf{X}_{a}$, given by

$\displaystyle\begin{split}(F_{\alpha}\varphi)(\lambda,\omega)&=(2\pi)^{-d_{a}/2}2^{-1/2}\lambda_{\alpha}^{(d_{a}-2)/4}\int{\mathrm{e}}^{-\mathrm{i}\lambda^{1/2}_{\alpha}\omega\cdot
x_{a}}\varphi(x_{a})\,{\mathrm{d}}x_{a};\\\
&\quad\quad\lambda_{\alpha}=\lambda-\lambda^{\alpha},\quad\lambda\in
I^{\alpha}.\end{split}$ (4.2)

We denote

$c_{\alpha}^{\pm}(\lambda)={\mathrm{e}}^{\pm\mathrm{i}\pi(d_{a}-3)/4}\pi^{-1/2}\lambda_{\alpha}^{1/4},$

and $F_{\rho}=F(\\{x\in{\mathbf{X}}\mid\lvert x\rvert<\rho\\})$ for $\rho>1$
(considered below as multiplication operators).

######  Proposition 4.1 ([Sk1]).

For any channel $\alpha=(a,\lambda^{\alpha},u^{\alpha})$,
$\lambda\in{\mathcal{E}}^{\alpha}$, $\psi\in{\mathcal{B}}$ and
$g\in{\mathcal{G}}_{a}$ there exist the weak limits

$\displaystyle\begin{split}\langle\Gamma^{\pm}_{\alpha}&(\lambda)\psi,g\rangle=\lim_{\rho\to\infty}\,\overline{c_{\alpha}^{\pm}(\lambda)}\rho^{-1}\\\
&\big{\langle}F_{\rho}R(\lambda\pm\mathrm{i}0)\psi,F_{\rho}\big{(}u^{\alpha}\otimes\lvert
x_{a}\rvert^{(1-d_{a})/2}{\mathrm{e}}^{\pm\mathrm{i}K_{a}(\lvert
x_{a}\rvert\cdot,\lambda_{\alpha})}g(\pm\cdot)\big{)}\big{\rangle}.\end{split}$
(4.3)

Here the limits $\Gamma^{\pm}_{\alpha}(\lambda)$ are weakly continuous
${\mathcal{L}}({\mathcal{B}},{\mathcal{G}}_{a})$-valued functions of
$\lambda\in{\mathcal{E}}^{\alpha}$.

The restrictions of the map $F_{\alpha}(W^{\pm}_{\alpha})^{*}$ have strong
almost everywhere interpretations, meaning more precisely

$\displaystyle
F_{\alpha}(W^{\pm}_{\alpha})^{*}\psi=\int^{\oplus}_{I^{\alpha}}\big{(}F_{\alpha}(W^{\pm}_{\alpha})^{*}\psi\big{)}(\lambda)\,{\mathrm{d}}\lambda;\quad\psi\in{\mathcal{H}}.$

When applied to $\psi\in{\mathcal{B}}\subseteq{\mathcal{H}}$ the following
relationship to Proposition 4.1 holds.

######  Theorem 4.2 ([Sk1]).

For any channel $(a,\lambda^{\alpha},u^{\alpha})$ and any
$\psi\in{\mathcal{B}}\subseteq{\mathcal{H}}$

$\big{(}F_{\alpha}(W^{\pm}_{\alpha})^{*}\psi\big{)}(\lambda)=\Gamma_{\alpha}^{\pm}(\lambda)\psi\quad\text{for
a.e. }\lambda\in{\mathcal{E}}^{\alpha}.$ (4.4)

In particular for any $\psi\in{\mathcal{B}}$ the restrictions
$\big{(}F_{\alpha}(W^{\pm}_{\alpha})^{*}\psi\big{)}(\cdot)$ are weakly
continuous ${\mathcal{G}}_{a}$-valued functions on ${\mathcal{E}}^{\alpha}$.

######  Definition 4.3.

An energy
$\lambda\in{\mathcal{E}}:=(\min{\mathcal{T}}(H),\infty)\setminus{\mathcal{T}}_{\mathrm{p}}(H)$
is stationary complete for $H$ if

$\forall\psi\in
L^{2}_{\infty}:\,\,\sum_{\lambda^{\beta}<\lambda}\,\lVert\Gamma_{\alpha}^{\pm}(\lambda)\psi\rVert^{2}=\langle\psi,\delta(H-\lambda){\psi}\rangle.$
(4.5)

Asymptotic completeness follows by integration provided (4.5) is known for
almost all $\lambda\in{\mathcal{E}}$ (motivating the used terminology). The
orthogonality of the channel wave operators (4.1) implies (as demonstrated in
[Sk1, Subsection 9.2]) that

$\forall\lambda\in{\mathcal{E}},\,\forall\psi\in
L^{2}_{\infty}:\,\,\sum_{\lambda^{\beta}<\lambda}\,\lVert\Gamma_{\alpha}^{\pm}(\lambda)\psi\rVert^{2}\leq\langle\psi,\delta(H-\lambda){\psi}\rangle.$
(4.6)

It is also known (see [Sk1, Proposition 9.16]) that a _sufficient and
necessary condition_ for $\lambda\in{\mathcal{E}}$ be stationary complete is
given as follows:

For all $\psi\in L^{2}_{\infty}$ there exists
$(g_{\beta})_{\beta}\in{\mathcal{G}}:=\Sigma^{\oplus}_{\beta}\,{\mathcal{G}}_{b}$
(here $\beta=(b,\lambda^{\beta},u^{\beta})$ runs over all channels) such that,
as an identity in ${\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$ (equipped with the
quotient topology),

$R(\lambda+\mathrm{i}0)\psi=2\pi\mathrm{i}\sum_{\lambda^{\beta}<\lambda}J_{\beta}v^{+}_{\beta,\lambda}[g_{\beta}],$
(4.7)

where (recalling) $J_{\beta}\phi=u^{\beta}\otimes\phi$, here with $\phi$ taken
to be the _outgoing one-body quasi-modes_ corresponding to the plus cases of

$v^{\pm}_{\beta,\lambda}[g](x_{b}):=\mp\tfrac{\mathrm{i}}{2\pi}\big{(}c_{\beta}^{\pm}(\lambda)\big{)}^{-1}\chi_{+}(\lvert
x_{b}\rvert)\lvert
x_{b}\rvert^{(1-d_{b})/2}{\mathrm{e}}^{\pm\mathrm{i}K_{b}(x_{b},\lambda_{\beta})}g(\pm\hat{x}_{b});\quad
g\in{\mathcal{G}}_{b}.$ (4.8)

It is an elementary property that the right-hand side of (4.7) is well-defined
in ${\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$ for any
$(g_{\beta})_{\lambda^{\beta}<\lambda}\in{\mathcal{G}}$, cf. (7.2c). In fact,
cf. [Sk1, (9.22)], the quasi-modes enjoy an ‘orthogonality property’ entailing

$\Big{\lVert}\sum_{\lambda^{\beta}<\lambda}J_{\beta}v^{+}_{\beta,\lambda}[g_{\beta}]\Big{\rVert}_{\rm
quo}^{2}=\sum_{\lambda^{\beta}<\lambda}\big{(}4\pi\lambda_{\beta}^{1/2}\big{)}^{-1}\,\lVert
g_{\beta}\rVert^{2}.$ (4.9)

From a practical point of view (here explaining our strategy of verifying
(4.7)) it suffices for any given $\lambda\in{\mathcal{E}},\,\psi\in
L^{2}_{\infty}$ and $\epsilon>0$ to produce a vector
$(g^{\epsilon}_{\beta})_{\lambda^{\beta}<\lambda}\in{\mathcal{G}}$ such that

$\big{\lVert}R(\lambda+\mathrm{i}0)\psi-2\pi\mathrm{i}\sum_{\lambda^{\beta}<\lambda}J_{\beta}v^{+}_{\beta,\lambda}[g^{\epsilon}_{\beta}]\big{\rVert}_{\rm
quo}\leq\epsilon.$

It is known that if (4.7) holds for some
$(g_{\beta})_{\lambda^{\beta}<\lambda}\in{\mathcal{G}}$, then necessarily
$g_{\beta}=\Gamma_{\beta}^{+}(\lambda)\psi$.

The scattering matrix
$S(\lambda)=\big{(}S_{\beta\alpha}(\lambda)\big{)}_{\beta\alpha}$ is given a
priori for almost all $\lambda\in{\mathcal{E}}$ by

$\hat{S}_{\beta\alpha}:=F_{\beta}(W_{\beta}^{+})^{*}W_{\alpha}^{-}F_{\alpha}^{-1}=\int^{\oplus}_{I_{\beta\alpha}}S_{\beta\alpha}(\lambda)\,{\mathrm{d}}\lambda,\quad
I_{\beta\alpha}=I^{\beta}\cap I^{\alpha}.$

(For $\lambda\notin I^{\beta}\cap I^{\alpha}$ we let
$S_{\beta\alpha}(\lambda)=0$.)

The scattering matrix is known from [Sk1] to be a weakly continuous
${\mathcal{L}}({\mathcal{G}})$-valued function (in fact contraction-valued) on
${\mathcal{E}}$. At stationary complete energies the scattering matrix is
characterized geometrically as follows.

######  Theorem 4.4 ([Sk1]).

Let $\lambda\in{\mathcal{E}}$ be stationary complete and
$\alpha=(a,\lambda^{\alpha},u^{\alpha})$ be any channel with
$\lambda^{\alpha}<\lambda$. Then the following existence and uniqueness
results hold for any $g\in{\mathcal{G}}_{a}$.

  1. 1)

Let $u=\Gamma^{-}_{\alpha}(\lambda)^{*}g$, and let
$(g_{\beta})_{\beta}\in{\mathcal{G}}$ be given by
$g_{\beta}=S_{\beta\alpha}(\lambda)g$. Then, as an identity in
${\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$,

$\displaystyle
u=J_{\alpha}v^{-}_{\alpha,\lambda}[g]+\sum_{\lambda^{\beta}<\lambda}J_{\beta}v^{+}_{\beta,\lambda}[g_{\beta}].$
(4.10)

  2. 2)

Conversely, if (4.10) is fulfilled for some $u\in{\mathcal{B}}^{*}\cap
H^{2}_{\mathrm{loc}}({\mathbf{X}})$ with $(H-\lambda)u=0$ and for some
$(g_{\beta})_{\lambda^{\beta}<\lambda}\in{\mathcal{G}}$, then
$u=\Gamma^{-}_{\alpha}(\lambda)^{*}g$ and
$g_{\beta}=S_{\beta\alpha}(\lambda)g$.

######  Theorem 4.5 ([Sk1]).

  1. 1)

For any channel $\alpha$

    1. a)

the operators
$\Gamma^{\pm}_{\alpha}(\lambda)\in{\mathcal{L}}({\mathcal{B}},{\mathcal{G}}_{a})$
are strongly continuous at any stationary complete energy
$\lambda\in{\mathcal{E}}^{\alpha}$.

    2. b)

the operators
$\Gamma^{\pm}_{\alpha}(\lambda)^{*}\in{\mathcal{L}}({\mathcal{G}}_{a},L^{2}_{-s})$,
$s>1/2$, are strongly continuous in $\lambda\in{\mathcal{E}}^{\alpha}$.

  2. 2)

The ${\mathcal{L}}({\mathcal{G}})$-valued function
$S(\lambda)=\big{(}S_{\beta\alpha}(\lambda)\big{)}_{\beta\alpha}$ is strongly
continuous at any stationary complete $\lambda\in{\mathcal{E}}$. Moreover the
restriction of $S(\lambda)$ to the energetically open sector of
${\mathcal{G}}$ is unitary at any such energy $\lambda$.

######  Remarks 4.6.

  1. i)

Although we have referred to various results of [Sk1], strictly speaking the
condition on the short-range potentials is somewhat stronger than those of
Condition 2.1. However the conditions of [Sk1] are used to deal with the long-
range case and the stronger condition imposed on the short-range pair-
potentials there are clearly not needed if the long-range part of the
potentials is absent.

  2. ii)

One can show the well-definedness of the channel wave operators (4.1) using
[Sk1]. Indeed to show the existence of the limits,

$\lim_{t\to\pm\infty}{\mathrm{e}}^{\mathrm{i}tH}J_{\alpha}{\mathrm{e}}^{-\mathrm{i}(S_{a}^{\pm}(p_{a},t)+\lambda^{\alpha}t)}\varphi=\lim_{t\to\pm\infty}{\mathrm{e}}^{\mathrm{i}tH}{\mathrm{e}}^{-\mathrm{i}tH_{a}}J_{\alpha}\varphi,$

it suffices to consider $\varphi$ localized as
$\varphi=f_{1}(k_{\alpha})\varphi$ with $f_{1}$ being any narrowly supported
standard support function obeying $f_{1}=1$ on a neighbourhood of any given
$\lambda_{0}\not\in{\mathcal{T}}_{{\mathrm{p}}}(H)$. Furthermore we can assume
that the Fourier transform $\hat{\varphi}$ is supported away from collision
planes, and with this assumption the proof reduces to the existence of the
limits

$\operatorname*{s-lim}_{t\to\pm\infty}\,{f_{2}}(H){\mathrm{e}}^{\mathrm{i}tH}M_{a}{\mathrm{e}}^{-\mathrm{i}tH_{a}}{f_{2}}(H_{a}),$
(4.11)

where $M_{a}$ are given by (3.1) for functions $m_{a}$ constructed as in
Section 3.2 and $f_{2}\succ f_{1}$, also given supported near $\lambda_{0}$.
At this point the freedom of adjusting the small parameter $\epsilon>0$ in
Lemmas 3.7 and 3.9 is important, cf. the beginning of [Sk1, Section 9]. Using
now Section 3.2 the limits are obtained by Kato smoothness bounds, obtained by
invoking the ‘propagation observable’ (3.28a) along with (3.27a)–(3.27c) and
the limiting absorption principle (2.4a), applicably for $H$ as well as for
$H_{a}$. This procedure is explained in detail for a more complicated setting
in [Sk1, Section 7.2].

##  5\. A partition of unity

Let $\lambda\notin{\mathcal{T}}_{\mathrm{p}}(H)$ and $\psi\in L^{2}_{\infty}$
be given. Recall that we aim at deriving the asymptotics of
$R(\lambda+\mathrm{i}0)\psi$ in agreement with (4.7).

In this section we explain the first step in a procedure that in Section 8
will be iterated to derive (4.7). As the reader will see a certain ‘effective
partition of unity’ $I\simeq\Sigma_{a\in{\mathcal{A}}^{\prime}}\,S_{a}$, will
allow us to reduce the problem of asymptotics to that of
$R_{a}(\lambda+\mathrm{i}0)\psi$, $a\in{\mathcal{A}}^{\prime}$. We let
$a_{0}=a_{\max}$ and consider the corresponding operator $M_{a}$ from
Subsection 3.2.

Recalling (2.5e) the problem is reduced to the asymptotics of
$\chi_{+}(B/\epsilon_{0})R(\lambda+\mathrm{i}0)\psi$. The problem is further
reduced thanks to the following elementary estimate, cf. [Sk1, Lemma 8.1] and
its proof.

######  Lemma 5.1.

Let the positive parameter $\epsilon$ in the construction of $M_{a_{0}}$ be
sufficiently small, and let $f\in C_{\mathrm{c}}^{\infty}({\mathbb{R}})$ be
given. Then

$\chi_{-}(2M_{a_{0}}/\epsilon_{0})\chi_{+}(B/\epsilon_{0})f(H)={\mathcal{O}}(\langle
x\rangle^{-1/2}).$ (5.1)

###### Proof.

We can assume $f=f_{1}$ is a standard support function. Let $f_{2}$ be another
support function with $f_{2}\succ f_{1}$.

Introducing

$S=f_{2}(H)\chi_{-}(2M_{a_{0}}/\epsilon_{0})\chi_{+}(B/\epsilon_{0})\langle
x\rangle^{1/2}f_{1}(H),$

it suffices to be show that $S={\mathcal{O}}(\langle x\rangle^{0})$.

We estimate for any ${\breve{\psi}}\in L^{2}_{\infty}\subseteq{\mathcal{H}}$
by repeatly commutation using tacitly (2.1) (recall also the generic notation
$\langle T\rangle_{\varphi}=\langle\varphi,T\varphi\rangle$):

$\displaystyle\begin{split}&\tfrac{1}{6}\lVert{S{\breve{\psi}}}\rVert^{2}\\\
&\leq\langle\epsilon_{0}I-M_{a_{0}}\rangle_{S{\breve{\psi}}}+C_{1}\lVert{\breve{\psi}}\rVert^{2}\\\
&\leq\langle(\epsilon_{0}+C\epsilon)I-B\rangle_{{S{\breve{\psi}}}}+C_{2}\lVert{\breve{\psi}}\rVert^{2}\quad\quad\quad(\text{by
}\eqref{eq:compa})\\\
&\leq\langle(\epsilon_{0}+C\epsilon-\tfrac{4}{3}\epsilon_{0})I\rangle_{{S{\breve{\psi}}}}+C_{3}\lVert{\breve{\psi}}\rVert^{2}\\\
&\leq
C_{3}\lVert{\breve{\psi}}\rVert^{2}\quad\quad\quad\quad\quad\quad\quad\quad\quad(\text{for
}3C\epsilon\leq\epsilon_{0}).\end{split}$ (5.2)

By repeating the estimation (5.2) with $S{\breve{\psi}}$ replaced by $\langle
x\rangle^{s}S\langle x\rangle^{-s}{\breve{\psi}}$,
$s\in{\mathbb{R}}\setminus\\{0\\}$, we conclude the estimate (2.2a) with $t=0$
as wanted. Note that the above constant $C$ works in this case also. ∎

Thanks to (2.5e) and Lemma 5.1 we are lead to studying the asymptotics of

$\chi^{2}_{+}(2M_{a_{0}}/\epsilon_{0})\chi_{+}(B/\epsilon_{0})R(\lambda+\mathrm{i}0)\psi.$

In turn, taking narrowly supported functions $f_{2}\succ f_{1}$ such that
$f_{1}=1$ in a small neighbourhood of $\lambda$, it suffices to consider

$f_{2}(H)\chi^{2}_{+}(2M_{a_{0}}/\epsilon_{0})f_{2}(H)\chi_{+}(B/\epsilon_{0})R(\lambda+\mathrm{i}0)\psi\text{
assuming in addition }\psi=f_{1}(H)\psi.$

For given $\phi_{1},\phi_{2}\in{\mathcal{B}}^{*}$ we write
$\phi_{1}\simeq\phi_{2}$, if $\phi_{1}-\phi_{2}\in{\mathcal{B}}^{*}_{0}$. With
this convention we are lead to rewrite, using in the second step Lemma 3.9 iv,
in the third step conveniently (3.29), commutation and (2.4a), and in the last
step the limiting absorption principle and the ‘$Q$-bounds’ (2.11b) and
(3.28c) for $H$ as well as for $H_{a}$ (see the comments after the
computation)

$\displaystyle R(\lambda+\mathrm{i}0)\psi$ $\displaystyle\simeq
f_{2}(H)\chi^{2}_{+}(2M_{a_{0}}/\epsilon_{0})f_{2}(H)\chi_{+}(B/\epsilon_{0})R(\lambda+\mathrm{i}0)\psi$
$\displaystyle=f_{2}(H)h(M_{a_{0}})\,\sum_{a\in{\mathcal{A}}^{\prime}}\,M_{a}f_{2}(H)\chi_{+}(B/\epsilon_{0})R(\lambda+\mathrm{i}0)\psi\quad(\text{with
}h(x)=\chi^{2}_{+}(2x/\epsilon_{0})/x)$
$\displaystyle\simeq\Sigma_{a\in{\mathcal{A}}^{\prime}}\,S_{a}R(\lambda+\mathrm{i}0)\psi\quad(\text{with
}S_{a}=f_{2}(H_{a})h(M_{a_{0}})M_{a}f_{2}(H)\chi_{+}(B/\epsilon_{0})f_{2}(H))$
$\displaystyle=\Sigma_{a\in{\mathcal{A}}^{\prime}}\,R_{a}(\lambda+\mathrm{i}0)\psi_{a};$
$\displaystyle\quad\quad\quad\quad\quad\quad\psi_{a}=S_{a}\psi-\mathrm{i}T_{a}R(\lambda+\mathrm{i}0)\psi,\quad
T_{a}=\mathrm{i}\big{(}H_{a}S_{a}-S_{a}H\big{)}.$

The last step is formally correct by the second resolvent equation, but it
needs justification eventually to be given in Appendix A. Here we give formal
justification. It is a straightforward application of (2.1) to show that
$S_{a}={\mathcal{O}}(\langle x\rangle^{0})$, thus in particular the operator
preserves $L^{2}_{\infty}$. Whence the first term $S_{a}\psi\in
L^{2}_{\infty}$. The second term of $\psi_{a}$ is a more subtle one. We
rewrite its contribution in terms of any support function $f_{3}\succ f_{2}$
as

$\displaystyle-\mathrm{i}R_{a}(\lambda+\mathrm{i}0)$
$\displaystyle({\mathbf{D}}_{a}S_{a})R(\lambda+\mathrm{i}0)\psi;$
$\displaystyle{\mathbf{D}}_{a}T=\mathrm{i}\big{(}H_{a}f_{3}(H_{a})T-THf_{3}(H)\big{)},\quad{\mathbf{D}}T=\mathrm{i}[Hf_{3}(H),T].$

We compute

$\displaystyle\begin{split}{\mathbf{D}}_{a}S_{a}&=T_{1}+T_{2}+T_{3};\\\
&T_{1}=f_{2}(H_{a})\big{(}{\mathbf{D}}_{a}h(M_{a_{0}})\big{)}M_{a}f_{2}(H)\chi_{+}(B/\epsilon_{0})f_{2}(H)\\\
&T_{2}=f_{2}(H_{a})h(M_{a_{0}})\big{(}{\mathbf{D}}M_{a}\big{)}f_{2}(H)\chi_{+}(B/\epsilon_{0})f_{2}(H)\\\
&T_{3}=f_{2}(H_{a})h(M_{a_{0}})M_{a}f_{2}(H)\big{(}{\mathbf{D}}\chi_{+}(B/\epsilon_{0})\big{)}f_{2}(H).\end{split}$
(5.3)

The contributions from $T_{1}$ and $T_{2}$ are treated similarly (note that
thanks to (3.29) we can freely replace ${\mathbf{D}}_{a}$ by ${\mathbf{D}}$).
For $T_{2}$ we expand the commutator into a sum of terms. Most of them may be
treated by yet another application of the limiting absorption principle (2.4a)
and (2.4b). In fact proceeding this way it only remains to examine

$-\mathrm{i}4R_{a}(\lambda+\mathrm{i}0)h(M_{a_{0}})f_{2}(H_{a})p\cdot\big{(}\chi_{+}(|x|)^{2}\nabla^{2}m_{a}(x)\big{)}pf_{2}(H)\chi_{+}(B/\epsilon_{0})f_{2}(H)R(\lambda+\mathrm{i}0)\psi,$

Using (3.27a) (and its notation) and (3.28c) this expression is given (up to a
well-defined term in ${\mathcal{B}}^{*}$ thanks to (2.4a)) by

$-\mathrm{i}4R_{a}(\lambda+\mathrm{i}0)h(M_{a_{0}})f_{2}(H_{a})\sum_{j\leq
J}\,\,G^{*}_{b_{j}}\xi_{j}(\hat{x})\chi_{+}(\lvert
x\rvert)\psi_{j};\quad\psi_{j}\in{\mathcal{H}}^{d}.$ (5.4)

By similar arguments for $H_{a}$ rather than for $H$ it follows that for
$j=1,\dots,J$

$R_{a}(\lambda+\mathrm{i}0)h(M_{a_{0}})f_{2}(H_{a})G^{*}_{b_{j}}\xi_{j}(\hat{x})\chi_{+}(\lvert
x\rvert)\in{\mathcal{L}}({\mathcal{H}}^{d},{\mathcal{B}}^{*}).$ (5.5)

Consequently
$-\mathrm{i}R_{a}(\lambda+\mathrm{i}0)T_{2}R(\lambda+\mathrm{i}0)\psi$ is a
well-defined element of ${\mathcal{B}}^{*}$.

The argument for $T_{1}$ is similar using Lemma 3.9 iv (and of course again
(2.1), (2.4a) and (2.4b)). The argument for $T_{3}$ is also similar, we use
the ‘$Q$-bound’ (2.11b) rather than the ‘$Q$-bound’ (3.28c).

In conclusion we have deduced an exact representation of the form

$R(\lambda+\mathrm{i}0)\psi=\Sigma_{a\in{\mathcal{A}}^{\prime}}\,R_{a}(\lambda+\mathrm{i}0)\psi_{a}\text{
in }{\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*},$ (5.6)

however we did not show that $\psi_{a}\in L^{2}_{\infty}$. In fact there is no
reason to believe that the constructed $\psi_{a}\in{\mathcal{B}}$, although
our procedure reveals that $\psi_{a}\in L^{2}_{1/2}$, cf. (5.4).

In Appendix A we give a rigorous derivation and interpretation of (5.6) and
prove in addition the following result.

######  Lemma 5.2.

Let $\lambda\notin{\mathcal{T}}_{\mathrm{p}}(H)$ and $\psi\in L^{2}_{\infty}$
be given. For the construction (5.6) there exists a sequence of vectors
$\psi_{a,n}\in L^{2}_{\infty}$, $a\in{\mathcal{A}}^{\prime}$, such that
$\psi_{a,n}\to\psi_{a}\in L^{2}_{1/2}$ (in particular $(\psi_{a,n})_{n}$ is
convergent in ${\mathcal{H}}$) and

$\big{\lVert}R_{a}(\lambda+\mathrm{i}0)\big{(}\psi_{a}-\psi_{a,n}\big{)}\big{\rVert}_{{\mathcal{B}}^{*}}\to
0\text{ for }n\to\infty.$ (5.7a) In particular
$\big{\lVert}R(\lambda+\mathrm{i}0)\psi-\sum_{a\in{\mathcal{A}}^{\prime}}\,R_{a}(\lambda+\mathrm{i}0)\psi_{a,n}\big{\rVert}_{{\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}}=o\big{(}n^{0}\big{)}.$
(5.7b)

We intend to iterate the assertion of Lemma 5.2 roughly by using the partition

$\chi_{+}(\lvert x\rvert)=\chi_{+}(\lvert
x\rvert)\chi^{2}_{-}(|x^{a}|/\kappa|x|)+\chi_{+}(\lvert
x\rvert)\chi^{2}_{+}(|x^{a}|/\kappa|x|),$

cf. the concept (2.3a) and (2.3b), in combination with Subsection 3.3. This is
eventually done in Section 8, while Sections 6 and 7 are devoted to some
preliminaries.

##  6\. Non-threshold analysis

The following bounds may be viewed as a stationary version of the so-called
minimal velocity bounds. In fact the latter bounds were used in [Is4, Is5] to
show the stationary ones for fastly decaying potentials in the $3$-body
problem. The proof given below is completely stationary and requires only a
minimum amount of regularity on the potentials (moreover it applies with long-
range potentials included).

######  Lemma 6.1.

Let $a\in{\mathcal{A}}\setminus\\{a_{\min},a_{\max}\\}$ and
$\lambda\notin{\mathcal{T}}_{\mathrm{p}}(H)$ be given. Then there exists
$c_{1}>0$ and for any $E\notin{\mathcal{T}}_{\mathrm{p}}(H^{a})$ an open
neighbourhood $U$ of $E$ such that with $\kappa=c_{1}d^{a}$, where

$d^{a}=d(E,H^{a})=4\operatorname{dist}\big{(}E,\\{\mu\in{\mathcal{T}}(H^{a})\mid\mu<E\\}\big{)}:$
$\displaystyle\text{For all real }f\in C^{\infty}_{\mathrm{c}}(U)\text{
\,and\, }\psi\in{\mathcal{B}},\text{ and with }C_{1}>0\text{ being independent
of }\psi,$ $\displaystyle\quad\quad\lVert
QR_{a}(\lambda+\mathrm{i}0)\psi\rVert_{{\mathcal{H}}}\leq
C_{1}\lVert\psi\rVert_{{\mathcal{B}}};$ (6.1a)
$\displaystyle\quad\quad\quad\quad\quad
Q=Q_{a,\kappa,f}=r^{-1/2}\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|)f(H^{a}).$ In fact, under the conditions
of (6.1) and with $f_{1}$ being any narrowly supported standard support
function with $f_{1}=1$ in a small neighbourhood of $\lambda$,
$\sup_{\operatorname{Im}z\neq 0}\lVert
Q_{a,\kappa,f}{f_{1}}(H_{a})R_{a}(z)\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}})}\leq
C_{1}<\infty.$ (6.1b)

###### Proof.

Clearly (6.1) is a consequence of (6.1b), cf. Remark 2.3 i. Fix any support
function $f_{1}$ as in (6.1b) and pick any support functions $f_{3}\succ
f_{2}\succ f_{1}$.

I (a positive commutator estimate). By the Mourre estimate for the operator
$\sqrt{r^{a}}B^{a}\sqrt{r^{a}}$ (the one referred to in (2.5a)–(2.5c) now with
$r$ and $H$ replaced by $r^{a}$ and $H^{a}$, respectively)

$1_{U}(H^{a})\mathrm{i}\Big{[}H^{a},A^{a}\Big{]}1_{U}(H^{a})\geq\tfrac{d^{a}}{2}1_{U}(H^{a});\quad
A^{a}=\sqrt{r^{a}}B^{a}\sqrt{r^{a}},\,B^{a}=2\operatorname{Re}\big{(}p^{a}\cdot\nabla^{a}r^{a}\big{)}.$

This is for a small open neighbourhood $U$ of $E$. We recall from (2.5a) that

$B=\mathrm{i}\big{[}H_{a},r\big{]}=2\operatorname{Re}\big{(}p\cdot\nabla
r\big{)}.$

The operators $B^{a}$ and $B$ are bounded relatively to $H_{a}$, as to be used
explicitly below. Fix any real $f\in C^{\infty}_{\mathrm{c}}(U)$. Now we
introduce for any small $\epsilon>0$ the ’propagation observable’

$\Psi=f_{1}(H_{a})f(H^{a})\zeta_{\epsilon}\big{(}r^{-1/2}A^{a}r^{-1/2}\big{)}f(H^{a})f_{1}(H_{a}),$

where $\zeta_{\epsilon}\in C^{\infty}({\mathbb{R}})$ is real and increasing
with $\zeta^{\prime}_{\epsilon}=1$ on $(-\epsilon,\epsilon)$ and
$\sqrt{\zeta^{\prime}_{\epsilon}}\in
C_{\mathrm{c}}^{\infty}((-2\epsilon,2\epsilon))$. Since the argument is small
on the support of the derivative $\zeta^{\prime}_{\epsilon}$ (more precisely
bounded by $2\epsilon$), we obtain the lower bound

$\displaystyle\mathrm{i}[H_{a},\Psi]\geq$
$\displaystyle\tfrac{d^{a}-C\epsilon}{2r}f_{1}(H_{a})f(H^{a})\zeta^{\prime}_{\epsilon}\big{(}r^{-1/2}A^{a}r^{-1/2}\big{)}f(H^{a})f_{1}(H_{a})$
$\displaystyle+f_{1}(H_{a}){\mathcal{O}}(r^{-1-2\mu})f_{1}(H_{a}).$

In fact the bound holds with $C=4\lVert Bf_{2}(H_{a})\rVert$. The error
${\mathcal{O}}(r^{-1-2\mu})$ arises from commutation when using the
representation formula (2.1).

We denote
$T_{\epsilon}=\sqrt{\zeta^{\prime}_{\epsilon}}\big{(}r^{-1/2}A^{a}r^{-1/2}\big{)}$
and conclude by Lemma 2.2 that for some positive constant $C_{1}^{\prime}$

$\sup_{\operatorname{Im}z\neq 0}\lVert
r^{-1/2}T_{\epsilon}f(H^{a})f_{1}(H_{a})R_{a}(z)\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}})}\leq
C_{1}^{\prime};\quad\epsilon=d^{a}/2(C+1).$ (6.2)

(Note that indeed $d^{a}>C\epsilon$, as needed.)

II (an estimate for the classically forbidden region). We recall that the
function $r^{a}$ has a scaling parameter, possibly large, to assure the Mourre
estimate used in Step I. This scaling might be depending on $E$, which could
cause a problem recalling that we need to produce a constant $c_{1}>0$ being
independent of $E$. This problem is easily cured by using (2.5b) as follows.
First we note that on the support of $\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|)$

$r^{-1/2}\sqrt{r^{a}}\leq 2\sqrt{\kappa}+C^{\prime}r^{-1/2},\quad
C^{\prime}=C^{\prime}(E).$

Secondly, another elementary direct computation shows that $\lVert
B^{a}f_{3}(H_{a})\rVert\leq\breve{C}$ for a positive constant $\breve{C}$
being independent of the considered
$E\notin{\mathcal{T}}_{\mathrm{p}}(H^{a})$.

Thanks to the above remarks we can now mimic the proof of Lemma 5.1 and
produce a constant $c>0$ being independent of $E$ such that

$\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|)(I-T_{\epsilon})f_{2}(H_{a})={\mathcal{O}}(\langle
x\rangle^{-1/2});\quad\kappa={c}\epsilon.$ (6.3)

Recall for comparison that $\epsilon$ has an explicit $E$-dependence, implying
the same for $\kappa$. Clearly (6.3) yields that for some positive constant
$C_{1}^{\prime\prime}$

$\sup_{\operatorname{Im}z\neq 0}\lVert r^{-1/2}\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|)(I-T_{\epsilon})f(H^{a})f_{1}(H_{a})R_{a}(z)\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}})}\leq
C_{1}^{\prime\prime}.$ (6.4)

III (combining bounds). Finally we insert $I=T_{\epsilon}+(I-T_{\epsilon})$ to
the left of the factor $f(H^{a})$ in $Q_{a,\kappa,f}$ and conclude from
(6.2)–(6.4) that for $\kappa={c}\epsilon$ the desired estimate (6.1b) holds
with $c_{1}=c/2(C+1)$ and $C_{1}=C_{1}^{\prime}+C_{1}^{\prime\prime}$. ∎

######  Corollary 6.2.

Let $a\in{\mathcal{A}}\setminus\\{a_{\min},a_{\max}\\}$ and
$\lambda\notin{\mathcal{T}}_{\mathrm{p}}(H)$ be given. Then there exists
$c_{2}>0$ such that for any compact subset $K$ of
${\mathbb{R}}\setminus{\mathcal{T}}_{\mathrm{p}}(H^{a})$ and with
$\kappa=c_{2}d^{a}_{K}$, where

$d^{a}_{K}=d(K,H^{a})=\inf_{E\in K}d(E,H^{a})=\inf_{E\in
K}4\operatorname{dist}\big{(}E,\\{\mu\in{\mathcal{T}}(H^{a})\mid\mu<E\\}\big{)}:$

$\displaystyle\text{For all real }f\in C^{\infty}_{\mathrm{c}}(K)\text{
\,and\, }\psi\in{\mathcal{B}},\text{ and with }C_{2}>0\text{ being independent
of }\psi,$ $\displaystyle\quad\quad\lVert
QR_{a}(\lambda+\mathrm{i}0)\psi\rVert_{{\mathcal{H}}}\leq
C_{2}\lVert\psi\rVert_{{\mathcal{B}}};$ (6.5a)
$\displaystyle\quad\quad\quad\quad\quad
Q=Q_{a,\kappa,f}=r^{-1/2}\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|)f(H^{a}).$ Under the conditions of (6.5)
and with $f_{1}$ being any narrowly supported standard support function with
$f_{1}=1$ in a small neighbourhood of $\lambda$, $\sup_{\operatorname{Im}z\neq
0}\lVert
Q_{a,\kappa,f}{f_{1}}(H_{a})R_{a}(z)\rVert_{{\mathcal{L}}({\mathcal{B}},{\mathcal{H}})}\leq
C_{2}<\infty.$ (6.5b)

###### Proof.

This is immediate from Lemma 6.1 (with $c_{2}=c_{1}/2$, for example) and a
compactness and partition of unity argument, writing $f=\Sigma_{1}^{J}f_{j}$
for suitable functions $f_{j}$ being narrowly supported near some
$E_{j}\in\operatorname{supp}f$, $j=1,\dots,J$, respectively. ∎

######  Corollary 6.3.

Let $a\in{\mathcal{A}}\setminus\\{a_{\min},a_{\max}\\}$,
$\lambda\notin{\mathcal{T}}_{\mathrm{p}}(H)$, $\epsilon>0$ and $\psi\in
L^{2}_{1}$ be given. Then there exists $\kappa>0$ such that

$\lVert\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|)1_{{\mathbb{R}}\setminus{\mathcal{T}}_{\mathrm{p}}(H^{a})}(H^{a})R_{a}(\lambda+\mathrm{i}0)\psi\rVert_{{\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}}\leq\epsilon.$
(6.6)

###### Proof.

We can assume that $\lambda\geq\inf\sigma(H^{a})$, and we can replace
${\mathbb{R}}\setminus{\mathcal{T}}_{\mathrm{p}}(H^{a})$ by a bounded open
subset $U\subseteq{\mathbb{R}}\setminus{\mathcal{T}}_{\mathrm{p}}(H^{a})$, say
explicitly by
$U=\big{(}{\mathbb{R}}\setminus{\mathcal{T}}_{\mathrm{p}}(H^{a})\big{)}\cap(\inf\sigma(H^{a})-1,\lambda+1)$.

Let $K_{m}\subseteq U$, $m\in{\mathbb{N}}$, be the set of $E\in U$ for which
$\operatorname{dist}\big{(}E,U^{\mathrm{c}}\big{)}\geq 1/m$. Then the sets
$K_{m}$ are compact and constitute an increasing sequence. Take any sequence
of standard support functions $f_{m}\in C^{\infty}_{\mathrm{c}}(U)$ obeying
and $f_{m}=1$ on $K_{m}$. Then

$\displaystyle\begin{split}\lVert(1_{U}-f_{m})&(H^{a})R_{a}(\lambda+\mathrm{i}0)\psi\rVert_{{\mathcal{B}}^{*}}\leq
C_{m}\lVert(1_{U}-f_{m})(H^{a})\langle
x_{a}\rangle\psi\rVert_{{\mathcal{H}}};\\\ &C_{m}=\sup_{\eta\in U\setminus
K_{m}}\,\lVert(p^{2}_{a}+\eta-\lambda-\mathrm{i}0)^{-1}\langle
x_{a}\rangle^{-1}\rVert_{{\mathcal{L}}(L^{2}({\mathbf{X}}_{a}),\,{\mathcal{B}}({\mathbf{X}}_{a})^{*})}.\end{split}$
(6.7)

Note that (6.7) follows by using the trivial inclusion $\\{\lvert
x\rvert\leq\rho\\}\subset\\{\lvert x_{a}\rvert\leq\rho\\}$ and the spectral
theorem on multiplication operator form for $H^{a}$ [RS, Theorem VIII.4]
(amounting to a partial diagonalization). The sequence $(C_{m})_{1}^{\infty}$
is bounded (since it is decreasing). Rewriting
${\mathcal{H}}=L^{2}\big{(}{\mathbf{X}}_{a},L^{2}({\mathbf{X}}^{a})\big{)}$
and invoking the dominated convergence theorem and the Borel calculus for
$H^{a}$, it follows that

$\lVert(1_{U}-f_{m})(H^{a})R_{a}(\lambda+\mathrm{i}0)\psi\rVert_{{\mathcal{B}}^{*}}\to
0\text{ for }m\to\infty.$

In particular we can fix $m$ such that

$\lVert(1_{U}-f_{m})(H^{a})R_{a}(\lambda+\mathrm{i}0)\psi\rVert_{{\mathcal{B}}^{*}}\leq\epsilon.$

Applying now Corollary 6.2 with $\kappa=c_{2}d^{a}_{\operatorname{supp}f_{m}}$
and $f=f_{m}$ we conclude that

$\displaystyle\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|)f_{m}(H^{a})R_{a}(\lambda+\mathrm{i}0)\psi=0\text{
in }{\mathcal{B}}_{0}^{*}$

as well as

$\displaystyle\lVert\chi_{+}(\lvert
x\rvert)\chi_{-}(|x^{a}|/2\kappa|x|)(1_{U}-f_{m})(H^{a})R_{a}(\lambda+\mathrm{i}0)\psi\rVert_{{\mathcal{B}}^{*}}\leq\epsilon,$

yielding in combination (6.6). ∎

##  7\. A squeezing argument

In this section our outset is Lemma 5.2. Whence we fix
$\lambda\notin{\mathcal{T}}_{\mathrm{p}}(H)$, $a\in{\mathcal{A}}^{\prime}$ and
$\psi\in L^{2}_{\infty}$. We will initiate a reduction procedure for deriving
the asymptotics of the constructed vectors
$R_{a}(\lambda+\mathrm{i}0)\psi_{a}\in{\mathcal{B}}^{*}$ from (5.6). Along
with a slightly more complicated version of the reduction procedure of Section
5 as well as the reduction from Corollary 6.2, this procedure can be iterated
as to be elaborated in Section 8.

######  Lemma 7.1.

There exists a sequence of standard support functions $f_{m}$ all supported in
${\mathbb{R}}\setminus{\mathcal{T}}_{\mathrm{p}}(H^{a})$ such that

$\big{(}I-f_{m}(H^{a})\big{)}R_{a}(\lambda+\mathrm{i}0)\psi_{a}=u_{m}+o(m^{0}),$
(7.1)

where the asymptotics of $u_{m}\in{\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$
agrees with the right-hand side of (4.7).

###### Proof.

We will use the approximations $R_{a}(\lambda+\mathrm{i}0)\psi_{a,n}$ from
(5.7a) (recalling $\psi_{a,n}\in L^{2}_{\infty}$) and the proof of Corollary
6.3. Let $f_{1}$ be a support function given as in (6.5b) and let $f_{2}\succ
f_{1}$, also given narrowly supported.

I (Kato-smoothness- and ${\mathcal{B}}^{*}$-estimates). We need a variation of
the bounds [Sk1, (7.3a) and (7.3b)]. By using the same ‘propagation
observables’ as for (A.7) (used also in Section 5 of course) we obtain by
mimicking the proof of [Sk1, Lemma 7.1] that

$Q_{l}{f_{2}}(H_{a})\delta(H_{a}-\lambda){f_{2}}(H_{a}){Q_{k}}^{*}\in{\mathcal{L}}({\mathcal{H}}).$

In combination with (2.4b) this leads to the following assertion for the
approximating sequence of (A.10b):

$\langle\delta(H_{a}-\lambda)\rangle_{\psi_{a}-\psi_{a,n}}\to 0\text{ for
}n\to\infty.$ (7.2a)

We are going to use the sequence of standard support functions $(f_{m})_{m}$
from the proof of Corollary 6.3. Let then $\bar{f}_{m}=1-f_{m}$ and consider
the following element of ${\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$:

$\displaystyle\phi_{a,m}:=\sum_{\alpha=(a,\lambda^{\alpha},u^{\alpha}),\,(H^{a}-\lambda^{\alpha})u^{\alpha}=0,\,\lambda^{\alpha}<\lambda}2\pi\mathrm{i}\,J_{\alpha}$
$\displaystyle
v^{+}_{\alpha,\lambda}[g_{\alpha}]-\bar{f}_{m}(H^{a})R_{a}(\lambda+\mathrm{i}0){f_{2}}(H_{a})\psi_{a};$
$\displaystyle\quad
g_{\alpha}:=\Gamma_{\alpha}^{a+}(\lambda)\psi_{a}\in{\mathcal{G}}_{a}.$

The operator $\Gamma_{\alpha}^{a+}(\lambda)$ is the outgoing restricted wave
operator for $H_{a}$ at energy $\lambda$, possibly defined as in Proposition
4.1 (with $H$ replaced by $H_{a}$). Here and henceforth we only consider
channels $\alpha$ specified as in the summation (in particular with the first
component fixed as $a$). At this point it should be mentioned that strictly
speaking Proposition 4.1 requires $\psi_{a}\in{\mathcal{B}}$, and we only know
$\psi_{a}\in L^{2}_{1/2}$. However thanks to [Sk1, Remark 9.15 1)]
$\Gamma_{\alpha}^{a+}(\lambda)\psi_{a}$ is still a well-defined element of
${\mathcal{G}}_{a}$, as given by the formula (4.3). More generally the above
sum is a well-defined element of ${\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$ due
to the following extended version of the Bessel inequality (4.6) for $H_{a}$,
see [Sk1, Subsection 9.2],

$\sum_{\lambda^{\alpha}<\lambda}\,\lVert
g_{\alpha}\rVert^{2}\leq\langle\delta(H_{a}-\lambda)\rangle_{\psi_{a}}<\infty,$
(7.2b)

and the general bounds, cf. (4.9),

$C_{1}\sum_{\lambda^{\alpha}<\lambda}\,\lVert
g_{\alpha}\rVert^{2}\leq\Big{\lVert}\sum_{\lambda^{\alpha}<\lambda}J_{\alpha}v^{+}_{\alpha,\lambda}[g_{\alpha}]\Big{\rVert}^{2}_{{\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}}\leq
C_{2}\sum_{\lambda^{\alpha}<\lambda}\,\lVert g_{\alpha}\rVert^{2}.$ (7.2c)

We conclude that $\phi_{a,m}$ is a well-defined element of
${\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$.

We introduce $\phi_{a,m,n}$ and $g_{\alpha,n}$ by replacing in the above
expressions for $\phi_{a,m}$ and $g_{\alpha}$ all appearances of $\psi_{a}$ by
$\psi_{a,n}$. Then similarly to (7.2b) we can record that

$\sum_{\lambda^{\alpha}<\lambda}\,\lVert
g_{\alpha}-g_{\alpha,n}\rVert^{2}\leq\langle\delta(H_{a}-\lambda)\rangle_{\psi_{a}-\psi_{a,n}}.$
(7.2d)

II (applying the estimates). Writing for given
$\phi_{1},\phi_{2}\in{\mathcal{B}}^{*}$, $\phi_{1}\simeq\phi_{2}$ if
$\phi_{1}-\phi_{2}\in{\mathcal{B}}^{*}_{0}$, we compute considering now
$\phi_{a,m}\in{\mathcal{B}}^{*}$ as a representative for the corresponding
coset (see below for further elaboration)

$\displaystyle\phi_{a,m}$
$\displaystyle\simeq\phi_{a,m,n}+o(n^{0})\quad\quad\big{(}\text{replacing
}\psi_{a,m}\text{ by }\psi_{a,m,n}\big{)}$ $\displaystyle\simeq
o(m^{0})+o(n^{0})\quad\quad(\text{by dominated convergence and spectral
theory})$ $\displaystyle\simeq o(m^{0}).$

In the last steps we first fixed a big $n$ and then $m=m(n)$ sufficiently big
to conclude that the distance from $\phi_{a,m}$ to ${\mathcal{B}}^{*}_{0}$
indeed is $o(m^{0})$. However we need to argue that for fixed (big) $n$ indeed
the approximation by taking $m$ correpondingly big works.

For this purpose we note, cf. the proof of Corollary 6.3, that with
$f^{a}_{m}:=\bar{f}_{m}-1_{{\mathcal{T}}_{\mathrm{p}}(H^{a})}$

$\displaystyle\begin{split}\lVert
f^{a}_{m}&(H^{a})R_{a}(\lambda+\mathrm{i}0)f_{2}(H_{a})\psi_{a,n}\rVert_{{\mathcal{B}}^{*}}\\\
&\leq C_{m}\lVert\big{(}f^{a}_{m}1_{(-\infty,\lambda+1)}\big{)}(H^{a})\langle
x_{a}\rangle f_{2}(H_{a})\psi_{a,n}\rVert_{{\mathcal{H}}};\\\
&\quad\quad\quad\quad C_{m}=\sup_{\eta\in
K_{m}^{c}}\,\lVert(p_{a}^{2}-\lambda+\eta-\mathrm{i}0)^{-1}\langle
x_{a}\rangle^{-1}\rVert_{{\mathcal{L}}(L^{2}({\mathbf{X}}_{a}),\,{\mathcal{B}}({\mathbf{X}}_{a})^{*})}.\end{split}$
(7.3a) As before the sequence $(C_{m})_{1}^{\infty}$ is bounded, and by
invoking the dominated convergence theorem and the Borel calculus for $H^{a}$
it follows that $\lVert
f^{a}_{m}(H^{a})R_{a}(\lambda+\mathrm{i}0)f_{2}(H_{a})\psi_{a,n}\rVert_{{\mathcal{B}}^{*}}\to
0\text{ for }m\to\infty.$ (7.3b)

Next we claim that

$2\pi\mathrm{i}\sum_{\lambda^{\alpha}<\lambda}J_{\alpha}v^{+}_{\alpha,\lambda}[\Gamma_{\alpha}^{a+}(\lambda)\psi_{a,n}]\simeq
1_{{\mathcal{T}}_{\mathrm{p}}(H^{a})}(H^{a})R_{a}(\lambda+\mathrm{i}0)f_{2}(H_{a})\psi_{a,n}.$
(7.3c)

Note that we consider the left-hand side as an element of ${\mathcal{B}}^{*}$
although its strict meaning is the correponding coset in
${\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$. Thanks to (7.3a), also the right-
hand side is well-defined in ${\mathcal{B}}^{*}$ (or in
${\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$).

To show (7.3c) we first consider the simplest case where there are only
finitely many channels $\alpha$ involved in the summation defining
$\phi_{a,m}$. We use that
$\Gamma_{\alpha}^{a+}(\lambda)=\gamma_{\alpha}^{+}(\lambda_{\alpha})J^{*}_{\alpha}$,
as expressed by the free one-body restriction operator
$\gamma_{\alpha}^{+}(\cdot)=\gamma_{a,0}(\cdot)$ for $p_{a}^{2}$ at the
positive energy $\lambda_{\alpha}=\lambda-\lambda^{\alpha}$ (cf. Theorem 4.2),
and stationary completeness of positive energies for $p^{2}_{a}$ (cf. [Sk1]).
This means concretely that

$(p^{2}_{a}-\lambda_{\alpha}-\mathrm{i}0)^{-1}\psi_{\alpha,n}-2\pi\mathrm{i}\,v^{+}_{\alpha,\lambda}[\gamma_{\alpha}^{+}(\lambda_{\alpha})\psi_{\alpha,n}]\in{\mathcal{B}}^{*}_{0}({\mathbf{X}}_{a});\quad\psi_{\alpha,n}=J^{*}_{\alpha}\psi_{a,n}\in{\mathcal{B}}({\mathbf{X}}_{a}).$

We then conclude (7.3c) by expanding the right-hand side into a (finite) sum.

In the remaining case where there are infinitely many channels $\alpha$ in the
summation we pick, given in terms by an arbitrary numbering of the eigenstates
of $H^{a}$ (with eigenvalues $\lambda^{\alpha}<\lambda$), an increasing
sequence of finite-rank projections $P^{a}_{j}\to
1_{{\mathcal{T}}_{\mathrm{p}}(H^{a})}(H^{a})$ (strongly). By using a
modification of (7.3a) (and arguing similarly) we then obtain as in (7.3b)
that

$\lVert\big{(}1_{{\mathcal{T}}_{\mathrm{p}}(H^{a})}(H^{a})-P^{a}_{j}\big{)}R_{a}(\lambda+\mathrm{i}0)f_{2}(H_{a})\psi_{a,n}\rVert_{{\mathcal{B}}^{*}}\to
0\text{ for }j\to\infty.$

Thanks to this property, a version of (7.2c) and the arguments for the finite
summation case also the infinite summation formula follows. We have proved
(7.3c).

Clearly the combination of (7.3b) and (7.3c) yields the wanted property
$\phi_{a,m}\simeq o(m^{0})$. ∎

##  8\. Proof of stationary completeness

The main result of the paper reads.

######  Theorem 8.1.

Suppose Condition 2.1. Then all
$\lambda\in{\mathcal{E}}=(\min{\mathcal{T}}(H),\infty)\setminus{{\mathcal{T}}_{{\mathrm{p}}}(H)}$
are stationary complete for $H$.

###### Proof.

Let $\lambda\notin{\mathcal{T}}_{\mathrm{p}}(H)$ and $\psi\in L^{2}_{\infty}$
be given. We need to show that
$R(\lambda+\mathrm{i}0)\psi\in{\mathcal{B}}^{*}/{\mathcal{B}}_{0}^{*}$ has
asymptotics agreeing with the right-hand side of (4.7). By combining (5.6) and
Lemma 7.1 we are left with analyzing the asymptotics of the vector
$f(H^{a})R_{a}(\lambda+\mathrm{i}0)\psi_{a}$, $a\in{\mathcal{A}}^{\prime}$,
for a fixed standard support function $f$ supported in
${\mathbb{R}}\setminus{\mathcal{T}}_{\mathrm{p}}(H^{a})$. In turn we can
assume that $a\neq 0$ and (by Lemma 5.2) that
$\psi_{a}=f_{1}(H_{a})\psi_{a}\in L^{2}_{\infty}$. Here $f_{1}$ is a support
function given as in (6.5b), and we let $f_{2}\succ f_{1}$, also given
narrowly supported.

With reference to Subsection 3.3 we denote from now on $a=a_{0}$ (reserving
the symbol ‘$a$’ for a different purpose) and repeat the partition of unity
procedure of Section 5 with $H$ replaced by $H_{a_{0}}$.

We write with
$u_{a_{0}}:=R_{a_{0}}(\lambda+\mathrm{i}0)f_{1}(H_{a_{0}})\psi_{a_{0}}$ and
for a small $\kappa>0$

$f(H^{a_{0}})u_{a_{0}}-\Sigma_{a\in{\mathcal{A}}_{a_{0}}^{\prime}}\,S^{a_{0}}_{a,\kappa}\,f(H^{a_{0}})u_{a_{0}}+Tf(H^{a_{0}})u_{a_{0}}\in{\mathcal{B}}_{0}^{*},$
(8.1)

where $T={\mathcal{O}}^{a_{0}}_{\kappa}(\langle x\rangle^{0})$ and

$S^{a_{0}}_{a,\kappa}=f_{2}(H_{a})h(M^{a_{0}}_{a_{0},\kappa})M^{a_{0}}_{a,\kappa}f_{2}(H_{a_{0}})\chi_{+}(B/\epsilon_{0})f_{2}(H_{a_{0}}).$

Indeed (8.1) may be obtained by mimicking Section 5 and by using various
commutation to treat the appearance of ‘$\kappa$-localization’ (the latter is
absent in Section 5); we skip the details.

Thanks to (6.5) and the assumed smallness of $\kappa>0$ the third term of
(8.1) is in $L^{2}_{-1/2}$, in particular in ${\mathcal{B}}_{0}^{*}$ as the
whole sum. Consequently it suffices to derive the good asymptotics of the
terms in the middle summation.

So let us for fixed $a\in{\mathcal{A}}_{a_{0}}^{\prime}$ look at
$S^{a_{0}}_{a,\kappa}\,f(H^{a_{0}})u_{a_{0}}$. Fixing any support function
$\tilde{f}\succ f$, also supported in
${\mathbb{R}}\setminus{\mathcal{T}}_{\mathrm{p}}(H^{a_{0}})$, it suffices to
derive asymptotics of
$\tilde{f}(H^{a_{0}}_{a})S^{a_{0}}_{a,\kappa}\,f(H^{a_{0}})u_{a_{0}}$. Recall
that we consider only sufficiently small $\kappa>0$ (the smallness may depend
on the choice of $\tilde{f}$).

In comparison with Section 5 and Appendix A we need to compute the operator

$T_{a,\kappa}:=\mathrm{i}\big{(}H_{a}\tilde{f}(H^{a_{0}}_{a})S^{a_{0}}_{a,\kappa}\,f(H^{a_{0}})-\tilde{f}(H^{a_{0}}_{a})S^{a_{0}}_{a,\kappa}\,f(H^{a_{0}})H_{a_{0}}\big{)}.$

Proceeding as before we expand $T_{a,\kappa}$ into a finite sum of terms on
the form

$f_{2}(H_{a}){Q_{k}}^{*}B_{k}Q_{k}f_{2}(H_{a_{0}}),$

where

$Q_{k}f_{2}(H_{a_{0}})={\mathcal{O}}(\langle
x\rangle^{-1/2}),\,\,Q_{k}f_{2}(H_{a})={\mathcal{O}}(\langle
x\rangle^{-1/2})\text{ \,and\, }B_{k}\text{ is bounded}.$

In all cases the appropriate versions of (2.7b) are at disposal. Note that we
use Remarks 2.3 and in particular that the scheme of Lemma 2.2 combines well
with (6.5b) and (3.36a)–(3.36d). Whence for a suitable vector
$\breve{\psi}_{a}\in L^{2}_{1/2}$

$\tilde{f}(H^{a_{0}}_{a})S^{a_{0}}_{a,\kappa}\,f(H^{a_{0}})u_{a_{0}}-R_{a}(\lambda+\mathrm{i}0)\breve{\psi}_{a}\in{\mathcal{B}}^{*}_{0}.$

As in Lemma 5.2, cf. Appendix A, we can approximate $\breve{\psi}_{a}$ by
vectors in $L^{2}_{\infty}$, and we can repeat the above procedure now for $a$
rather than for $a_{0}$.

When iterated clearly the procedure terminates and eventually we thus conclude
asymptotics agreeing with the right-hand side of (4.7), first up to an
arbitrary small correction and then finally without a correction term. ∎

######  Remark.

The notion of a stationary complete energy $\lambda$ requires
$\lambda\notin{\mathcal{T}}_{\mathrm{p}}(H)$. In particular if
$\lambda\notin{\mathcal{T}}(H)$ is an eigenvalue this
$R(\lambda+\mathrm{i}0)\psi$ ill-defined. However in this case the
corresponding eigenprojection, say denoted $P_{\lambda}$, maps to
$L^{2}_{\infty}$ and the expression $(H-P_{\lambda}-\lambda-\mathrm{i}0)^{-1}$
(along with its imaginary part) does have an interpretation, cf. [AHS], and
the scattering theories for $H$ and $H-P_{\lambda}$ coincide. Whence we could
deal with $\lambda\in\sigma_{{\mathrm{pp}}}(H)\setminus{\mathcal{T}}(H)$ upon
modifying the Parseval formula (4.5) by using $\delta(H-P_{\lambda}-\lambda)$
rather than $\delta(H-\lambda)$ in the formula, cf. [Sk1, Remark 9.3]. With
this modified definition of stationary completeness at such $\lambda$ our
procedure of proof applies (we leave out the details). On the other hand
$\lambda\notin{\mathcal{T}}(H)$ is essential.

##  Appendix A Strong resolvent bounds, proof of Lemma 5.2

We shall in this appendix justify our somewhat formal derivation of (5.6) and
prove Lemma 5.2. For that purpose we need certain strong resolvent bounds.

The Mourre estimate (2.5c) implies (2.4a), (2.4b) as well as (2.5e). Let us
here recall partially stronger versions of these assertions from [AIIS] (used
also in [Sk1]).

In agreement with the stated Mourre estimate we first fix
$\lambda\not\in{\mathcal{T}}_{{\mathrm{p}}}(H)$ and any positive
$c<d(\lambda,H)$ (the latter given by (2.5d)). Then for a suitable rescaled
version of $r(x)$ (also denoted by $r$) and a small open neighbourhood $U$ of
$\lambda$, it follows that

$\forall\text{ real }f\in
C_{\mathrm{c}}^{\infty}(U):\quad{f}(H)\mathrm{i}[H,r^{1/2}Br^{1/2}]{f}(H)\geq
c\,{f}(H)^{2}.$ (A.1)

We let then ${\mathcal{N}}$ denote any compact neighbourhood of $\lambda$ with
${\mathcal{N}}\subseteq U$ and introduce
${\mathcal{N}}_{\pm}=\\{z\in{\mathbb{C}}\mid\operatorname{Re}z\in{\mathcal{N}}\text{
\,and\, }0<\pm\operatorname{Im}z\leq 1\\}$.

######  Theorem A.1.

  1. 1)

There exists $C>0$ such that for all $z\in{\mathcal{N}}_{\pm}$

$\|R(z)\|_{{\mathcal{L}}({\mathcal{B}},{\mathcal{B}}^{*})}\leq C.$ (A.2)

  2. 2)

For any $s>1/2$ and $\alpha\in(0,\min\\{\mu,s-1/2\\})$ there exists $C>0$ such
that for $k\in\\{0,1\\}$, $z\in{\mathcal{N}}_{\pm}$ and
$z^{\prime}\in{\mathcal{N}}_{\pm}$, respectively,

$\displaystyle\|p^{k}R(z)-p^{k}R(z^{\prime})\|_{\mathcal{L}(L^{2}_{s},L^{2}_{-s})}\leq
C|z-z^{\prime}|^{\alpha}.$ (A.3)

In particular, for any $\lambda\in{\mathcal{N}}$ and $s>1/2$ the following
boundary values exist:

$\displaystyle p^{k}R(\lambda\pm\mathrm{i}0):=\lim_{\epsilon\to
0_{+}}p^{k}R(\lambda\pm\mathrm{i}\epsilon)\text{\ \ in
}\mathcal{L}(L^{2}_{s},L^{2}_{-s}),$

respectively (here the limits are taken in the operator-norm topology). The
same boundary values are realized (in an extended form) as

$\displaystyle
p^{k}R(\lambda\pm\mathrm{i}0)=\operatorname*{s-w^{\star}-lim}_{\epsilon\to
0_{+}}p^{k}R(\lambda\pm\mathrm{i}\epsilon)\text{\ \ in
}{\mathcal{L}}(\mathcal{B},\mathcal{B}^{*}),$

respectively (here the right-hand side operators act on any
$\psi\in\mathcal{B}$ as the $\epsilon$-limits of
$p^{k}R(\lambda\pm\mathrm{i}\epsilon)\psi$ in the weak-star topology of
$\mathcal{B}^{*}$).

  3. 3)

For any $\beta\in(0,\mu)$ and $F\in C^{\infty}(\mathbb{R})$ with

$\mathop{\mathrm{supp}}F\subset\Big{(}-\infty,\sqrt{c}\Big{)}\text{ \,and\,
}F^{\prime}\in C^{\infty}_{\mathrm{c}}(\mathbb{R}),$

there exists $C>0$, such that for all $z\in{\mathcal{N}}_{\pm}$ and $\psi\in
L^{2}_{1/2+\beta}$

$\|F(\pm B)R(z)\psi\|_{L^{2}_{-1/2+\beta}}\leq C\|\psi\|_{L^{2}_{1/2+\beta}},$
(A.4)

respectively.

Recall from Section 5 that

$S_{a}=f_{2}(H_{a})h(M_{a_{0}})M_{a}f_{2}(H)\chi_{+}(B/\epsilon_{0})f_{2}(H).$

We need to justify the convincing, although formal, computation

$\displaystyle\begin{split}&S_{a}R(\lambda+\mathrm{i}0)\psi=\phi_{a}:=R_{a}(\lambda+\mathrm{i}0)\psi_{a};\\\
&\quad\quad\quad\quad\quad\quad\psi_{a}=S_{a}\psi-\mathrm{i}T_{a}R(\lambda+\mathrm{i}0)\psi,\quad
T_{a}=\mathrm{i}\big{(}H_{a}S_{a}-S_{a}H\big{)}.\end{split}$ (A.5)

We will prove that the function $\phi_{a}$ in (A.5) has a well-defined meaning
as an element in ${\mathcal{B}}^{*}$ using an a priori interpretation
different from (2.4b). Rather from the outset $\phi_{a}$ is the weak limit,
say in $L^{2}_{-1}$,

$\displaystyle\phi_{a}=\lim_{\epsilon\to
0_{+}}R_{a}(\lambda+\mathrm{i}\epsilon)$ $\displaystyle
S_{a}\psi-\mathrm{i}\lim_{\epsilon\to
0_{+}}R_{a}(\lambda+\mathrm{i}\epsilon)T_{a}R(\lambda+\mathrm{i}\epsilon)\psi$
$\displaystyle=R_{a}(\lambda+\mathrm{i}0)S_{a}\psi-\mathrm{i}\lim_{\epsilon\to
0_{+}}$ $\displaystyle
R_{a}(\lambda+\mathrm{i}\epsilon)\chi^{2}_{-}(2B/\epsilon_{0})T_{a}R(\lambda+\mathrm{i}\epsilon)\psi$
$\displaystyle-\mathrm{i}\lim_{\epsilon\to
0_{+}}R_{a}(\lambda+\mathrm{i}\epsilon)\chi^{2}_{+}(2B/\epsilon_{0})T_{a}R(\lambda+\mathrm{i}\epsilon)\psi.$

It follows from (2.4b) (or from Theorem A.1) that the first term to the right
is an element in ${\mathcal{B}}^{*}$ (since $S_{a}\psi\in L^{2}_{\infty}$). By
commutation we can write (for the second term)

$\chi^{2}_{-}(2B/\epsilon_{0})T_{a}=\langle x\rangle^{-1}B\langle
x\rangle^{-1}\text{ with }B\text{ bounded}.$

This allows us to compute

$\displaystyle\lim_{\epsilon\to 0_{+}}$ $\displaystyle
R_{a}(\lambda+\mathrm{i}\epsilon)\chi^{2}_{-}(2B/\epsilon_{0})T_{a}R(\lambda+\mathrm{i}\epsilon)\psi$
$\displaystyle=\lim_{\epsilon\to
0_{+}}R_{a}(\lambda+\mathrm{i}\epsilon)\chi^{2}_{-}(2B/\epsilon_{0})T_{a}R(\lambda+\mathrm{i}0)\psi\quad(\text{by
}\eqref{eq:LAPbnda}$
$\displaystyle=R_{a}(\lambda+\mathrm{i}0)\chi^{2}_{-}(2B/\epsilon_{0})T_{a}R(\lambda+\mathrm{i}0)\psi\in{\mathcal{B}}^{*}\quad(\text{by
}\eqref{eq:BB^*a}.$

We conclude that the first and second terms are on a form consistent with
(2.4b),

$R_{a}(\lambda+\mathrm{i}0){\breve{\psi}}\in{\mathcal{B}}^{*}\text{ for some
}{\breve{\psi}}\in(L^{2}_{1}\subseteq)\,{\mathcal{B}}.$

The third term is different. The expression $T_{a}R(\lambda+\mathrm{i}0)\psi$
defines an element of $L_{s}^{2}$, $s=1/2$, as to be demonstrated below,
however we do not prove better decay. Thanks to (2.4a) and Theorem A.1 (for
$H_{a}$ rather than for $H$) there exists the operator-norm-limit

$\operatorname*{{\mathcal{L}(\mathcal{H})}-lim}_{\epsilon\to 0_{+}}\,\,\langle
x\rangle^{-1}R_{a}(\lambda+\mathrm{i}\epsilon)\chi^{2}_{+}(2B/\epsilon_{0})\langle
x\rangle^{-s},\quad s=1/2.$ (A.6)

Next we invoke (5.3) and further expand $T_{a}$ into a sum of terms on the
form $f_{2}(H_{a}){Q_{k}}^{*}B_{k}Q_{k}f_{2}(H)$ for suitable
$Q_{k}f_{2}(H)\in{\mathcal{O}}(\langle x\rangle^{-s})$ and bounded $B_{k}$,
say for $k=1,\dots,K$, cf. (2.9). This is doable along the line outlined in
Section 5. The most important property reads, cf. (2.4a), (2.11b) and (3.28c),

$\displaystyle\begin{split}&\sup_{\operatorname{Im}z\neq 0}\lVert
Q_{k}{f_{2}}(H){R(z)}\rVert_{{\mathcal{L}}({\mathcal{B}},\widetilde{{\mathcal{H}}})}<\infty,\\\
&\sup_{\operatorname{Im}z\neq 0}\lVert