# Eigenvalue asymptotics for the one-particle kinetic energy operator

Alexander V. Sobolev  Department of Mathematics
University College London
Gower Street
London
WC1E 6BT UK<EMAIL_ADDRESS>
###### Abstract.

The kinetic energy of a multi-particle system is described by the one-particle
kinetic energy matrix $\tau(x,y)$. Alongside the one-particle density matrix
$\gamma(x,y)$ it is one of the key objects in the quantum-mechanical
approximation schemes. We prove the asymptotic formula
$\lambda_{k}\sim(Bk)^{-2}$, $B\geq 0$, as $k\to\infty$, for the eigenvalues
$\lambda_{k}$ of the self-adjoint operator $\boldsymbol{{\sf{T}}}\geq 0$ with
kernel $\tau(x,y)$.

###### Key words and phrases:

Multi-particle Schrödinger operator, one-particle density matrix, eigenvalues,
integral operators

###### 2010 Mathematics Subject Classification:

Primary 35J10; Secondary 47G10, 81Q10

##  1\. Introduction

Consider on $\textup{{{L}}}^{2}(\mathbb{R}^{3N})$ the Schrödinger operator

(1.1) $\displaystyle
H=\sum_{k=1}^{N}\bigg{(}-\Delta_{k}-\frac{Z}{|x_{k}|}\bigg{)}+\sum_{1\leq
j<k\leq N}\frac{1}{|x_{j}-x_{k}|},$

describing an atom with $N$ charged particles with coordinates
$\mathbf{x}=(x_{1},x_{2},\dots,x_{N}),\,x_{k}\in\mathbb{R}^{3}$,
$k=1,2,\dots,N$, and a nucleus with charge $Z>0$. The notation $\Delta_{k}$ is
used for the Laplacian w.r.t. the variable $x_{k}$. The operator $H$ acts on
the Hilbert space $\textup{{{L}}}^{2}(\mathbb{R}^{3N})$ and it is self-adjoint
on the domain $D(H)=\textup{{{H}}}^{2}(\mathbb{R}^{3N})$, since the potential
in (1.1) is an infinitesimal perturbation relative to the unperturbed operator
$-\Delta=-\sum_{k}\Delta_{k}$, see e.g. [18, Theorem X.16]. Let
$\psi=\psi(\mathbf{x})$ be an eigenfunction of the operator $H$ with an
eigenvalue $E\in\mathbb{R}$, i.e. $\psi\in D(H)$ and

$\displaystyle(H-E)\psi=0.$

For each $j=1,\dots,N$, we represent

$\displaystyle\mathbf{x}=(\hat{\mathbf{x}}_{j},x_{j}),\quad\textup{where}\
\hat{\mathbf{x}}_{j}=(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{N}),$

with obvious modifications if $j=1$ or $j=N$. The one-particle density matrix
is defined as the function

(1.2)
$\displaystyle\gamma(x,y)=\sum_{j=1}^{N}\int\limits_{\mathbb{R}^{3N-3}}\overline{\psi(\hat{\mathbf{x}}_{j},x)}\psi(\hat{\mathbf{x}}_{j},y)\
d\hat{\mathbf{x}}_{j},\quad(x,y)\in\mathbb{R}^{3}\times\mathbb{R}^{3}.$

Introduce also the function

(1.3)
$\displaystyle\tau(x,y)=\sum_{j=1}^{N}\int\limits_{\mathbb{R}^{3N-3}}\overline{\nabla_{x}\psi(\hat{\mathbf{x}}_{j},x)}\cdot\nabla_{y}\psi(\hat{\mathbf{x}}_{j},y)\
d\hat{\mathbf{x}}_{j},$

that we call the one-particle kinetic energy matrix. The choice of this term
does not seem to be standard, but it is partly motivated by the fact that the
trace

$\displaystyle\operatorname{{tr}}\boldsymbol{{\sf{T}}}=\int_{\mathbb{R}^{3}}\tau(x,x)\,dx$

gives the kinetic energy of the $N$ particles, see e.g. [17, Chapter 3] or
[16, Section 4]. Denote by $\boldsymbol{\sf\Gamma}=\operatorname{{\sf
Int}}(\gamma)$ and $\boldsymbol{{\sf{T}}}=\operatorname{{\sf Int}}(\tau)$ the
integral operators with kernels $\gamma(x,y)$ and $\tau(x,y)$ respectively. We
call $\boldsymbol{\sf\Gamma}$ the one-particle density operator, and
$\boldsymbol{{\sf{T}}}$ \- the one-particle kinetic energy operator. If we
assume that the particles are fermions (resp. bosons), i.e. that the
eigenfunction is antisymmetric (resp. symmetric) with respect to the
permutations $x_{k}\leftrightarrow x_{j}$, then the formulas for $\gamma(x,y)$
and $\tau(x,y)$ take a more compact form:

(1.4)
$\displaystyle\begin{cases}\gamma(x,y)=N\int\limits_{\mathbb{R}^{3N-3}}\overline{\psi(\hat{\mathbf{x}},x)}\psi(\hat{\mathbf{x}},y)\
d\hat{\mathbf{x}},\\\\[14.22636pt]
\tau(x,y)=N\int\limits_{\mathbb{R}^{3N-3}}\overline{\nabla_{x}\psi(\hat{\mathbf{x}},x)}\cdot\nabla_{y}\psi(\hat{\mathbf{x}},y)\,d\hat{\mathbf{x}},\end{cases}$

where $\hat{\mathbf{x}}=\hat{\mathbf{x}}_{N}$. In this paper, however, we do
not make these assumptions and work with the general definitions (1.2) and
(1.3). Since $\psi,\nabla\psi\in\textup{{{L}}}^{2}(\mathbb{R}^{3N})$, both
$\boldsymbol{\sf\Gamma}$ and $\boldsymbol{{\sf{T}}}$ are trace class. We are
interested in the exact decay rate of the eigenvalues
$\lambda_{k},k=1,2,\dots,$ for $\boldsymbol{\sf\Gamma}$ and
$\boldsymbol{{\sf{T}}}$. Both operators play a central role in quantum
chemistry computations of atomic and molecular bound states, see e.g. the
papers [11, 12, 14, 15] and the book [1]. The knowledge of the eigenvalue
behaviour should serve to estimate the errors due to finite-dimensional
approximations.

The eigenvalue asymptotics for the operator $\boldsymbol{\sf\Gamma}$ were
studied in [21] under the assumption that the function $\psi$ satisfies an
exponential bound

$\displaystyle\sup_{\mathbf{x}\in\mathbb{R}^{3N}}e^{\varkappa_{\scaleto{0}{3pt}}|\mathbf{x}|}\,|\psi(\mathbf{x})|<$

with some $\varkappa_{0}>0$. This condition always holds if $E$ is a discrete
eigenvalue. For detailed discussion and bibliography on the property (1) we
refer to [19]. It was shown in [21] that

(1.5)
$\displaystyle\lim_{k\to\infty}k^{\frac{8}{3}}\,\lambda_{k}(\boldsymbol{\sf\Gamma})=A^{\frac{8}{3}}$

with some coefficient $A\geq 0$, see also [7] and [8] for relevant quantum
chemistry calculations. The subject of the current paper is to establish for
the eigenvalues $\lambda_{k}(\boldsymbol{{\sf{T}}})$ the formula

(1.6)
$\displaystyle\lim_{k\to\infty}k^{2}\lambda_{k}(\boldsymbol{{\sf{T}}})=B^{2},$

with a coefficient $B\geq 0$. The precise statement, including the formula for
the coefficient $B$, requires further properties of the function $\psi$ and it
is given in Theorem 2.3. Moreover, in Sect. 2 for comparison we also provide a
formula for the coefficient $A$.

The proof splits into three steps that are briefly outlined below.

Step 1: factorization. First we represent the operator $\boldsymbol{{\sf{T}}}$
as the product
$\boldsymbol{{\sf{T}}}=\boldsymbol{{\sf{V}}}^{*}\boldsymbol{{\sf{V}}}$, where
$\boldsymbol{{\sf{V}}}:\textup{{{L}}}^{2}(\mathbb{R}^{3})\to\textup{{{L}}}^{2}(\mathbb{R}^{3N-3})$
is a suitable compact operator. This representation implies that
$\lambda_{k}(\boldsymbol{{\sf{T}}})=s_{k}(\boldsymbol{{\sf{V}}})^{2}$,
$k=1,2\dots,$, where $s_{k}(\boldsymbol{{\sf{V}}})$ are the singular values of
the operator $\boldsymbol{{\sf{V}}}$. Therefore the asymptotic formula (1.6)
takes the form

(1.7) $\displaystyle\lim_{k\to\infty}k\,s_{k}(\boldsymbol{{\sf{V}}})=B.$

If we take formula (1.4) as the definition of $\tau(x,y)$ (which is the case
for fermions or bosons), then the operator $\boldsymbol{{\sf{V}}}$ can be
taken to be the integral operator with the vector-valued kernel
$\sqrt{N}\,\nabla_{x}\psi(\hat{\mathbf{x}},x)$, i.e.

(1.8)
$\displaystyle(\boldsymbol{{\sf{V}}}u)(\hat{\mathbf{x}})=\sqrt{N}\int_{\mathbb{R}^{3}}\nabla_{x}\psi(\hat{\mathbf{x}},x)u(x)dx,\
u\in\textup{{{L}}}^{2}(\mathbb{R}^{3}).$

For the general case the operator $\boldsymbol{{\sf{V}}}$ has a more
complicated form and it is defined in Subsect. 2.2. The factorization
$\boldsymbol{{\sf{T}}}=\boldsymbol{{\sf{V}}}^{*}\boldsymbol{{\sf{V}}}$ reduces
the study of the asymptotics to the operator $\boldsymbol{{\sf{V}}}$ which is
convenient since the properties of the functions $\psi$ have been very well
studied in the literature. For the rest of the introduction we will assume for
simplicity that $\boldsymbol{{\sf{V}}}$ is given by the formula (1.8).

For the sake of comparison we note that the operator $\boldsymbol{\sf\Gamma}$
also admits a factorization:
$\boldsymbol{\sf\Gamma}=\boldsymbol{\Psi}^{*}\boldsymbol{\Psi}$. Under the
assumption that $\gamma(x,y)$ is given by the formula (1.4) the scalar
operator
$\boldsymbol{\Psi}:\textup{{{L}}}^{2}(\mathbb{R}^{3})\to\textup{{{L}}}^{2}(\mathbb{R}^{3N-3})$
has the form

(1.9)
$\displaystyle(\boldsymbol{\Psi}u)(\hat{\mathbf{x}})=\sqrt{N}\int_{\mathbb{R}^{3}}\psi(\hat{\mathbf{x}},x)u(x)dx,\
u\in\textup{{{L}}}^{2}(\mathbb{R}^{3}).$

This factorization was a key observation in the study of the operator
$\boldsymbol{\sf\Gamma}$ in [21].

Step 2: estimates for singular values. The asymptotic analysis of the operator
$\boldsymbol{{\sf{V}}}$ begins with effective bounds for the singular values
of $\boldsymbol{{\sf{V}}}$. By effective bounds we mean bounds for the
operator $\boldsymbol{{\sf{V}}}$ with weights $a=a(x),x\in\mathbb{R}^{3}$,
$b=b(\hat{\mathbf{x}}),\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}$, of the form

$\displaystyle\limsup_{k\to\infty}k\,s_{k}(b\,\boldsymbol{{\sf{V}}}a)\leq
C(a,b),$

where the right-hand side depends exlpicitly on $a$ and $b$. It is important
that $C(a,b)\to 0$ as some natural integral norm of $a$ (or $b$) tends to
zero. The precise statement is the subject of Theorem 4.1. Our proof of the
effective bound follows the method used by the author in [20] to establish
analogous bounds for the operator $\boldsymbol{\Psi}$. It relies on the
results by M.S. Birman and M. Z. Solomyak on estimates for singular values of
integral operators via suitable Sobolev norms of their kernels, see [4], and
on the regularity results for the function $\psi$. By an ellipticity argument,
the function $\psi$ is real analytic away from the coalescence points of the
particles, i.e. from $x_{j}\not=x_{k},1\leq j<k\leq N$ and $x_{j}\not=0$,
$j=1,2,\dots,N$. The classical T. Kato’s paper [13] shows that at the
coalescence points the function $\psi$ is Lipschitz. In our proofs we use the
global estimates for the derivatives of $\psi$ obtained by S. Fournais and
T.Ø. Sørensen in the recent paper [9]. In combination with the estimates from
[4] they lead to the required effective bounds for $\boldsymbol{{\sf{V}}}$. In
article [9] one can find further references to the vast bibliography on the
regularity of solutions to the Schrödinger equation.

Step 3: asymptotics. This step follows the approach of [21] where the
asymptotic formula (1.5) was proved. In order to obtain an asymptotic formula
for $\boldsymbol{{\sf{V}}}$ one needs to know the behaviour of the kernel
$\nabla_{x}\psi(\hat{\mathbf{x}},x)$ near the coalescence points. An
appropriate representation for the function $\psi$ was obtained in [10]. To
simplify the explanation we assume that the system consists of two particles,
i.e. that $N=2$. Under this assumption the problem retains all its crucial
features, but permits to avoid some tedious technical details. For $N=2$ we
have $\mathbf{x}=(t,x)\in\mathbb{R}^{3}\times\mathbb{R}^{3}$, and the operator
$\boldsymbol{{\sf{V}}}$ as defined in (1.8) acts from
$\textup{{{L}}}^{2}(\mathbb{R}^{3})$ into
$\textup{{{L}}}^{2}(\mathbb{R}^{3})$. According to [10], there exists a
neighbourhood
$\Omega_{1,2}\subset\big{(}\mathbb{R}^{3}\setminus\\{0\\}\big{)}\times\big{(}\mathbb{R}^{3}\setminus\\{0\\}\big{)}$
of the diagonal set $\\{(x,x):x\in\mathbb{R}^{3}\setminus\\{0\\}\\}$ and two
functions $\xi_{1,2},\eta_{1,2}$, real analytic in $\Omega_{1,2}$, such that
the eigenfunction $\psi=\psi(t,x)$ admits the representation

(1.10)
$\displaystyle\psi(t,x)=\xi_{1,2}(t,x)+|t-x|\,\eta_{1,2}(t,x),\quad\textup{for
all}\quad(t,x)\in\Omega_{1,2},$

and hence the kernel of the operator $\boldsymbol{{\sf{V}}}$ has the form

$\displaystyle\nabla_{x}\psi(t,x)=$ $\displaystyle\
\nabla_{x}\xi_{1,2}(t,x)+|t-x|\,\nabla_{x}\eta_{1,2}(t,x)+\frac{t-x}{|t-x|}\eta_{1,2}(t,x),$
(1.11) $\displaystyle\ \qquad\qquad\textup{for all}\quad(t,x)\in\Omega_{1,2}.$

Each of the terms on the right-hand side of (1) gives a different contribution
to the asymptotics of the singular values. According to [4], infinitely smooth
kernels lead to the decay rate which is faster than any negative power of the
singular value number. This allows us to say that the first term on the right-
hand side of (1) gives a zero contribution. The other two terms contain
homogeneous factors of order one and zero respectively. Spectral asymptotics
for operators with homogeneous kernels were studied extensively by M. Birman
and M. Solomyak in [2, 3] and [5], see also [4]. A summary of the results
needed for our purposes here can be found in [21].

The homogeneity order one kernels played a central role in paper [21] in the
study of the operator $\boldsymbol{\Psi}$ defined in (1.9). In this case the
Birman-Solomyak theory led to the asymptotics

$\displaystyle\lim_{k\to\infty}k^{\frac{4}{3}}s_{k}(\boldsymbol{\Psi})=A^{\frac{4}{3}},$

which, in its turn, implied (1.5). In the current paper the decay of the
singular values of the operator $\boldsymbol{{\sf{V}}}$ is determined by the
term of homogeneity order zero in (1). Precisely, according to the Birman-
Solomyak results (see [21] for a summary), kernels of order zero produce the
asymptotics of order $k^{-1}$ , which agrees with (1.7). However, as in [21],
the known formulas for integral operators with homogeneous kernels are not
directly applicable, since we have information neither on the smoothness of
the functions $\xi_{1,2},\eta_{1,2}$ on the closure $\overline{\Omega_{1,2}}$
nor on the integrability of these functions and their derivatives on
$\Omega_{1,2}$. To resolve this problem we approximate $\xi_{1,2},\eta_{1,2}$
by suitable $\textup{{{C}}}^{\infty}_{0}$-functions supported inside
$\Omega_{1,2}$. The estimates obtained in Step 2 ensure that the induced error
in the spectral asymptotics tends to zero as the smooth approximations
converge. In the limit this leads to the formula (1.7) with the constant

$\displaystyle
B=\frac{4}{3\pi}\int_{\mathbb{R}^{3}}|2^{1/2}\eta_{1,2}(x,x)|dx.$

Interestingly, the finiteness of the above integral is a by-product of the
spectral bounds. We emphasize that the coalescence points $x=0$ or $t=0$ do
not contribute to the asymptotics.

For $N\geq 3$ the proof needs to be modified to incorporate representations of
the form (1.10) in the neighbourhood of all pair coalescence points
$x_{j}=x_{k}$, $j,k=1,2,\dots,N$, $j\not=k$, see (2.5). The kernel of the
operator $\boldsymbol{{\sf{V}}}$ near the coalescence points becomes more
complex and as a result the proof requires an extra step involving an integral
operator whose structure mimics that of the operator $\boldsymbol{{\sf{V}}}$.
Such a model operator was studied in the author’s paper [21]. In order to make
it adaptable for the use in different situations, the model operator was
allowed to have arbitrary order of homogeneity at the pair coalescence points.
For order one the model operator was the key point in the analysis for the
operator $\boldsymbol{\Psi}$ conducted in [21]. In the current paper we use
the same model operator but with homogeneity of order zero, which is relevant
to the operator $\boldsymbol{{\sf{V}}}$. The required results are collected in
Subsect. 2.5. Using these facts together with approximations of
$\boldsymbol{{\sf{V}}}$ similar to the ones in the case $N=2$, we arrive at
the asymptotics (1.7) with the coefficient $B$ given in (2.7), concluding the
proof.

The paper is organized as follows. Section 2 contains the main result and
provides the preliminaries. First we describe the representation of the
function $\psi$ near the pair coalescence points and state the main result as
Theorem 2.3, which includes the formula (2.7) for the coefficient $B$. The
important Theorem 2.2 ensures that the coefficient $B$ is finite. In Subsect.
2.2 we detail the factorization
$\boldsymbol{{\sf{T}}}=\boldsymbol{{\sf{V}}}^{*}\boldsymbol{{\sf{V}}}$. In
Subsect. 2.3, 2.4 we put together the necessary facts about classes of compact
operators with power-like spectral behaviour and collect the estimates for
singular values of integral operators borrowed from [4]. Subsect. 2.5 is
focused on spectral asymptotics of the model integral operator that is
instrumental to the case $N\geq 3$. Using the factorization
$\boldsymbol{{\sf{T}}}=\boldsymbol{{\sf{V}}}^{*}\boldsymbol{{\sf{V}}}$, in
Sect. 3 the main Theorem 2.3 is restated in terms of the operator
$\boldsymbol{{\sf{V}}}$, see Theorem 3.1. In Lemma 3.2 we construct an
approximation for $\boldsymbol{{\sf{V}}}$ which is adapted for the use of the
model operator from Subsect. 2.5. These results allow us to prove Theorem 3.1
(and hence Theorem 2.3) in Subsect. 3.4. The rest of the paper is focused on
the proof of the approximation Lemma 3.2. It begins with derivation of
effective spectral bounds for the operator $\boldsymbol{{\sf{V}}}$ in Sect. 4.
With the help of the obtained bounds in Sect. 5 we show that the error in the
asymptotics induced by the approximations of $\boldsymbol{{\sf{V}}}$ tends to
zero as the approximations converge, thereby proving Lemma 3.2.

We conclude the introduction with some general notational conventions.

Coordinates. As mentioned earlier, we use the following standard notation for
the coordinates: $\mathbf{x}=(x_{1},x_{2},\dots,x_{N})$, where
$x_{j}\in\mathbb{R}^{3}$, $j=1,2,\dots,N$. The vector $\mathbf{x}$ is often
represented in the form

$\displaystyle\mathbf{x}=(\hat{\mathbf{x}}_{j},x_{j})\quad\textup{with}\quad\hat{\mathbf{x}}_{j}=(x_{1},x_{2},\dots,x_{j-1},x_{j+1},\dots,x_{N})\in\mathbb{R}^{3N-3},$

for arbitrary $j=1,2,\dots,N$. Most frequently we use this notation with
$j=N$, and write $\hat{\mathbf{x}}=\hat{\mathbf{x}}_{N}$, so that
$\mathbf{x}=(\hat{\mathbf{x}},x_{N})$. In order to write formulas in a more
compact and unified way, we sometimes use the notation $x_{0}=0$.

In the space $\mathbb{R}^{d},d\geq 1,$ the notation $|x|$ stands for the
Euclidean norm. In Sect. 4 in some calculations it is convenient to use the
$\ell^{1}$-norm of $x$ which we denote by $|x|_{1}$.

For $N\geq 3$ it is also useful to introduce the notation for $\mathbf{x}$
with $x_{j}$ and $x_{k}$ taken out:

(1.12)
$\displaystyle\tilde{\mathbf{x}}_{k,j}=\tilde{\mathbf{x}}_{j,k}=(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{k-1},x_{k+1},\dots,x_{N}),\quad\textup{for}\
j<k.$

If $j<k$, then we write $\mathbf{x}=(\tilde{\mathbf{x}}_{j,k},x_{j},x_{k})$.
For any $j\leq N-1$ the vector $\hat{\mathbf{x}}$ can be represented as
$\hat{\mathbf{x}}=(\tilde{\mathbf{x}}_{j,N},x_{j})$.

The notation $B_{R}$ is used for the ball $\\{x\in\mathbb{R}^{3}:|x|<R\\}$.

Indicators. For any set $\Lambda\subset\mathbb{R}^{d}$ we denote by
$\mathbbm{1}_{\Lambda}$ its indicator function (or indicator).

Derivatives. Let $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. If
$x=(x^{\prime},x^{\prime\prime},x^{\prime\prime\prime})\in\mathbb{R}^{3}$ and
$m=(m^{\prime},m^{\prime\prime},m^{\prime\prime\prime})\in\mathbb{N}_{0}^{3}$,
then the derivative $\partial_{x}^{m}$ is defined in the standard way:

$\displaystyle\partial_{x}^{m}=\partial_{x^{\prime}}^{m^{\prime}}\partial_{x^{\prime\prime}}^{m^{\prime\prime}}\partial_{x^{\prime\prime\prime}}^{m^{\prime\prime\prime}}.$

Bounds. For two non-negative numbers (or functions) $X$ and $Y$ depending on
some parameters, we write $X\lesssim Y$ (or $Y\gtrsim X$) if $X\leq CY$ with
some positive constant $C$ independent of those parameters. To avoid confusion
we may comment on the nature of (implicit) constants in the bounds.

Cut-off functions. We systematically use the following smooth cut-off
functions. Let

(1.13)
$\displaystyle\theta\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}),\quad\zeta(t)=1-\theta(t),$

be functions such that $0\leq\theta\leq 1$ and

(1.14) $\displaystyle\theta(t)=0,\quad\textup{if}\quad|t|>1;\
\quad\theta(t)=1,\quad\textup{if}\quad|t|<\frac{1}{2}.\ $

Integral operators. The notation $\operatorname{{\sf Int}}(\mathcal{K})$ is
used for the integral operator with kernel $\mathcal{K}$, e.g.
$\Gamma=\operatorname{{\sf Int}}(\gamma)$. The functional spaces, where
$\operatorname{{\sf Int}}(\mathcal{K})$ acts are obvious from the context.

##  2\. Main result, factorization of $\boldsymbol{{\sf{T}}}$, compact
operators

###  2.1. Representation formula

In order to state the main result we need a formula for the function $\psi$
describing its behaviour in a neighbourhood of the pair coalescence points.
Such a formula was obtained in [10]. For convenience, along with the standard
notation $\mathbf{x}=(x_{1},x_{2},\dots,x_{N})\in\mathbb{R}^{3N}$ we use the
notation $x_{0}=0$. Thus, unless otherwise stated, the indices labeling the
particles, run from $0$ to $N$.

Denote

(2.1) $\displaystyle{\sf
S}_{l,s}=\\{\mathbf{x}\in\mathbb{R}^{3N}:x_{l}\not=x_{s}\\},\ l\not=s.$

The function $\psi$ is real-analytic on the set

$\displaystyle{\sf{U}}=\bigcap_{0\leq l<s\leq N}{\sf S}_{l,s}.$

For each pair $j,k:j\not=k$, the set

(2.2) $\displaystyle{\sf{U}}_{j,k}=\bigcap_{\begin{subarray}{c}l\not=s\\\
(l,s)\not=(j,k)\end{subarray}}{\sf S}_{l,s}.$

contains the coalescence point $x_{j}=x_{k}$, whereas the other pairs of
variables are separated from each other. For the main result we are interested
in the shape of function $\psi$ near the diagonal

(2.3) $\displaystyle{\sf{U}}^{(\rm
d)}_{j,k}=\\{\mathbf{x}\in{\sf{U}}_{j,k}:x_{j}=x_{k}\\}.$

The sets introduced above are obviously symmetric with respect to permutations
of indices, e.g. ${\sf{U}}_{j,k}={\sf{U}}_{k,j}$, ${\sf{U}}^{(\rm
d)}_{j,k}={\sf{U}}^{(\rm d)}_{k,j}$.

The following property follows from [10, Theorem 1.4].

######  Proposition 2.1.

For each pair of indices $j,k=0,1,\dots,N$ such that $j\not=k$, there exists
an open connected set $\Omega_{j,k}=\Omega_{k,j}\subset\mathbb{R}^{3N}$, such
that

(2.4)
$\displaystyle{\sf{U}}_{j,k}^{(d)}\subset\Omega_{j,k}\subset{\sf{U}}_{j,k},$

and two uniquely defined functions $\xi_{j,k},\eta_{j,k}$, real analytic on
$\Omega_{j,k}$, such that for all $\mathbf{x}\in\Omega_{j,k}$ the following
representation holds:

(2.5)
$\displaystyle\psi(\mathbf{x})=\xi_{j,k}(\mathbf{x})+|x_{j}-x_{k}|\eta_{j,k}(\mathbf{x}).$

Because of the uniqueness of functions $\xi_{j,k},\eta_{j,k}$, they are
symmetric with respect to the permutations $j\leftrightarrow k$, i.e.
$\xi_{j,k}=\xi_{k,j}$, $\eta_{j,k}=\eta_{k,j}$ for all $j\not=k$.

The asymptotic coefficient $B$ in the formula (1.6) is defined via the
functions $\eta_{j,k}$, $j,k=1,2,\dots,N,j<k$, on the sets (2.3). Using the
notation (1.12) we write the function $\eta_{j,k}(\mathbf{x})$ on
${\sf{U}}_{j,k}^{(\operatorname{d})}$ as
$\eta_{j,k}(\tilde{\mathbf{x}}_{j,k},x,x)$. Proposition 2.1 gives no
information on the properties of $\xi_{j,k},\eta_{j,k}$ on the closure
$\overline{\Omega_{j,k}}$, but while proving the asymptotic formula (1.6) we
find the following integrability property of
$\eta_{j,k}(\tilde{\mathbf{x}}_{j,k},x,x)$.

######  Theorem 2.2.

If $N\geq 3$, then each function $\eta_{j,k}(\ \cdot\ ,x,x)$, $1\leq j<k\leq
N$, belongs to $\textup{{{L}}}^{2}(\mathbb{R}^{3N-6})$ a.e.
$x\in\mathbb{R}^{3}$ and the function

(2.6) $\displaystyle H(x):=\bigg{[}2\sum\limits_{1\leq j<k\leq
N}\int_{\mathbb{R}^{3N-6}}\big{|}\eta_{j,k}(\tilde{\mathbf{x}}_{j,k},x,x)\big{|}^{2}d\tilde{\mathbf{x}}_{j,k}\bigg{]}^{\frac{1}{2}},$

belongs to $\textup{{{L}}}^{1}(\mathbb{R}^{3})$.

If $N=2$, then the function $H(x):=\sqrt{2}|\eta_{1,2}(x,x)|$ belongs to
$\textup{{{L}}}^{1}(\mathbb{R}^{3})$.

Now we are in a position to state the main result of the paper.

######  Theorem 2.3.

Suppose that the eigenfunction $\psi$ satisfies the bound (1). Then the
eigenvalues $\lambda_{k}(\boldsymbol{{\sf{T}}}),k=1,2,\dots,$ of the operator
$\boldsymbol{{\sf{T}}}$ satisfy the asymptotic formula (1.6) with the constant

(2.7) $\displaystyle B=\frac{4}{3\pi}\int_{\mathbb{R}^{3}}H(x)\,dx.$

######  Remark.

  1. (1)

Theorem 2.3 extends to the case of a molecule with several nuclei whose
positions are fixed. The modifications are straightforward.

  2. (2)

The formula (2.7) shows that the spectral asymptotics depend only on the
behaviour of the eigenfunction $\psi$ near the pair coalescence points
$x_{j}=x_{k}$, $j,k=1,2,\dots,N$, $j\not=k$. Neither the points
$x_{j}=0,j=1,2,\dots,N$, nor the coalescence points of higher orders (e.g.
$x_{j}=x_{k}=x_{l}$ with pair-wise distinct $j,k,l$) contribute to the
asymptotics (1.6).

  3. (3)

For the sake of comparison we give here the formula for the coefficient $A$ in
the asymptotic formula (1.5) for the operator $\boldsymbol{\sf\Gamma}$. As
shown in [21, Theorem 2.2], the function $H$ also belongs to
$\textup{{{L}}}^{3/4}(\mathbb{R}^{3})$ and

$\displaystyle
A=\frac{1}{3}\bigg{(}\frac{2}{\pi}\bigg{)}^{\frac{5}{4}}\int_{\mathbb{R}^{3}}H(x)^{\frac{3}{4}}\,dx.$

Thus in both asymptotic formulas (1.5) and (1.6) are determined only by the
functions $\eta_{j,k}$.

###  2.2. Factorization of $\boldsymbol{{\sf{T}}}$: change of variables
$(\hat{\mathbf{x}}_{j},x)\mapsto(\hat{\mathbf{x}},x)$

Let us describe the factorization of the operator $\boldsymbol{{\sf{T}}}$
which is central to the proof of Theorem 2.3, and the associated change of
variables. Rewrite the definition (1.3) in the form:

$\displaystyle\tau(x,y)=$ $\displaystyle\
\sum_{j=1}^{N}\int_{\mathbb{R}^{3N-3}}\overline{{\sf{v}}_{j}(\hat{\mathbf{x}},x)}\cdot{\sf{v}}_{j}(\hat{\mathbf{x}},y)d\hat{\mathbf{x}},\quad\textup{where}$
(2.8) $\displaystyle{\sf{v}}_{j}(\hat{\mathbf{x}},x)=$ $\displaystyle\
\nabla_{x}\psi(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad
j=1,2,\dots,N.$

Therefore $\boldsymbol{{\sf{T}}}$ can be represented as a product
$\boldsymbol{{\sf{T}}}=\boldsymbol{{\sf{V}}}^{*}\boldsymbol{{\sf{V}}}$, where
$\boldsymbol{{\sf{V}}}:\textup{{{L}}}^{2}(\mathbb{R}^{3})\to\textup{{{L}}}^{2}(\mathbb{R}^{3N-3})$
is the integral operator with the vector-valued kernel
${\sf{V}}(\hat{\mathbf{x}},x):\mathbb{C}\to\mathbb{C}^{3N}$ given by

(2.9)
$\displaystyle{\sf{V}}(\hat{\mathbf{x}},x)=\\{{\sf{v}}_{j}(\hat{\mathbf{x}},x)\\}_{j=1}^{N}.$

As explained in the Introduction, given this factorization, the asymptotic
relation (1.6) translates to the formula (1.7). Later we state this fact again
as Theorem 3.1 using a more convenient notation.

The change of variables $(\hat{\mathbf{x}}_{j},x)\mapsto(\hat{\mathbf{x}},x)$
plays an important role throughout the paper. In particular, it is crucial to
rewrite Proposition 2.1 in terms of the new variables $(\hat{\mathbf{x}},x)$.
Let $j\not=k$, and let $\Omega_{j,k}$ be the sets and $\xi_{j,k}(\mathbf{x})$,
$\eta_{j,k}(\mathbf{x})$ be the functions from Proposition 2.1. For all
$j=1,2,\dots,N$ and all $k=0,1,\dots,N-1,$ denote

$\displaystyle\tilde{\Omega}_{j,k}=\begin{cases}\\{(\hat{\mathbf{x}},x)\in\mathbb{R}^{3N}:(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1})\in\Omega_{j,k}\\},\quad\textup{if}\
j\geq k+1,\\\
\\{(\hat{\mathbf{x}},x)\in\mathbb{R}^{3N}:(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1})\in\Omega_{j,k+1}\\},\quad\textup{if}\
j\leq k.\end{cases}$

According to (2.4) we have

(2.10)
$\displaystyle{\sf{U}}_{N,k}^{(\operatorname{d})}\subset\tilde{\Omega}_{j,k}\subset{\sf{U}}_{N,k},\quad
k=0,1,\dots,N-1,$

for all $j=1,2,\dots,N$. Together with functions $\xi_{j,k},\eta_{j,k}$ define

$\displaystyle\tilde{\xi}_{j,k}(\hat{\mathbf{x}},x)=\begin{cases}\xi_{j,k}(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad\textup{if}\
j\geq k+1,\\\\[5.69046pt]
\xi_{j,k+1}(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad\textup{if}\ j\leq
k,\end{cases}$

and

$\displaystyle\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x)=\begin{cases}\eta_{j,k}(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad\textup{if}\
j\geq k+1,\\\\[5.69046pt]
\eta_{j,k+1}(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad\textup{if}\
j\leq k.\end{cases}$

By Proposition 2.1, for each $j=1,2,\dots,N$, and each $k=0,1,\dots,N-1$, we
have

$\displaystyle{\sf{v}}_{j}(\hat{\mathbf{x}},x)=\nabla_{x}\tilde{\xi}_{j,k}(\hat{\mathbf{x}},x)+$
$\displaystyle\ |x_{k}-x|\nabla_{x}\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x)$
(2.11) $\displaystyle\
+(x_{k}-x)|x_{k}-x|^{-1}\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x),\quad\textup{for
all}\ (\hat{\mathbf{x}},x)\in\tilde{\Omega}_{j,k}.$

Observe that the sets $\tilde{\Omega}_{j,k}$ and the functions
$\tilde{\xi}_{j,k},\tilde{\eta}_{j,k}$ are not symmetric under the permutation
$j\leftrightarrow k$.

The function (2.6) can be easily rewritten via the new functions
$\tilde{\eta}_{j,k}$:

(2.12) $\displaystyle
H(x)=\begin{cases}\big{(}|\tilde{\eta}_{1,1}(x,x)|^{2}+|\tilde{\eta}_{2,1}(x,x)|^{2}\big{)}^{1/2},\quad\textup{if}\
N=2;\\\\[8.5359pt]
\bigg{[}\sum\limits_{j=1}^{N}\sum\limits_{k=1}^{N-1}\int_{\mathbb{R}^{3N-6}}\big{|}\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},x,x)\big{|}^{2}d\tilde{\mathbf{x}}_{k,N}\bigg{]}^{\frac{1}{2}},\quad\textup{if}\
N\geq 3.\end{cases}$

This calculation is done in [21, Sect. 2].

###  2.3. Compact operators

Here we provide necessary information about compact operators. Most of the
listed facts can be found in [6, Chapter 11]. Let $\mathcal{H}$ and
$\mathcal{G}$ be separable Hilbert spaces. Let $T:\mathcal{H}\to\mathcal{G}$
be a compact operator. If $\mathcal{H}=\mathcal{G}$ and $T=T^{*}\geq 0$, then
$\lambda_{k}(T)$, $k=1,2,\dots$, denote the positive eigenvalues of $T$
numbered in descending order counting multiplicity. For arbitrary spaces
$\mathcal{H}$, $\mathcal{G}$ and compact $T$, by $s_{k}(T)>0$, $k=1,2,\dots$,
we denote the singular values of $T$ defined by
$s_{k}(T)^{2}=\lambda_{k}(T^{*}T)=\lambda_{k}(TT^{*})$. Note the useful
inequality

(2.13) $\displaystyle s_{2k}(T_{1}+T_{2})\leq s_{2k-1}(T_{1}+T_{2})\leq
s_{k}(T_{1})+s_{k}(T_{2}),$

which holds for any two compact $T_{1},T_{2}$. If $s_{k}(T)\lesssim
k^{-1/p},k=1,2,\dots$, with some $p>0$, then we say that
$T\in\mathbf{S}_{p,\infty}$ and denote

(2.14) $\displaystyle\|T\|_{p,\infty}=\sup_{k}s_{k}(T)k^{\frac{1}{p}}.$

The class $\mathbf{S}_{p,\infty}$ is a complete linear space with the quasi-
norm $\|T\|_{p,\infty}$. For all $p>0$ the functional $\|T\|_{p,\infty}$
satisfies the following “triangle” inequality for operators
$T_{1},T_{2}\in\mathbf{S}_{p,\infty}$:

(2.15)
$\displaystyle\|T_{1}+T_{2}\|_{{\scaleto{p}{4pt}},{\scaleto{\infty}{3pt}}}^{{\frac{{\scaleto{p}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\leq\|T_{1}\|_{{\scaleto{p}{4pt}},{\scaleto{\infty}{3pt}}}^{{\frac{{\scaleto{p}{3pt}}}{{\scaleto{p+1}{4pt}}}}}+\|T_{2}\|_{{\scaleto{p,\infty}{4pt}}}^{{\frac{{\scaleto{p}{3pt}}}{{\scaleto{p+1}{4pt}}}}}.$

For $T\in\mathbf{S}_{p,\infty}$ the following numbers are finite:

(2.16)
$\displaystyle\begin{cases}{\sf{G}}_{p}(T)=\big{(}\limsup\limits_{k\to\infty}k^{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p}{3pt}}}}s_{k}(T)\big{)}^{p},\\\\[8.5359pt]
{\sf{g}}_{p}(T)=\big{(}\liminf\limits_{k\to\infty}k^{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p}{3pt}}}}s_{k}(T)\big{)}^{p},\end{cases}$

and they clearly satisfy the inequalities

$\displaystyle{\sf{g}}_{p}(T)\leq{\sf{G}}_{p}(T)\leq\|T\|_{p,\infty}^{p}.$

Note that if ${\sf{G}}_{q}(T)=0$ for all $q>p$. Observe that

(2.17)
$\displaystyle{\sf{g}}_{p}(TT^{*})={\sf{g}}_{p}(T^{*}T)={\sf{g}}_{2p}(T),\quad{\sf{G}}_{p}(TT^{*})={\sf{G}}_{p}(T^{*}T)={\sf{G}}_{2p}(T).$

If ${\sf{G}}_{p}(T)={\sf{g}}_{p}(T)$, then the singular values of $T$ satisfy
the asymptotic formula

$\displaystyle
s_{n}(T)=\big{(}{\sf{G}}_{p}(T)\big{)}^{\frac{1}{p}}n^{-\frac{1}{p}}+o(n^{-\frac{1}{p}}),\
n\to\infty.$

The functionals ${\sf{g}}_{p}(T)$, ${\sf{G}}_{p}(T)$ also satisfy the
inequalities of the type (2.15):

(2.18)
$\displaystyle\begin{cases}{\sf{G}}_{p}(T_{1}+T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\leq{\sf{G}}_{p}(T_{1})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}+{\sf{G}}_{p}(T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}},\\\\[8.5359pt]
{\sf{g}}_{p}(T_{1}+T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\leq{\sf{g}}_{p}(T_{1})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}+{\sf{G}}_{p}(T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}.\end{cases}$

We need a version of the first inequality for infinitely many operators.

######  Lemma 2.4.

Suppose that $T_{j}\in\mathbf{S}_{p,\infty}$, $j=1,2,\dots,$ with some $p>0$
and that

(2.19)
$\displaystyle\sum_{j}\|T_{j}\|_{{\scaleto{p}{4pt}},{\scaleto{\infty}{3pt}}}^{{\frac{{\scaleto{p}{3pt}}}{{\scaleto{p+1}{4pt}}}}}<\infty.$

Then

(2.20)
$\displaystyle{\sf{G}}_{p}\big{(}\sum_{j}T_{j}\big{)}^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\leq\sum_{j}{\sf{G}}_{p}(T_{j})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}.$

###### Proof.

The bound (2.20) follows from the triangle inequality (2.18) by virtue of
completeness of $\mathbf{S}_{p,\infty}$. ∎

It also follows from (2.18) that the functionals ${\sf{G}}_{p}$ and
${\sf{g}}_{p}$ are continuous on $\mathbf{S}_{p,\infty}$:

$\displaystyle\big{|}{\sf{G}}_{p}(T_{1})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}-{\sf{G}}_{p}(T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\big{|}\leq$
$\displaystyle\
{\sf{G}}_{p}(T_{1}-T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}},$
$\displaystyle\big{|}{\sf{g}}_{p}(T_{1})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}-{\sf{g}}_{p}(T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\big{|}\leq$
$\displaystyle\
{\sf{G}}_{p}(T_{1}-T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}.$

We need the following two corollaries of this fact:

######  Corollary 2.5.

Suppose that ${\sf{G}}_{p}(T_{1}-T_{2})=0$. Then

$\displaystyle{\sf{G}}_{p}(T_{1})={\sf{G}}_{p}(T_{2}),\quad{\sf{g}}_{p}(T_{1})={\sf{g}}_{p}(T_{2}).$

The next corollary is more general:

######  Corollary 2.6.

Suppose that $T\in\mathbf{S}_{p,\infty}$ and that for every $\nu>0$ there
exists an operator $T_{\nu}\in\mathbf{S}_{p,\infty}$ such that
${\sf{G}}_{p}(T-T_{\nu})\to 0$, $\nu\to 0$. Then the functionals
${\sf{G}}_{p}(T_{\nu}),{\sf{g}}_{p}(T_{\nu})$ have limits as $\nu\to 0$ and

$\displaystyle\lim_{\nu\to
0}{\sf{G}}_{p}(T_{\nu})={\sf{G}}_{p}(T),\quad\lim_{\nu\to
0}{\sf{g}}_{p}(T_{\nu})={\sf{g}}_{p}(T).$

We also need to make a remark about “block-vector” operators. Let
$T_{j}\in\mathbf{S}_{p,\infty}$ be a finite collection of compact operators.
Define the operator $\mathbf{T}:\mathcal{H}\to\oplus_{j}\mathcal{G}$ by
$\mathbf{T}=\\{T_{j}\\}_{j}$. Since

$\displaystyle\mathbf{T}^{*}\mathbf{T}=\sum_{j}T_{j}^{*}T_{j},$

by (2.18) and (2.17) we have
$\mathbf{T}^{*}\mathbf{T}\in\mathbf{S}_{q,\infty}$, $q=p/2$ and

$\displaystyle{\sf{G}}_{q}(\mathbf{T}^{*}\mathbf{T})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{q+1}{4pt}}}}}\leq\sum_{j}{\sf{G}}_{q}(T_{j}^{*}T_{j})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{q+1}{4pt}}}}}.$

Therefore, in view of (2.17) again,

(2.21)
$\displaystyle{\sf{G}}_{p}(\mathbf{T})^{{\frac{{\scaleto{2}{3pt}}}{{\scaleto{p+2}{4pt}}}}}\leq\sum_{j}{\sf{G}}_{p}(T_{j})^{{\frac{{\scaleto{2}{3pt}}}{{\scaleto{p+2}{4pt}}}}}.$

The same bound holds if one replaces ${\sf{G}}_{p}(\ \cdot\ )$ with the quasi-
norm $\|\ \cdot\ \|_{p,\infty}$. Consequently, in order to estimate the
singular values of $\mathbf{T}$ it suffices to estimate those of its
components $T_{j}$. We use this fact throughout the paper.

###  2.4. Singular values of integral operators

The final ingredient of the proof is the result due to M.Birman and M.
Solomyak, investigating the membership of integral operators in the class
$\mathbf{S}_{p,\infty}$ with some $p>0$. For estimates of the singular values
we rely on [4, Propositions 2.1, 2.3], see also [6, Theorem 11.8.4], which we
state here in a form convenient for our purposes. Let
$T_{ba}:\textup{{{L}}}^{2}(X)\to\textup{{{L}}}^{2}(\mathbb{R}^{n}),X\subset\mathbb{R}^{d},$
be the integral operator of the form

$\displaystyle(T_{ba}u)(t)=b(t)\int_{X}T(t,x)a(x)u(x)\,dx.$

We consider this operator in two cases: when $X$ is the unit cube
$\mathcal{C}=(0,1)^{d}$, or $X=\mathbb{R}^{d}$.

######  Proposition 2.7.

Let $X=\mathcal{C}$. Assume that the kernel
$T(t,x),t\in\mathbb{R}^{n},x\in\mathcal{C},$ is such that $T(t,\ \cdot\
)\in\textup{{{H}}}^{l}(\mathcal{C})$ with some $l=1,2,\dots$, $2l\geq d$, a.e.
$t\in\mathbb{R}^{n}$. Assume that $a\in\textup{{{L}}}^{r}(\mathcal{C})$, where
$r=2$ if $2l>d$ and $r>1$ is arbitrary if $2l=d$. Then

$\displaystyle s_{k}(T_{ba})\lesssim
k^{-\frac{1}{2}-\frac{l}{d}}\biggl{[}\int_{\mathbb{R}^{n}}\|T(t,\ \cdot\
)\|_{\textup{{{H}}}^{l}}^{2}\,|b(t)|^{2}\,dt\biggr{]}^{\frac{1}{2}}\|a\|_{\textup{{{L}}}^{r}(\mathcal{C})},$

$k=1,2,\dots$. In other words, $T_{ba}\in\mathbf{S}_{q,\infty}$ with
$q^{-1}=2^{-1}+ld^{-1}$ and

$\displaystyle\|T_{ba}\|_{q,\infty}\lesssim\biggl{[}\int_{\mathbb{R}^{n}}\|T(t,\
\cdot\
)\|_{\textup{{{H}}}^{l}}^{2}\,|b(t)|^{2}\,dt\biggr{]}^{\frac{1}{2}}\|a\|_{\textup{{{L}}}^{r}(\mathcal{C})}.$

It is straightforward to check that if one replaces the cube $\mathcal{C}$
with its translate $\mathcal{C}_{n}=\mathcal{C}+n,n\in\mathbb{Z}^{d}$, then
the bounds of Proposition 2.7 still hold with implicit constants independent
of $n$.

######  Proposition 2.8.

Let $X=\mathbb{R}^{d}$. Assume that the kernel
$T(t,x),t\in\mathbb{R}^{n},x\in\mathbb{R}^{d},$ is such that $T(t,\ \cdot\
)\in\textup{{{H}}}^{l}(\mathbb{R}^{d})$ with some $l=1,2,\dots$, $2l<d$, a.e.
$t\in\mathbb{R}^{n}$. Assume that $a\in\textup{{{L}}}^{r}(\mathbb{R}^{d})$,
where $r=dl^{-1}$. Then

$\displaystyle s_{k}(T_{ba})\lesssim
k^{-\frac{1}{2}-\frac{l}{d}}\biggl{[}\int_{\mathbb{R}^{n}}\|T(t,\ \cdot\
)\|_{\textup{{{H}}}^{l}}^{2}\,|b(t)|^{2}\,dt\biggr{]}^{\frac{1}{2}}\|a\|_{\textup{{{L}}}^{r}(\mathbb{R}^{d})},$

$k=1,2,\dots$. In other words, $T_{ba}\in\mathbf{S}_{q,\infty}$ with
$q^{-1}=2^{-1}+ld^{-1}$ and

$\displaystyle\|T_{ba}\|_{q,\infty}\lesssim\biggl{[}\int_{\mathbb{R}^{n}}\|T(t,\
\cdot\
)\|_{\textup{{{H}}}^{l}}^{2}\,|b(t)|^{2}\,dt\biggr{]}^{\frac{1}{2}}\|a\|_{\textup{{{L}}}^{r}(\mathbb{R}^{d})}.$

###  2.5. Model operator

Let $a,b_{j,k},\beta_{j,k}$, $j=1,2,\dots,N$, $k=1,2,\dots,N-1$, be scalar
functions such that

(2.22)
$\displaystyle\begin{cases}a\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3}),&\
\quad b_{j,k}\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N-3}),\\\\[5.69046pt]
\beta_{j,k}\in\textup{{{C}}}^{\infty}(\mathbb{R}^{3N}),\end{cases}$

for all $j=1,2,\dots,N$, $k=1,2,\dots,N-1$. Let
$\Phi\in\textup{{{C}}}^{\infty}(\mathbb{R}^{3}\setminus\\{0\\})$ be a vector
function with $m$ scalar components, homogeneous of order $\alpha>-3$:

(2.23) $\displaystyle\Phi(tx)=t^{\alpha}\Phi(x),\quad x\not=0,t>0.$

Consider the vector-valued kernel
$\mathcal{M}(\hat{\mathbf{x}},x):\mathbb{C}\to\mathbb{C}^{mN}$:

(2.24)
$\displaystyle\begin{cases}\mathcal{M}(\hat{\mathbf{x}},x)=\\{\mathcal{M}_{j}(\hat{\mathbf{x}},x)\\}_{j=1}^{N},\
\quad\mathcal{M}_{j}(\hat{\mathbf{x}},x)=\sum_{k=1}^{N-1}\mathcal{M}_{j,k}(\hat{\mathbf{x}},x),\\\\[8.5359pt]
\mathcal{M}_{j,k}(\hat{\mathbf{x}},x)=b_{j,k}(\hat{\mathbf{x}})\Phi(x_{k}-x)a(x)\beta_{j,k}(\hat{\mathbf{x}},x).\end{cases}$

The eigenvalue asymptotics for the operator $\operatorname{{\sf
Int}}(\mathcal{M})$ was found in [21] for general functions $\Phi$ of the form
(2.23). Below we state this result for the function
$\Phi(x)=\nabla|x|=x|x|^{-1}$, which is homogeneous of order $0$. This is the
case needed for the study of the operator $\boldsymbol{{\sf{T}}}$. We use the
representations $(\hat{\mathbf{x}},x)=(\tilde{\mathbf{x}}_{k,N},x_{k},x)$
introduced in (1.12). Denote

(2.25)
$\displaystyle\begin{cases}h(t)=\bigg{[}\sum_{j=1}^{N}\sum_{k=1}^{N-1}\int_{\mathbb{R}^{3N-6}}|b_{j,k}(\tilde{\mathbf{x}}_{k,N},t)\beta_{j,k}(\tilde{\mathbf{x}}_{k,N},t,t)|^{2}d\tilde{\mathbf{x}}_{k}\bigg{]}^{\frac{1}{2}},\
\textup{if}\ N\geq 3;\\\\[8.5359pt]
h(t)=\big{(}|b_{1,1}(t)\beta_{1,1}(t,t)|^{2}+|b_{2,1}(t)\beta_{2,1}(t,t)|^{2}\big{)}^{\frac{1}{2}},\
\textup{if}\ N=2.\end{cases}$

The next proposition summarizes the required results from [21].

######  Proposition 2.9.

If $\Phi(x)=\nabla|x|=x|x|^{-1}$, then $\operatorname{{\sf
Int}}(\mathcal{M})\in\mathbf{S}_{1,\infty}$ and

(2.26) $\displaystyle{\sf{G}}_{1}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}={\sf{g}}_{1}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}=\nu_{0,3}\int_{\mathbb{R}^{3}}|a(x)h(x)|\,dx,$

where $\nu_{0,3}=4(3\pi)^{-1}$.

If $\Phi(x)$ is homogeneous of order $\alpha>0$, then
${\sf{G}}_{1}\big{(}\operatorname{{\sf Int}}(\mathcal{M})\big{)}=0$.

##  3\. Factorization of $\boldsymbol{{\sf{T}}}$: operator
$\boldsymbol{{\sf{V}}}$

###  3.1. Reformulation of the problem

Using the functionals (2.16), one can rewrite the sought formula (1.6) as

$\displaystyle{\sf{G}}_{2}(\boldsymbol{{\sf{T}}})={\sf{g}}_{2}(\boldsymbol{{\sf{T}}})=B.$

Since $\boldsymbol{{\sf{T}}}=\boldsymbol{{\sf{V}}}^{*}\boldsymbol{{\sf{V}}}$
with the operator
$\boldsymbol{{\sf{V}}}:\textup{{{L}}}^{2}(\mathbb{R}^{3})\to\textup{{{L}}}^{2}(\mathbb{R}^{3N-3})$
defined in (2.9), by (2.17) the above equalities rewrite as

(3.1)
$\displaystyle{\sf{G}}_{1}(\boldsymbol{{\sf{V}}})={\sf{g}}_{1}(\boldsymbol{{\sf{V}}})=B.$

Thus the main Theorem 2.3 can be recast as follows:

######  Theorem 3.1.

Under the conditions of Theorem 2.3 the formula (3.1) holds with the constant
$B$ which is defined in (2.7).

The rest of the paper is focused on the proof of Theorem 3.1.

As explained in the Introduction, at the heart of the proof is the formula
(2.5) for the function $\psi$, which translates to the representation (2.2)
for the kernels ${\sf{v}}_{j}$ defined in (2.2). This representation allows us
to reduce the problem to the model operator considered in Subsect. 2.5 with
the function $\Phi(x)=x|x|^{-1}$. At the first stage of this reduction we
construct convenient approximations of the kernels
${\sf{v}}_{j}(\hat{\mathbf{x}},x)$.

###  3.2. Cut-off functions

Firt we construct appropriate cut-offs. Fix a $\delta>0$. Along with the sets
(2.1) introduce

(3.2) $\displaystyle{\sf S}_{l,s}(\delta)={\sf
S}_{s,l}(\delta)=\\{\mathbf{x}\in\mathbb{R}^{3N}:|x_{l}-x_{s}|>\delta\\},\
0\leq l<s\leq N,$

and for all $k=0,1,\dots,N-1$, define

$\displaystyle{\sf{U}}_{k}(\delta)=\bigg{(}\bigcap_{0\leq l<s\leq N-1}{\sf
S}_{l,s}(\delta)\bigg{)}\bigcap\bigg{(}\bigcap_{\begin{subarray}{c}0\leq s\leq
N-1\\\ s\not=k\end{subarray}}{\sf S}_{s,N}(\delta)\bigg{)}.$

Comparing with (2.2) we see that ${\sf{U}}_{k}(\delta)\subset{\sf{U}}_{k,N}$,
and for $\mathbf{x}\in{\sf{U}}_{k}(\delta)$ all the coordinate pairs, except
for $x_{k}$ and $x_{N}$, are separated by a distance $\delta$. Similarly to
(2.3) define the diagonal set

$\displaystyle{\sf{U}}^{(\rm
d)}_{k}(\delta)=\\{\mathbf{x}\in{\sf{U}}_{k}(\delta):x_{j}=x_{N}\\}\subset{\sf{U}}_{k,N}^{(\operatorname{d})}.$

Recall that the representation (2.2) holds on the domain
$\tilde{\Omega}_{j,k}$ which satisfies (2.10) for all $j=1,2,\dots,N$,
$k=0,1,\dots,N-1$. We construct a compact subset of $\tilde{\Omega}_{j,k}$ in
the following way. For $R>0$ let

$\displaystyle{\sf{U}}_{k}(\delta,R)=$ $\displaystyle\
{\sf{U}}_{k}(\delta)\bigcap\ (B_{R})^{N},$ $\displaystyle{\sf{U}}^{(\rm
d)}_{k}(\delta,R)=$ $\displaystyle\
\\{\mathbf{x}\in{\sf{U}}_{k}(\delta,R):x_{k}=x_{N}\\},$

where $B_{R}=\\{x\in\mathbb{R}^{3}:|x|<R\\}$. The set ${\sf{U}}^{(\rm
d)}_{k}(\delta,R)$ is bounded and its closure belongs to
$\tilde{\Omega}_{j,k}$ for all $\delta>0,R>0$. Therefore, there exists an
$\varepsilon_{0}=\varepsilon_{0}(\delta,R)>0$ such that the $j$-independent
$\varepsilon$-neighbourhood

(3.3)
$\displaystyle\tilde{\Omega}_{k}(\delta,R,\varepsilon):=\\{\mathbf{x}\in{\sf{U}}_{k}(\delta,R):|x_{k}-x_{N}|<\varepsilon\\},$

together with its closure, belongs to $\tilde{\Omega}_{j,k}$ for all
$\varepsilon\in(0,\varepsilon_{0})$:

(3.4)
$\displaystyle\overline{\tilde{\Omega}_{k}(\delta,R,\varepsilon)}\subset\tilde{\Omega}_{j,k},\quad\forall\varepsilon\in(0,\varepsilon_{0}).$

Now we specify $\textup{{{C}}}^{\infty}_{0}$ cutoffs supported on the domains
$\tilde{\Omega}_{k}(\delta,R,\varepsilon)$. Let
$\theta\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R})$ and $\zeta=1-\theta$ be as
defined in (1.13), (1.14). Denote

(3.5) $\displaystyle Y_{\delta}(\hat{\mathbf{x}})=\prod_{0\leq l<s\leq
N-1}\zeta\big{(}|x_{l}-x_{s}|(4\delta)^{-1}\big{)}.$

By the definition of $\zeta$,

$\displaystyle\operatorname{{supp}}Y_{\delta}\subset\bigcap_{0\leq l<s\leq
N-1}{\sf S}_{l,s}(2\delta),$

where ${\sf S}_{l,s}(\ \cdot\ )$ is defined in (3.2). As $\theta\geq 0$ and
$\zeta\leq 1$, we have

(3.6) $\displaystyle 1-Y_{\delta}(\hat{\mathbf{x}})\leq\sum_{0\leq l<s\leq
N-1}\theta\big{(}|x_{l}-x_{s}|(4\delta)^{-1}\big{)},$

Furthermore, due to (1.14), for any $\varepsilon\leq\delta$ we have

(3.7) $\displaystyle
Y_{\delta}(\hat{\mathbf{x}})\sum_{j=0}^{N-1}\theta\big{(}|x-x_{j}|\varepsilon^{-1}\big{)}+Y_{\delta}(\hat{\mathbf{x}})\prod_{j=0}^{N-1}\zeta\big{(}|x-x_{j}|\varepsilon^{-1}\big{)}=Y_{\delta}(\hat{\mathbf{x}}).$

Define also cut-offs at infinity. Denote

(3.8) $\displaystyle Q_{R}(\hat{\mathbf{x}})=\prod_{1\leq l\leq
N-1}\theta\big{(}|x_{l}|R^{-1}\big{)},\quad
K_{R}(x)=\theta\big{(}|x|R^{-1}\big{)}.$

As shown in [21, Lemma 5.2],

for all $\displaystyle\
\varepsilon<\min\\{\varepsilon_{0},\delta\\}\quad\textup{the support of the
function}$ (3.9) $\displaystyle\
Q_{R}(\hat{\mathbf{x}})K_{R}(x)Y_{\delta}(\hat{\mathbf{x}})\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\
\ \textup{belongs to}\ \ \tilde{\Omega}_{k}(\delta,R,\varepsilon)$

for all $k=0,1,\dots,N-1$.

### 3.3.

Using the cut-offs introduced above we construct a convenient approximation
for the kernels ${\sf{v}}_{j}(\hat{\mathbf{x}},x)$. Taking if necessary, a
smaller $\varepsilon_{0}$ in (3.4), we will assume that
$\varepsilon_{0}(\delta,R)\leq\delta$, and hence for all
$\varepsilon<\varepsilon_{0}(\delta,R)$, apart from the inclusion (3.4) we
also have (3.2). Thus, for these values of $\varepsilon$ the real analytic
functions $\tilde{\xi}_{j,k},\tilde{\eta}_{j,k}$ are well-defined on the
support of the function (3.2), and hence the vector-valued kernel

(3.10)
$\displaystyle\Upsilon_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})K_{R}(x)\sum_{k=1}^{N-1}\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\frac{x-x_{k}}{|x-x_{k}|}\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x),$

is well-defined for all $(\hat{\mathbf{x}},x)\in\mathbb{R}^{3N}$, and each of
the functions

$\displaystyle
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})K_{R}(x)\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x),\quad
k=1,2,\dots,N-1,$

is $\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N})$. Our objective is to prove
that the vector-valued kernel

$\displaystyle\boldsymbol{\Upsilon}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=\big{\\{}\Upsilon_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)\big{\\}}_{j=1}^{N}$

is an approximation for $\boldsymbol{{\sf{V}}}(\hat{\mathbf{x}},x)$(see (2.9))
in the following sense.

######  Lemma 3.2.

The following relations hold:

$\displaystyle{\sf{G}}_{1}(\boldsymbol{{\sf{V}}})=\lim\limits_{\begin{subarray}{c}\delta\to
0\\\ R\to\infty\end{subarray}}\lim_{\varepsilon\to
0}{\sf{G}}_{1}\big{(}\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])\big{)},\quad{\sf{g}}_{1}(\boldsymbol{{\sf{V}}})=\lim\limits_{\begin{subarray}{c}\delta\to
0\\\ R\to\infty\end{subarray}}\lim_{\varepsilon\to
0}{\sf{g}}_{1}\big{(}\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])\big{)},$

where the limits on the right-hand side exist.

This lemma in combination with Proposition 2.9 is sufficient for the proof of
Theorem 3.1, which is the subject of the next subsection.

###  3.4. Proof of Theorems 2.2 and 3.1, 2.3

First we apply Proposition 2.9 to the operator $\operatorname{{\sf
Int}}\big{(}\boldsymbol{\Upsilon}[\delta,R,\varepsilon]\big{)}$.

######  Lemma 3.3.

The operator $\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])$ belongs to
$\mathbf{S}_{1,\infty}$ for all
$\delta>0,R>0,\varepsilon<\varepsilon_{0}(\delta,R)$ and

(3.11) $\displaystyle{\sf{G}}_{1}\big{(}\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])\big{)}={\sf{g}}_{1}\big{(}\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])\big{)}=\frac{4}{3\pi}\int
K_{R}(t)H_{\delta,R}(t)dt,$

where

$\displaystyle
H_{\delta,R}(t)=Q_{R}(t)Y_{\delta}(t)\,\big{(}|\tilde{\eta}_{1,1}(t,t)|^{2}+\tilde{\eta}_{1,2}(t,t)|^{2}\big{)}^{1/2},\
\textup{if}\ N=2,$

and

(3.12) $\displaystyle
H_{\delta,R}(t)=\bigg{[}\sum_{j=1}^{N}\sum_{k=1}^{N-1}\int_{\mathbb{R}^{3N-6}}\big{|}Q_{R}(\tilde{\mathbf{x}}_{k,N},t)Y_{\delta}(\tilde{\mathbf{x}}_{k,N},t)\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},t,t)\big{|}^{2}d\tilde{\mathbf{x}}_{k,N}\bigg{]}^{\frac{1}{2}},\
\textup{if}\ N\geq 3.$

###### Proof.

The kernel $\boldsymbol{\Upsilon}[\delta,R,\varepsilon]$ (see (3.10)) has the
form (2.24) with

$\displaystyle a(x)=K_{2R}(x),\ $ $\displaystyle\
b_{j,k}(\hat{\mathbf{x}})=Q_{2R}(\hat{\mathbf{x}})Y_{\delta/2}(\hat{\mathbf{x}}),$
$\displaystyle\beta_{j,k}(\hat{\mathbf{x}},x)=$ $\displaystyle\
\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x)Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})K_{R}(x),$

and the homogeneous function $\Phi(x)=x|x|^{-1}$. Here we have used the fact
that

$\displaystyle
Q_{R}(\hat{\mathbf{x}})Q_{2R}(\hat{\mathbf{x}})=Q_{R}(\hat{\mathbf{x}}),\quad
Y_{\delta}(\hat{\mathbf{x}})Y_{\delta/2}(\hat{\mathbf{x}})=Y_{\delta}(\hat{\mathbf{x}})\quad\textup{and}\quad
K_{R}(x)K_{2R}(x)=K_{R}(x).$

Thus defined functions $a,b_{j,k}$ and $\beta_{j,k}$ satisfy conditions
(2.22). Therefore we can use Proposition 2.9. It is immediate to see that in
this case the function $h$ defined in (2.25), coincides with $H_{\delta,R}$,
so that (2.26) entails (3.11), as required. ∎

###### Proof of Theorems 2.2, 3.1 and 2.3.

By Lemma 3.2 each term in the relation (3.11) has a limit as $\delta\to
0,R\to\infty$. Therefore the integral on the right-hand side of (3.11) is
bounded uniformly in $\delta>0,R>0$. Assume for convenience that the function
$\theta$ defined in (1.14) is monotone decreasing for $t\geq 0$. Therefore the
poitwise convergencies

$\displaystyle Y_{\delta}(\tilde{\mathbf{x}}_{k,N},t)\to 1,\ \delta\to
0\quad\textup{and}\quad K_{R}(t)\to 1,Q_{R}(\tilde{\mathbf{x}}_{k,N},t)\to 1,\
R\to\infty,$

are monotone increasing. By the Monotone Convergence Theorem, the integrand
$K_{R}(t)H_{\delta,R}(t)$ on the right-hand side of (3.11) converges for a.e.
$t\in\mathbb{R}^{3}$ as $\delta\to 0,R\to\infty$ to an
$\textup{{{L}}}^{1}(\mathbb{R})$-function, which we denote by $\tilde{H}(t)$,
and the integral in (3.11) converges to

(3.13) $\displaystyle\frac{4}{3\pi}\int\tilde{H}(t)dt.$

If $N=2$, then this concludes the proof of Theorem 2.2, since in this case

$\displaystyle
H_{\delta,R}(t)\to\big{(}|\tilde{\eta}_{1,1}(t,t)|^{2}+\tilde{\eta}_{2,1}(t,t)|^{2}\big{)}^{1/2},$

a.e. $t\in\mathbb{R}^{3}$, and by virtue of (2.12) this limit coincides with
$H(t)$.

If $N\geq 3$, then the convergence to $\tilde{H}(t)$ implies that for a.e.
$t\in\mathbb{R}^{3}$ the function $K_{R}(t)H_{\delta,R}(t)$, and hence
$H_{\delta,R}(t)$, is bounded uniformly in $\delta$ and $R$. Applying the
Monotone Convergence Theorem to the integral (3.12), we conclude that the
a.e.-limit

$\displaystyle|\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},t,t)|=\lim_{\delta\to
0,R\to\infty}\big{|}Q_{R}(\tilde{\mathbf{x}}_{k,N},t)Y_{\delta}(\tilde{\mathbf{x}}_{k,N},t)\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},t,t)\big{|},\
$

belongs to $\textup{{{L}}}^{2}(\mathbb{R}^{3N-6})$, a.e. $t\in\mathbb{R}^{3}$,
and

$\displaystyle\lim_{\delta\to
0,R\to\infty}H_{\delta,R}(t)=H(t),\quad\textup{a.e.}\quad t\in\mathbb{R}^{3},$

where we have used the formula (2.12) for $H$. Thus
$H=\tilde{H}\in\textup{{{L}}}^{1}(\mathbb{R}^{3})$. As (2.12) is equivalent to
(2.6), this completes the proof of Theorem 2.2. Furthermore, the integral
(3.13) coincides with the coefficient $B$ in (2.7). Together with Lemma 3.2
this completes the proof of Theorem 3.1. As explained before, Theorem 3.1 is
equivalent to Theorem 2.3. ∎

The rest of the paper focuses on the proof of Lemma 3.2. The pivotal role in
the proof is played by the singular value estimates for the weighted operator
$\boldsymbol{{\sf{V}}}$, that are derived in the next section.

##  4\. Spectral bounds for the operator $\boldsymbol{{\sf{V}}}$

Recall that the vector-valued kernel ${\sf{V}}(\hat{\mathbf{x}},x)$ of the
operator $\boldsymbol{{\sf{V}}}$ is given by (2.9) with the kernels
${\sf{v}}_{j}(\hat{\mathbf{x}},x)$ defined in (2.2).

###  4.1. The weighted operator $\boldsymbol{{\sf{V}}}$

We assume that the weights $a=a(x),x\in\mathbb{R}^{3},$ and
$b=b(\hat{\mathbf{x}})$, $\hat{\mathbf{x}}\in\mathbb{R}^{3N-3},$ satisfy the
following properties. Denote $\mathcal{C}_{n}=(0,1)^{3}+n$,
$n\in\mathbb{Z}^{3}$. Let $\varkappa_{l}>0$ be the constants in the
exponential bounds (1) and (4.5). We assume that
$b\in\textup{{{L}}}^{\infty}(\mathbb{R}^{3N-3})$ and that the weight
$a\in\textup{{{L}}}^{3}_{\textup{\tiny loc}}(\mathbb{R}^{3})$ is such that

$\displaystyle\sup_{n\in\mathbb{Z}^{3}}\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}<\infty,$

so that both

(4.1) $\displaystyle
R_{\varkappa}(a)=\bigg{[}\sum_{n\in\mathbb{Z}^{3}}e^{-\frac{1}{2}\varkappa|n|_{\scaleto{1}{3pt}}}\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}^{\frac{1}{2}}\bigg{]}^{2},$

and

(4.2) $\displaystyle
M_{\varkappa}(b)=\biggl{[}\int_{\mathbb{R}^{3N-3}}|b(\hat{\mathbf{x}})|^{2}e^{-2\varkappa|\hat{\mathbf{x}}|_{\scaleto{1}{3pt}}}d\hat{\mathbf{x}}\biggr{]}^{\frac{1}{2}},$

are finite for any $\varkappa>0$. Note that in (4.1) and (4.2) we use the
$\ell^{1}$-norm $|\mathbf{x}|_{1}$ instead of the Eucledian norm
$|\mathbf{x}|$. We do this for computational convenience later in the proofs.

Recall that the functional ${\sf{G}}_{p}$ is defined in (2.16). Our objective
is to prove the following theorem.

######  Theorem 4.1.

Let the weights $a$ and $b$ be as described above. Then $b\,\operatorname{{\sf
Int}}({\sf{v}}_{j})a\in\mathbf{S}_{1,\infty}$, $j=1,2,\dots,N$, and there
exists a $\varkappa>0$ such that

(4.3) $\displaystyle\|b\,\operatorname{{\sf
Int}}({\sf{v}}_{j})a\|_{1,\infty}\lesssim$ $\displaystyle\
\|b\|_{\textup{{{L}}}^{\infty}}R_{\varkappa}(a),$ (4.4)
$\displaystyle{\sf{G}}_{1}(b\,\operatorname{{\sf
Int}}({\sf{v}}_{j})a)\lesssim$ $\displaystyle\
M_{\varkappa}(b)R_{\varkappa}(a),$

for each $j=1,2,\dots,N$.

By (2.21) the above bounds imply the same bounds for the operator
$b\,\boldsymbol{{\sf{V}}}a$. We prove Theorem 4.1 for the kernel
${\sf{v}}_{N}(\hat{\mathbf{x}},x)=:{\sf{v}}(\hat{\mathbf{x}},x)$. Bounds for
the remaining $j$’s follow by a simple permutation of variables.

###  4.2. Regularity of the eigenfunction

We need some of the efficient bounds for the derivatives of the eigenfunction
away from the coalescence points, obtained by S. Fournais and T.Ø. Sørensen in
[9]. Let

$\displaystyle\operatorname{d}(\hat{\mathbf{x}},x)=\min\\{|x|,|x-x_{j}|,\
j=1,2,\dots,N-1\\}.$

The following proposition is a consequence of [9, Corollary 1.3] Recall that
$|\ \cdot\ |_{{\scaleto{1}{3pt}}}$ denotes the $\ell^{1}$-norm.

######  Proposition 4.2.

Assume that $\psi$ satisfies (1). Then for all multi-indices
$m\in\mathbb{N}_{0}^{3}$, $|m|_{1}\geq 1$, we have

(4.5)
$\displaystyle|\partial_{x}^{m}\psi(\hat{\mathbf{x}},x)|\lesssim\operatorname{d}(\hat{\mathbf{x}},x)^{1-l}e^{-\varkappa_{{\scaleto{l}{3pt}}}{|\mathbf{x}|_{\scaleto{1}{3pt}}}},\
l=|m|_{1},$

with some $\varkappa_{l}>0$.

The precise values of the constants $\varkappa_{l}>0$ are insignificant for
us, and therefore we may assume that

(4.6) $\displaystyle\varkappa_{1}\geq\varkappa_{2}\geq\dots>0.$

Let us rewrite the bounds (4.5) using the notation $x_{0}=0$. With this
convention, we have

$\displaystyle\operatorname{d}(\hat{\mathbf{x}},x)=\min\\{|x-x_{j}|,\
j=0,1,2,\dots,N-1\\}$

and

$\displaystyle\operatorname{d}(\hat{\mathbf{x}},x)^{-1}\leq\sum_{0\leq j\leq
N-1}|x-x_{j}|^{-1}.$

Therefore (4.5) implies that

$\displaystyle|\partial_{x}^{m}{\sf{v}}(\hat{\mathbf{x}},x)|\lesssim$
$\displaystyle
e^{-\varkappa_{{\scaleto{l+1}{3pt}}}|\mathbf{x}|_{\scaleto{1}{3pt}}}\big{(}\min_{0\leq
j\leq N-1}|x-x_{j}|\big{)}^{-l}$ (4.7) $\displaystyle\lesssim$ $\displaystyle
e^{-\varkappa_{{\scaleto{l+1}{3pt}}}|\mathbf{x}|_{\scaleto{1}{3pt}}}\sum_{0\leq
j\leq N-1}|x-x_{j}|^{-l},\quad\ l=|m|_{1},$

for all $m\in\mathbb{N}_{0}^{3}$.

The plan of the proof of Theorem 4.1 follows the approach of [20], where a
similar theorem was proved for the operator $\boldsymbol{\Psi}$. First we
study the operators $\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}$, $n\in\mathbb{Z}^{3}$. With the
help of various cut-offs including $Y_{\nu}$, for each fixed $n$ this operator
is split in the sum of several operators depending on the parameter $\nu>0$,
whose singular values are estimated in different ways using Propositions 2.7
and 2.8. A key point is that we keep track of the explicit dependence on the
parameter $\nu$. Thus, although none of these bounds is sharp, in the end,
when collecting all of them together in Subsect. 4.4, we get the sharp bounds
(4.3), (4.4) by making an optimal choice of the parameter $\nu$.

Whenever we consider the operators $b\operatorname{{\sf Int}}(\ \cdot\
)\mathbbm{1}_{\mathcal{C}_{n}}a$ with weights $a,b$, the constants in all the
bounds are independent on the weights or on the parameter
$n\in\mathbb{Z}^{3}$.

Recall also that we use the notation $x_{0}=0$. The symbol $\sum_{j}$ (resp.
$\prod_{j}$) assumes summation (resp. product) over all $j=0,1,\dots,N-1$.

###  4.3. Partition of the kernel ${\sf{v}}(\hat{\mathbf{x}},x)$

Let the function $Y_{\nu}$, $\nu>0$, be as defined in (3.5). We use the new
parameter $\nu$ in order not to confuse it with the parameter $\delta$ in
Lemma 3.2. Represent ${\sf{v}}$ as follows:

(4.8) $\displaystyle{\sf{v}}=$ $\displaystyle\
{\sf{v}}_{1}^{(\nu)}+{\sf{v}}_{2}^{(\nu)},$
$\displaystyle{\sf{v}}_{1}^{(\nu)}(\hat{\mathbf{x}},x)=$ $\displaystyle\
Y_{\nu}(\hat{\mathbf{x}}){\sf{v}}(\hat{\mathbf{x}},x),$
$\displaystyle{\sf{v}}_{2}^{(\nu)}(\hat{\mathbf{x}},x)=$ $\displaystyle\
\big{(}1-Y_{\nu}(\hat{\mathbf{x}})\big{)}{\sf{v}}(\hat{\mathbf{x}},x).$

It follows from (3.6) and (4.7) that

(4.9)
$\displaystyle|\partial_{x}^{m}{\sf{v}}^{(\nu)}_{2}(\hat{\mathbf{x}},x)|\lesssim
e^{-\varkappa_{|m|_{\scaleto{1}{3pt}}+1}|\mathbf{x}|_{\scaleto{1}{3pt}}}\sum_{0\leq
j\leq N-1}|x-x_{j}|^{-|m|_{\scaleto{1}{3pt}}}\sum_{0\leq l<s\leq
N-1}\mathbbm{1}_{\\{|x_{l}-x_{s}|<4\nu\\}}(\hat{\mathbf{x}}),$

for all $m\in\mathbb{N}_{0}^{3}$, with implicit constants independent of
$\nu>0$. In the next lemma we consider the operator $\operatorname{{\sf
Int}}({\sf{v}}_{2}^{(\nu)})$ the weight $b=1$ and arbitrary
$a\in\textup{{{L}}}^{3}(\mathcal{C}_{n})$. Further on we use the
straightforward inequality

(4.10) $\displaystyle\max_{x:|x-n|\leq
6}e^{-\varkappa_{{\scaleto{l}{3pt}}}|\mathbf{x}|_{\scaleto{1}{3pt}}}\lesssim
e^{-\varkappa_{{\scaleto{l}{3pt}}}|\hat{\mathbf{x}}|_{\scaleto{1}{3pt}}}e^{-\varkappa_{{\scaleto{l}{3pt}}}|n|_{\scaleto{1}{3pt}}}.$

######  Lemma 4.3.

The operator $\operatorname{{\sf
Int}}({\sf{v}}_{2}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}$ belongs to
$\mathbf{S}_{6/5,\infty}$ and

(4.11) $\displaystyle\|\operatorname{{\sf
Int}}({\sf{v}}^{(\nu)}_{2})\,a\,\mathbbm{1}_{\mathcal{C}_{n}}\|_{6/5,\infty}\lesssim$
$\displaystyle\
e^{-\varkappa_{{\scaleto{2}{3pt}}}|n|_{\scaleto{1}{3pt}}}\nu^{\frac{3}{2}}\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})},$

for all $n\in\mathbb{Z}^{3}$ and all $\nu>0$.

###### Proof.

By (4.9), ${\sf{v}}_{2}^{(\nu)}\in\textup{{{H}}}^{1}(\mathbb{R}^{3})$ a.e.
$\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}$, so in order to estimate the singular
values, we consider the operator on the left-hand side of (4.11) as an
operator on $\textup{{{L}}}^{2}(\mathbb{R}^{3})$, and hence we use Proposition
2.8 with the weight $a(x)\mathbbm{1}_{\mathcal{C}_{n}}(x)$, and parameters
$l=1$, $d=3$ and $r=3$.

Let $\theta$ be the cut-off defined in (1.13). Since

${\sf{v}}_{2}^{(\nu)}(\hat{\mathbf{x}},x)\theta\big{(}|x-n|/6\big{)}\mathbbm{1}_{\mathcal{C}_{n}}(x)={\sf{v}}_{2}^{(\nu)}(\hat{\mathbf{x}},x)\mathbbm{1}_{\mathcal{C}_{n}}(x),$

instead of the function ${\sf{v}}_{2}^{(\nu)}$ we study the kernel
$T(\hat{\mathbf{x}},x)={\sf{v}}_{2}^{(\nu)}(\hat{\mathbf{x}},x)\theta\big{(}|x-n|/6\big{)}$.
According to (4.9) and (4.10), $T(\hat{\mathbf{x}},\ \cdot\
)\in\textup{{{H}}}^{1}(\mathbb{R}^{3})$, for a.e.
$\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}$ and

$\displaystyle
e^{2\varkappa_{{\scaleto{2}{3pt}}}|\hat{\mathbf{x}}|_{\scaleto{1}{3pt}}}$
$\displaystyle\ \|T(\hat{\mathbf{x}},\ \cdot\ )\|_{\textup{{{H}}}^{1}}^{2}$
$\displaystyle\lesssim$ $\displaystyle\
e^{-2\varkappa_{{\scaleto{2}{3pt}}}|n|_{\scaleto{1}{3pt}}}\bigg{(}\int\limits_{|x-n|<6}\sum_{0\leq
j\leq N-1}|x-x_{j}|^{-2}\,dx\bigg{)}\sum_{0\leq l<s\leq
N-1}\mathbbm{1}_{\\{|x_{l}-x_{s}|<4\nu\\}}(\hat{\mathbf{x}})$
$\displaystyle\lesssim$ $\displaystyle\
e^{-2\varkappa_{{\scaleto{2}{3pt}}}|n|_{\scaleto{1}{3pt}}}\sum_{0\leq l<s\leq
N-1}\mathbbm{1}_{\\{|x_{l}-x_{s}|<4\nu\\}}(\hat{\mathbf{x}}).$

Using Proposition 2.8 with $l=1,d=3,r=3$ we get that the operator on the left-
hand side of (4.11) belongs to $\mathbf{S}_{q,\infty}$ with $1/q=1/2+1/3=5/6$
and

$\displaystyle\|\operatorname{{\sf
Int}}({\sf{v}}^{(\nu)}_{2})\,a\,\mathbbm{1}_{\mathcal{C}_{n}}\|_{6/5,\infty}\lesssim$
$\displaystyle\ \bigg{[}\int_{\mathbb{R}^{3N-3}}\|T(\hat{\mathbf{x}},\ \cdot\
)\|_{\textup{{{H}}}^{1}}^{2}\,d\hat{\mathbf{x}}\biggl{]}^{\frac{1}{2}}\,\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}$
$\displaystyle\lesssim$ $\displaystyle\
e^{-\varkappa_{{\scaleto{2}{3pt}}}|n|_{\scaleto{1}{3pt}}}\bigg{[}\int_{\mathbb{R}^{3N-3}}e^{-2\varkappa_{{\scaleto{2}{3pt}}}|\hat{\mathbf{x}}|_{\scaleto{1}{3pt}}}\sum_{0\leq
l<s\leq
N-1}\mathbbm{1}_{\\{|x_{l}-x_{s}|<4\nu\\}}(\hat{\mathbf{x}})\,d\hat{\mathbf{x}}\bigg{]}^{\frac{1}{2}}\,\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}$
$\displaystyle\lesssim$ $\displaystyle\
e^{-\varkappa_{{\scaleto{2}{3pt}}}|n|_{\scaleto{1}{3pt}}}\nu^{\frac{3}{2}}\,\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})},$

which gives (4.11). ∎

In its turn, the kernel ${\sf{v}}_{1}^{(\nu)}$ splits into two components.
Consider separately the following components of ${\sf{v}}_{1}^{(\nu)}$:

(4.12) $\displaystyle{\sf{v}}_{1}^{(\nu)}=$ $\displaystyle\
{\sf{v}}_{11}^{(\nu)}+{\sf{v}}_{12}^{(\nu)},$
$\displaystyle{\sf{v}}_{11}^{(\nu)}(\hat{\mathbf{x}},x)=$ $\displaystyle\
Y_{\nu}(\hat{\mathbf{x}})\sum_{0\leq j\leq
N-1}\theta\big{(}|x-x_{j}|\nu^{-1}\big{)}{\sf{v}}(\hat{\mathbf{x}},x),$
$\displaystyle{\sf{v}}^{(\nu)}_{12}(\hat{\mathbf{x}},x)=$ $\displaystyle\
Y_{\nu}(\hat{\mathbf{x}})\big{[}1-\sum_{0\leq j\leq
N-1}\theta\big{(}|x-x_{j}|\nu^{-1}\big{)}\big{]}{\sf{v}}(\hat{\mathbf{x}},x).$

By (3.7) we have

$\displaystyle{\sf{v}}_{12}^{(\nu)}(\hat{\mathbf{x}},x)=$ $\displaystyle\
Y_{\nu}(\hat{\mathbf{x}}){\sf{v}}(\hat{\mathbf{x}},x)\prod_{j=0}^{N-1}\zeta\big{(}|x-x_{j}|\nu^{-1}\big{)}.$

To estimate the derivatives of the above functions first observe that

$\displaystyle\big{|}\partial_{x}^{m}\theta\big{(}|x|\nu^{-1}\big{)}\big{|}\lesssim|x|^{-|m|_{\scaleto{1}{3pt}}}\mathbbm{1}_{\\{|x|<\nu\\}}(x),\quad\big{|}\partial_{x}^{m}\zeta\big{(}|x|\nu^{-1}\big{)}\big{|}\lesssim|x|^{-|m|_{\scaleto{1}{3pt}}}\mathbbm{1}_{\\{|x|>\nu/2\\}}(x),\quad
m\in\mathbb{N}_{0}^{3},$

uniformly in $\nu>0$, and hence

$\displaystyle\big{|}\partial_{x}^{m}\sum_{j=0}^{N-1}\theta\big{(}|x-x_{j}|\nu^{-1}\big{)}\big{|}\lesssim$
$\displaystyle\
\sum_{j=0}^{N-1}|x-x_{j}|^{-|m|_{\scaleto{1}{3pt}}}\mathbbm{1}_{\\{|x-x_{j}|<\nu\\}}(\hat{\mathbf{x}},x),$
$\displaystyle\big{|}\partial_{x}^{m}\prod_{j=0}^{N-1}\zeta\big{(}|x-x_{j}|\nu^{-1}\big{)}\big{|}\lesssim$
$\displaystyle\
\bigg{(}\sum_{j=0}^{N-1}|x-x_{j}|^{-|m|_{\scaleto{1}{3pt}}}\bigg{)}\prod_{j=0}^{N-1}\mathbbm{1}_{\\{|x-x_{j}|>\nu/2\\}}(\hat{\mathbf{x}},x),\quad
m\in\mathbb{N}_{0}^{3}.$

Together with (4.7) these give

(4.13)
$\displaystyle\begin{cases}\big{|}\partial_{x}^{m}{\sf{v}}_{11}^{(\nu)}(\hat{\mathbf{x}},x)\big{|}\lesssim
e^{-\varkappa_{|m|_{\scaleto{1}{3pt}}+1}|\mathbf{x}|_{\scaleto{1}{3pt}}}\sum_{j=0}^{N-1}|x-x_{j}|^{-|m|_{\scaleto{1}{3pt}}}\mathbbm{1}_{\\{|x-x_{j}|<\nu\\}}(\hat{\mathbf{x}},x),\\\\[11.38092pt]
\big{|}\partial_{x}^{m}{\sf{v}}_{12}^{(\nu)}(\hat{\mathbf{x}},x)\big{|}\lesssim
e^{-\varkappa_{|m|_{\scaleto{1}{3pt}}+1}|\mathbf{x}|_{\scaleto{1}{3pt}}}\sum_{j=0}^{N-1}|x-x_{j}|^{-|m|_{\scaleto{1}{3pt}}}\mathbbm{1}_{\\{|x-x_{j}|>\nu/2\\}}(\hat{\mathbf{x}},x),\end{cases}$

for all $m\in\mathbb{N}_{0}^{3}$, uniformly in $\nu>0$. For the first (resp.
second) bound in (4.13) we have used the first (resp. second) bound in (4.7).

Recall that the quantity $M_{\varkappa}(b)$ is defined in (4.2).

######  Lemma 4.4.

For any $l\geq 2$ the operator $b\operatorname{{\sf
Int}}({\sf{v}}_{12}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}$ belongs to
$\mathbf{S}_{q,\infty}$ with

(4.14) $\displaystyle\frac{1}{q}=\frac{1}{2}+\frac{l}{3},$

and

(4.15) $\displaystyle\|b\operatorname{{\sf
Int}}({\sf{v}}_{12}^{(\nu)})\,a\,\mathbbm{1}_{\mathcal{C}_{n}}\|_{q,\infty}\lesssim
e^{-\varkappa|n|_{\scaleto{1}{3pt}}}M_{\varkappa}(b)\nu^{-l+\frac{3}{2}}\|a\|_{\textup{{{L}}}^{2}(\mathcal{C}_{n})},\quad\varkappa=\varkappa_{l+1},$

for all $\nu>0$, with an implicit constant depending on $l$ only.

###### Proof.

According to (4.13), ${\sf{v}}_{12}^{(\nu)}(\hat{\mathbf{x}},\ \cdot\
)\in\textup{{{H}}}^{l}(\mathcal{C}_{n})$, for a.e.
$\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}$ with an arbitrary $l\geq 2$ and we have

$\displaystyle\|{\sf{v}}^{(\nu)}_{12}(\hat{\mathbf{x}},\cdot\
)\|_{\textup{{{H}}}^{l}}^{2}\lesssim$ $\displaystyle\
e^{-2\varkappa_{{\scaleto{l+1}{3pt}}}|n|_{\scaleto{1}{3pt}}}e^{-2\varkappa_{{\scaleto{l+1}{3pt}}}|\hat{\mathbf{x}}|_{\scaleto{1}{3pt}}}\int_{\mathcal{C}_{n}}\sum_{j=0}^{N-1}|x-x_{j}|^{-2l}\mathbbm{1}_{\\{|x-x_{j}|>\nu/2\\}}(\hat{\mathbf{x}},x)dx$
$\displaystyle\lesssim$ $\displaystyle\
e^{-2\varkappa_{{\scaleto{l+1}{3pt}}}|n|_{\scaleto{1}{3pt}}}e^{-2\varkappa_{{\scaleto{l+1}{3pt}}}|\hat{\mathbf{x}}|_{\scaleto{1}{3pt}}}\nu^{-2l+3}.$

Thus by Proposition 2.7 with $l\geq 2$, $d=3$ (so that $2l>d$) and $r=2$, for
$q$ defined in (4.14) we have

$\displaystyle\|b\operatorname{{\sf
Int}}({\sf{v}}_{12}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}\|_{q,\infty}\lesssim
e^{-\varkappa_{{\scaleto{l+1}{3pt}}}|n|_{{\scaleto{1}{3pt}}}}\nu^{-l+\frac{3}{2}}\biggl{[}\int_{\mathbb{R}^{3N-3}}e^{-2\varkappa_{{\scaleto{l+1}{3pt}}}|\hat{\mathbf{x}}|_{{\scaleto{1}{3pt}}}}|b(\hat{\mathbf{x}})|^{2}d\hat{\mathbf{x}}\biggr{]}^{\frac{1}{2}}\|a\|_{\textup{{{L}}}^{2}(\mathcal{C}_{n})}.$

By definition (4.2) the bound (4.15) follows. ∎

######  Lemma 4.5.

The operator $b\operatorname{{\sf
Int}}({\sf{v}}_{11}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}$ belongs to
$\mathbf{S}_{6/5,\infty}$ and

(4.16) $\displaystyle\|b\operatorname{{\sf
Int}}({\sf{v}}^{(\nu)}_{11})\,a\,\mathbbm{1}_{\mathcal{C}_{n}}\|_{6/5,\infty}\lesssim$
$\displaystyle\
e^{-\varkappa|n|_{\scaleto{1}{3pt}}}M_{\varkappa}(b)\nu^{\frac{1}{2}}\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})},\quad\varkappa=\varkappa_{2},$

for all $n\in\mathbb{Z}^{3}$ and all $\nu>0$.

###### Proof.

Similarly to the proof of Lemma 4.3, we use Proposition 2.8 with the kernel
$T(\hat{\mathbf{x}},x)={\sf{v}}_{11}^{(\nu)}(\hat{\mathbf{x}},x)\theta\big{(}|x-n|/6\big{)}$,
the weight $a(x)\mathbbm{1}_{\mathcal{C}_{n}}(x)$ and parameters $l=1,d=3$ and
$r=3$.

According to (4.13), $T(\hat{\mathbf{x}},\ \cdot\
)\in\textup{{{H}}}^{1}(\mathbb{R}^{3})$, for a.e.
$\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}$, and we have

$\displaystyle
e^{2\varkappa_{{\scaleto{2}{3pt}}}|\hat{\mathbf{x}}|_{\scaleto{1}{3pt}}}\|T(\hat{\mathbf{x}},\cdot\
)\|_{\textup{{{H}}}^{1}}^{2}\lesssim$ $\displaystyle\
e^{-2\varkappa_{{\scaleto{2}{3pt}}}|n|_{\scaleto{1}{3pt}}}\int\limits_{|x-n|<6}\sum_{j=0}^{N-1}|x-x_{j}|^{-2}\mathbbm{1}_{\\{|x-x_{j}|<\nu\\}}(\hat{\mathbf{x}},x)dx$
$\displaystyle\lesssim$ $\displaystyle\
e^{-2\varkappa_{{\scaleto{2}{3pt}}}|n|_{\scaleto{1}{3pt}}}\nu.$

Use Proposition 2.7 with $d=3,l=1$, so that $q^{-1}=5/6$ and $r=3$:

$\displaystyle\|b\operatorname{{\sf
Int}}({\sf{v}}_{11}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}\|_{6/5,\infty}\lesssim$
$\displaystyle\ \bigg{[}\int_{\mathbb{R}^{3N-3}}\|T(\hat{\mathbf{x}},\ \cdot\
)\|_{\textup{{{H}}}^{1}}^{2}|b(\hat{\mathbf{x}})|^{2}\,d\hat{\mathbf{x}}\biggl{]}^{\frac{1}{2}}\,\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}$
$\displaystyle\lesssim$ $\displaystyle\
e^{-\varkappa_{{\scaleto{2}{3pt}}}|n|_{{\scaleto{1}{3pt}}}}\nu^{\frac{1}{2}}\biggl{[}\int_{\mathbb{R}^{3N-3}}e^{-2\varkappa_{{\scaleto{2}{3pt}}}|\hat{\mathbf{x}}|_{{\scaleto{1}{3pt}}}}|b(\hat{\mathbf{x}})|^{2}d\hat{\mathbf{x}}\biggr{]}^{\frac{1}{2}}\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}.$

By (4.2) this leads to the bound (4.16). ∎

###  4.4. Estimates for $\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}$

Now we can put together the bounds obtained above for the parts of the
operator $\operatorname{{\sf Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}$.

######  Lemma 4.6.

Suppose that $b\in\textup{{{L}}}^{\infty}(\mathbb{R}^{3N-3})$ and
$a\in\textup{{{L}}}^{3}(\mathcal{C}_{n})$. Then $b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\in\mathbf{S}_{1,\infty}$ and

(4.17) $\displaystyle\|b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\|_{1,\infty}\lesssim$
$\displaystyle\
e^{-\varkappa|n|_{{\scaleto{1}{3pt}}}}\|b\|_{\textup{{{L}}}^{\infty}}\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})},$
(4.18) $\displaystyle{\sf{G}}_{1}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\big{)}\lesssim$ $\displaystyle\
e^{-\varkappa|n|_{\scaleto{1}{3pt}}}M_{\varkappa}(b)\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})},\quad\varkappa=\varkappa_{3}.$

###### Proof.

Without loss of generality assume that $\|b\|_{\textup{{{L}}}^{\infty}}\leq 1$
and $\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}\leq 1$.

By (4.8) and (4.12) for each $\nu>0$ we have

$\displaystyle{\sf{v}}={\sf{v}}_{11}^{(\nu)}+{\sf{v}}_{12}^{(\nu)}+{\sf{v}}_{2}^{(\nu)}.$

According to (4.11),(4.16) and the inequality (2.15),

$\displaystyle\|b\,\operatorname{{\sf
Int}}({\sf{v}}_{2}^{(\nu)}+{\sf{v}}_{11}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}\|_{{\scaleto{6}{4pt}}/{\scaleto{5}{4pt}},\infty}^{{\scaleto{6}{4pt}}/{\scaleto{11}{4pt}}}\leq$
$\displaystyle\ \|b\,\operatorname{{\sf
Int}}({\sf{v}}_{2}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}\|_{{\scaleto{6}{4pt}}/{\scaleto{5}{4pt}},\infty}^{{\scaleto{6}{4pt}}/{\scaleto{11}{4pt}}}+\|\operatorname{{\sf
Int}}({\sf{v}}_{11}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}\|_{{\scaleto{6}{4pt}}/{\scaleto{5}{4pt}},\infty}^{\frac{{\scaleto{6}{4pt}}}{{\scaleto{11}{4pt}}}}$
$\displaystyle\lesssim$ $\displaystyle\
\big{[}e^{-\varkappa_{{\scaleto{2}{3pt}}}|n|_{\scaleto{1}{3pt}}}\big{(}\nu^{\frac{1}{2}}M_{\varkappa_{{\scaleto{2}{3pt}}}}(b)+\nu^{\frac{3}{2}}\big{)}\big{]}^{{\scaleto{6}{4pt}}/{\scaleto{11}{4pt}}},$

so that, by definition (2.14),

$\displaystyle s_{k}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}}_{2}^{(\nu)}+{\sf{v}}_{11}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}\big{)}$
(4.19) $\displaystyle\lesssim$ $\displaystyle\
e^{-\varkappa_{{\scaleto{2}{3pt}}}|n|_{\scaleto{1}{3pt}}}\big{(}\nu^{\frac{1}{2}}M_{\varkappa_{{\scaleto{2}{3pt}}}}(b)+\nu^{\frac{3}{2}}\big{)}k^{-\frac{5}{6}},\quad
k=1,2,\dots.$

Since
$\|a\|_{\textup{{{L}}}^{2}(\mathcal{C}_{n})}\leq\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}\leq
1$, it follows from (4.15) that

(4.20) $\displaystyle s_{k}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}}_{12}^{(\nu)})a\mathbbm{1}_{\mathcal{C}_{n}}\big{)}\lesssim
e^{-\varkappa_{{\scaleto{l+1}{3pt}}}|n|_{\scaleto{1}{3pt}}}M_{\varkappa_{{\scaleto{l+1}{3pt}}}}(b)\delta^{-l+\frac{3}{2}}k^{-\frac{1}{2}-\frac{l}{3}},$

for all $l\geq 2$. Due to (2.13) and (4.6), combining (4.4) and (4.20), we get
the estimate

$\displaystyle s_{2k}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\big{)}\leq$ $\displaystyle\
s_{2k-1}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\big{)}$ (4.21)
$\displaystyle\lesssim$ $\displaystyle\
e^{-\varkappa|n|_{\scaleto{1}{3pt}}}\bigg{[}\big{(}\nu^{\frac{1}{2}}M_{\varkappa}(b)+\nu^{\frac{3}{2}}\big{)}k^{-\frac{5}{6}}+\nu^{-l+\frac{3}{2}}M_{\varkappa}(b)k^{-\frac{1}{2}-\frac{l}{3}}\bigg{]},\quad\varkappa=\varkappa_{l+1},$

where we have used that $M_{\varkappa_{{\scaleto{2}{3pt}}}}(b)\leq
M_{\varkappa_{{\scaleto{l+1}{3pt}}}}(b)$, $l\geq 2$. Rewrite the expression in
the square brackets, gathering the terms containing $M_{\varkappa}(b)$:

$\displaystyle M_{\varkappa}(b)\big{(}\nu^{\frac{1}{2}}k^{-\frac{5}{6}}+$
$\displaystyle\
\nu^{-l+\frac{3}{2}}k^{-\frac{1}{2}-\frac{l}{3}}\big{)}+\nu^{\frac{3}{2}}k^{-\frac{5}{6}}$
$\displaystyle=$ $\displaystyle\
M_{\varkappa}(b)\nu^{\frac{1}{2}}k^{-\frac{5}{6}}\big{(}1+\nu^{-l+1}k^{\frac{1-l}{3}}\big{)}+\nu^{\frac{3}{2}}k^{-\frac{5}{6}}.$

Since $\nu>0$ is arbitrary, we can pick $\nu=\nu_{k}=k^{-1/3}$ so that

$\displaystyle\nu^{-l+1}k^{\frac{1-l}{3}}=1,\quad\nu^{\frac{1}{2}}k^{-\frac{5}{6}}=k^{-1},\quad\nu^{\frac{3}{2}}k^{-\frac{5}{6}}=k^{-\frac{4}{3}}.$

Thus the bound (4.4) rewrites as

(4.22) $\displaystyle s_{2k}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\big{)}\leq$ $\displaystyle\
s_{2k-1}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\big{)}\lesssim
e^{-\varkappa|n|_{{\scaleto{1}{3pt}}}}\big{(}M_{\varkappa}(b)k^{-1}+k^{-\frac{4}{3}}\big{)}.$

Using the bound $M_{\varkappa}(b)\lesssim\|b\|_{\textup{{{L}}}^{\infty}}\leq
1$ we conclude that

$\displaystyle s_{k}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\big{)}\lesssim
e^{-\varkappa|n|_{{\scaleto{1}{3pt}}}}k^{-1}.$

This leads to (4.17) if one takes $l=2$, so that $\varkappa=\varkappa_{3}$.

In order to obtain (4.18), we use (4.22) to write

$\displaystyle\limsup_{k\to\infty}ks_{k}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\big{)}\lesssim
e^{-\varkappa|n|_{\scaleto{1}{3pt}}}\limsup_{k\to\infty}\big{(}M_{\varkappa}(b)+k^{-\frac{1}{3}}\big{)}=e^{-\varkappa|n|_{\scaleto{1}{3pt}}}M_{\varkappa}(b).$

Taking again $l=2$, we obtain (4.18). ∎

###  4.5. Proof of Theorem 4.1

Here we sum up over $n\in\mathbb{Z}^{3}$ the estimates in Lemma 4.6 to
complete the proof of Theorem 4.1. Recall again that the quantities
$R_{\varkappa}(a)$ and $M_{\varkappa}(b)$ are defined in (4.1) and (4.2)
respectively.

As noted earlier, it suffices to prove the bounds (4.3) and (4.4) for the
operator $b\,\operatorname{{\sf Int}}({\sf{v}})a$, where
${\sf{v}}:={\sf{v}}_{N}$. Since
${\sf{v}}=\sum_{n\in\mathbb{Z}^{3}}{\sf{v}}\mathbbm{1}_{\mathcal{C}_{n}}$, we
have, by (2.15) and (4.17),

$\displaystyle\|b\,\operatorname{{\sf
Int}}({\sf{v}})a\|_{1,\infty}^{\frac{{\scaleto{1}{4pt}}}{{\scaleto{2}{4pt}}}}\leq$
$\displaystyle\ \sum_{n\in\mathbb{Z}^{3}}\|b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\|_{1,\infty}^{\frac{{\scaleto{1}{4pt}}}{{\scaleto{2}{4pt}}}}$
$\displaystyle\lesssim$ $\displaystyle\
\|b\|_{\textup{{{L}}}^{\infty}}^{\frac{1}{2}}\sum_{n\in\mathbb{Z}^{3}}e^{-\frac{1}{2}\varkappa|n|_{\scaleto{1}{3pt}}}\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}^{\frac{1}{2}}=\|b\|_{\textup{{{L}}}^{\infty}}^{\frac{1}{2}}\big{(}R_{\varkappa}(a)\big{)}^{\frac{1}{2}},\quad\varkappa=\varkappa_{3}.$

This proves (4.3).

To prove (4.4) observe that the previous calculation shows that condition
(2.19) holds for the operators $T_{n}={\sf{v}}\mathbbm{1}_{\mathcal{C}_{n}}$,
and hence we can use Lemma 2.4. According to (2.20) and (4.18),

$\displaystyle{\sf{G}}_{1}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}})a\big{)}^{\frac{1}{2}}\leq$ $\displaystyle\
\sum_{n\in\mathbb{Z}^{3}}{\sf{G}}_{1}\big{(}b\,\operatorname{{\sf
Int}}({\sf{v}})\mathbbm{1}_{\mathcal{C}_{n}}a\big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim$ $\displaystyle\
\big{(}M_{\varkappa}(b)\big{)}^{\frac{1}{2}}\sum_{n\in\mathbb{Z}^{3}}e^{-\frac{1}{2}\varkappa|n|_{\scaleto{1}{3pt}}}\|a\|_{\textup{{{L}}}^{3}(\mathcal{C}_{n})}^{\frac{1}{2}}=\big{(}M_{\varkappa}(b)\big{)}^{\frac{1}{2}}\big{(}R_{\varkappa}(a)\big{)}^{\frac{1}{2}},\quad\varkappa=\varkappa_{3}.$

This completes the proof of Theorem 4.1. ∎

##  5\. Proof of Lemma 3.2

The proof of Lemma 3.2 is divided in several steps each of which is justified
using either Corollary 2.5 or Corollary 2.6.

The first stage is described in the next lemma. Recall that the cut-offs
$Y_{\delta}$ and $Q_{R}$ are defined in (3.5) and (3.8).

######  Lemma 5.1.

The following relations hold:

(5.1)
$\displaystyle{\sf{G}}_{1}(\boldsymbol{{\sf{V}}})=\lim\limits_{\begin{subarray}{c}\delta\to
0\\\
R\to\infty\end{subarray}}{\sf{G}}_{1}(Q_{R}Y_{\delta}\boldsymbol{{\sf{V}}}K_{R}),\quad{\sf{g}}_{1}(\boldsymbol{{\sf{V}}})=\lim\limits_{\begin{subarray}{c}\delta\to
0\\\
R\to\infty\end{subarray}}{\sf{g}}_{1}(Q_{R}Y_{\delta}\boldsymbol{{\sf{V}}}K_{R}),$

where the limits on the right-hand side exist.

###### Proof.

First we check that

(5.2) $\displaystyle\begin{cases}\lim\limits_{\delta\to
0}{\sf{G}}_{1}\big{(}(I-Y_{\delta})\boldsymbol{{\sf{V}}}\big{)}=0,\\\\[8.5359pt]
\lim\limits_{R\to\infty}{\sf{G}}_{1}\big{(}(I-Q_{R})\boldsymbol{{\sf{V}}}\big{)}=0,\
\lim\limits_{R\to\infty}{\sf{G}}_{1}\big{(}\boldsymbol{{\sf{V}}}(I-K_{R})\big{)}=0.\end{cases}$

Consider first $(I-Y_{\delta})\boldsymbol{{\sf{V}}}$. By virtue of (3.6), it
follows from (4.4) that for some $\varkappa>0$, we have

$\displaystyle{\sf{G}}_{1}\big{(}(1-Y_{\delta})\boldsymbol{{\sf{V}}}\big{)}\lesssim$
$\displaystyle\ M_{\varkappa}(1-Y_{\delta})$ $\displaystyle\lesssim$
$\displaystyle\ \sum_{0\leq l<s\leq
N-1}\bigg{[}\int\theta\big{(}|x_{l}-x_{s}|(4\delta)^{-1}\big{)}^{2}e^{-2\varkappa|\hat{\mathbf{x}}|}d\hat{\mathbf{x}}\bigg{]}^{1/2}\lesssim\delta^{3/2}\to
0,\ \delta\to 0,$

and hence the first relation in (5.2) holds.

In a similar way one estimates $(I-Q_{R})\boldsymbol{{\sf{V}}}$ and
$\boldsymbol{{\sf{V}}}(I-K_{R})$. Estimate, for example, the first of these
operators. Since

$\displaystyle 1-Q_{R}(\hat{\mathbf{x}})\leq\sum_{1\leq l\leq
N-1}\zeta\big{(}|x_{l}|R^{-1}\big{)},$

it follows from (4.4) again that

$\displaystyle{\sf{G}}_{1}\big{(}(I-Q_{R})\boldsymbol{{\sf{V}}}\big{)}\lesssim$
$\displaystyle\ M_{\varkappa}(1-Q_{R})$ $\displaystyle\lesssim$
$\displaystyle\ \sum_{0\leq l\leq
N-1}\bigg{[}\int_{\mathbb{R}^{3}}\zeta(|x_{l}|R^{-1})^{2}e^{-2\varkappa|x_{l}|}\,dx_{l}\bigg{]}^{1/2}\lesssim
e^{-\varkappa R/2}\to 0,\ R\to\infty,$

whence the second equality in (5.2). The third one is derived similarly.

Represent $\Psi$ in the form

$\displaystyle\boldsymbol{{\sf{V}}}=Q_{R}Y_{d}\boldsymbol{{\sf{V}}}K_{R}+(I-Q_{R})\boldsymbol{{\sf{V}}}+Q_{R}(1-Y_{\delta})\boldsymbol{{\sf{V}}}+Q_{R}Y_{\delta}\boldsymbol{{\sf{V}}}(I-K_{R}),$

According to (2.18),

$\displaystyle{\sf{G}}_{1}\big{(}\boldsymbol{{\sf{V}}}-Q_{R}Y_{d}\boldsymbol{{\sf{V}}}K_{R}\big{)}^{\frac{{\scaleto{1}{4pt}}}{{\scaleto{2}{4pt}}}}\leq$
$\displaystyle\
{\sf{G}}_{1}\big{(}(I-Q_{R})\boldsymbol{{\sf{V}}}\big{)}^{\frac{{\scaleto{1}{4pt}}}{{\scaleto{2}{4pt}}}}$
$\displaystyle\
+{\sf{G}}_{1}\big{(}Q_{R}(1-Y_{\delta})\boldsymbol{{\sf{V}}}\big{)}^{\frac{{\scaleto{1}{4pt}}}{{\scaleto{2}{4pt}}}}+{\sf{G}}_{1}\big{(}Q_{R}Y_{\delta}\boldsymbol{{\sf{V}}}(I-K_{R})\big{)}^{\frac{{\scaleto{1}{4pt}}}{{\scaleto{2}{4pt}}}}$
$\displaystyle\leq$ $\displaystyle\
{\sf{G}}_{1}\big{(}(I-Q_{R})\boldsymbol{{\sf{V}}}\big{)}^{\frac{{\scaleto{1}{4pt}}}{{\scaleto{2}{4pt}}}}$
$\displaystyle\
+{\sf{G}}_{1}\big{(}(1-Y_{\delta})\boldsymbol{{\sf{V}}}\big{)}^{\frac{{\scaleto{1}{4pt}}}{{\scaleto{2}{4pt}}}}+{\sf{G}}_{1}\big{(}\Psi(I-K_{R})\boldsymbol{{\sf{V}}}\big{)}^{\frac{{\scaleto{1}{4pt}}}{{\scaleto{2}{4pt}}}}.$

By virtue of (5.2) the right-hand side tends to zero as $\delta\to
0,R\to\infty$. By Corollary 2.6 this implies (5.1). ∎

In the next stage we partition the kernel

(5.3) $\displaystyle
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}}){\sf{V}}(\hat{\mathbf{x}},x)K_{R}(x)$

of the operator $Q_{R}Y_{\delta}\boldsymbol{{\sf{V}}}K_{R}$ on the right-hand
side of the formulas in (5.1). We do this by introducing the cut-offs
$\theta\big{(}|x-~{}x_{j}|\varepsilon^{-1}\big{)}$, $j=0,1,\dots,N-1$,
assuming that $\varepsilon<\delta$. In view of (3.7) the $j$’th component of
the kernel (5.3) can be represented as follows:

(5.4) $\displaystyle
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}}){\sf{v}}_{j}(\hat{\mathbf{x}},x)K_{R}(x)=\sum_{k=0}^{N-1}\phi_{j,k}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)+\rho_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)$

with

$\displaystyle\phi_{j,k}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=$
$\displaystyle\
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\theta\big{(}|x-~{}x_{k}|\varepsilon^{-1}\big{)}{\sf{v}}_{j}(\hat{\mathbf{x}},x)K_{R}(x),\quad
k=0,1,\dots,N-1,$
$\displaystyle\rho_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=$
$\displaystyle\
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\prod_{k=0}^{N-1}\zeta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}{\sf{v}}_{j}(\hat{\mathbf{x}},x)K_{R}(x).$

First we show that the kernels $\rho_{j}[\delta,R,\varepsilon]$ and
$\phi_{j,0}[\delta,R,\varepsilon]$ give negligible contributions to the
asymptotics.

######  Lemma 5.2.

For each $\delta>0,R>0$ and $\varepsilon<\delta$ one has

(5.5) $\displaystyle{\sf{G}}_{1}\big{(}\operatorname{{\sf
Int}}\big{(}\rho_{j}[\delta,R,\varepsilon]\big{)}\big{)}=0,\ j=1,2,\dots,N.$

###### Proof.

By the definitions (3.5) and (1.14), the support of the kernel
$\rho_{j}[\delta,R,\varepsilon]$ belongs to the bounded domain

$\displaystyle{\bigcap_{0\leq l<s\leq N}{\sf
S}_{l,s}(\varepsilon/2)\cap(B_{R})^{N}}.$

The function ${\sf{v}}_{j}$ is real-analytic on this domain and it is
uniformly bounded together with all its derivatives, so that
$\rho_{j}[\delta,R,\varepsilon]\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N})$.
By Proposition 2.7, ${\sf{G}}_{p}(\operatorname{{\sf
Int}}(\rho_{j}[\delta,\mathbb{R},\varepsilon]))=0$ for all $p>0$, and in
particular, for $p=1$, as claimed. ∎

######  Lemma 5.3.

For each $\delta>0,R>0$ one has

(5.6) $\displaystyle\lim_{\varepsilon\to
0}{\sf{G}}_{1}\big{(}\operatorname{{\sf
Int}}\big{(}\phi_{j,0}[\delta,R,\varepsilon]\big{)}\big{)}=0,\ j=1,2,\dots,N.$

###### Proof.

As $x_{0}=0$ by definition, the kernel $\phi_{j,0}[\delta,R,\varepsilon]$ has
the form

$\displaystyle\phi_{j,0}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}}){\sf{v}}_{j}(\hat{\mathbf{x}},x)\theta\big{(}|x|\varepsilon^{-1}\big{)}K_{R}(x).$

Estimating $Q_{R}Y_{\delta}\leq 1$, $K_{R}\leq 1$, one sees that the singular
values of $\operatorname{{\sf Int}}(\phi_{j,0}[\delta,R,\varepsilon])$ do not
exceed those of the operator $\operatorname{{\sf Int}}({\sf{v}}_{j})a$ with
the weight $a(x)=\theta(|x|\varepsilon^{-1})$. By (4.4),

$\displaystyle{\sf{G}}_{1}(\operatorname{{\sf Int}}({\sf{v}}_{j})a)\lesssim
R_{\varkappa}(a)\lesssim\bigg{(}\int_{\mathbb{R}^{3}}\theta\big{(}|x|\varepsilon^{-1}\big{)}^{3}dx\bigg{)}^{1/3}\lesssim\varepsilon\to
0,\ \varepsilon\to 0.$

This implies (5.6). ∎

######  Corollary 5.4.

Denote by
$\boldsymbol{\alpha}[\delta,R,\varepsilon]=\\{\alpha_{j}[\delta,R,\varepsilon]\\}_{j=1}^{N}$
the vector-valued kernel with the components

$\displaystyle\alpha_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=\sum_{j=1}^{N-1}\phi_{j,k}[\delta,R,\varepsilon](\hat{\mathbf{x}},x).$

Then for all $\delta>0$ and $R>0$, we have

(5.7)
$\displaystyle\begin{cases}{\sf{G}}_{1}(Q_{R}Y_{\delta}\boldsymbol{{\sf{V}}}K_{R})=&\
\lim\limits_{\varepsilon\to 0}{\sf{G}}_{1}(\operatorname{{\sf
Int}}(\boldsymbol{\alpha}[\delta,R,\varepsilon])),\\\\[5.69046pt]
{\sf{g}}_{1}(Q_{R}Y_{\delta}\boldsymbol{{\sf{V}}}K_{R})=&\
\lim\limits_{\varepsilon\to 0}{\sf{g}}_{1}(\operatorname{{\sf
Int}}(\boldsymbol{\alpha}[\delta,R,\varepsilon])),\end{cases}$

where the limits on the right-hand side exist.

###### Proof.

By (5.4), the kernel $Q_{R}Y_{\delta}{\sf{v}}_{j}K_{R}$ has the form

$\displaystyle\alpha_{j}[\delta,R,\varepsilon]+\phi_{j,0}[\delta,R,\varepsilon]+\rho_{j}[\delta,R,\varepsilon].$

By virtue of (2.20) and (5.5), (5.6), we have

$\displaystyle\lim_{\varepsilon\to 0}{\sf{G}}_{1}\big{(}\operatorname{{\sf
Int}}\big{(}\phi_{j,0}[\delta,R,\varepsilon]+\rho_{j}[\delta,R,\varepsilon]\big{)}\big{)}=0.$

Now (5.7) follows from Corollary 2.6. ∎

###### Completion of the proof of Lemma 3.2.

According to (3.2), under the condition
$\varepsilon<\varepsilon_{0}(\delta,R)$, the support of each kernel

$\displaystyle\phi_{j,k}[\delta,R,\varepsilon],\quad j=1,2,\dots,N,\quad
k=1,2,\dots,N-1,$

belongs to $\tilde{\Omega}_{k}(\delta,R,\varepsilon)$, see (3.3) for the
definition. Therefore one can use the representation (2.2) for the function
${\sf{v}}_{j}$:

$\displaystyle\alpha_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=\sum_{k=1}^{N-1}$
$\displaystyle\
\phi_{j,k}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=\sum_{k=1}^{N-1}Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\nabla_{x}\tilde{\xi}_{j,k}(\hat{\mathbf{x}},x)K_{R}(x)$
$\displaystyle\
\quad+\sum_{k=1}^{N-1}Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}|x_{k}-x|\nabla_{x}\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x)K_{R}(x)$
$\displaystyle\
\quad+\sum_{k=1}^{N-1}Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\frac{x_{k}-x}{|x_{k}-x|}\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x)K_{R}(x).$

Each term in the first sum on the right-hand side is
$\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N})$. Thus, by Proposition 2.7, the
functional ${\sf{G}}_{p}$ for the associated operator equals zero for all
$p>0$, and in particular, for $p=1$. Each term in the second sum contains the
function $\Phi(x)=|x|$, which is homogeneous of order $1$. Thus, by
Proposition 2.9 the functional ${\sf{G}}_{1}$ for the associated operator
equals zero. The third sum coincides with the kernel
$\Upsilon_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)$, defined in (3.10).
Therefore, by Corollary 2.5,

(5.8) $\displaystyle\begin{cases}{\sf{G}}_{1}(\operatorname{{\sf
Int}}(\boldsymbol{\alpha}[\delta,R,\varepsilon]))={\sf{G}}_{1}(\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])),\\\\[5.69046pt]
{\sf{g}}_{1}(\operatorname{{\sf
Int}}(\boldsymbol{\alpha}[\delta,R,\varepsilon]))={\sf{g}}_{1}(\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])),\end{cases}$

for each $\delta>0,R>0$ and $\varepsilon<\varepsilon_{0}(\delta,R)$. Putting
together (5.1), (5.7) and (5.8), and using Corollary 2.6, we conclude the
proof of Lemma 3.2. ∎

Recall that the main Theorem 2.3 was derived from Lemma 3.2 in Subsect. 3.4.
Thus all proofs are complete.

Acknowledgments. The author is grateful to J. Cioslowski for stimulating
discussions, and to M. Lewin for pointing out the asymptotic problem for the
kinetic energy operator.

The author was supported by the EPSRC grant EP/P024793/1.

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