Completeness of SoV Representation for $\mathrm{SL}(2,\mathbb{R})$ Spin Chains

Completeness of SoV Representation
for $\boldsymbol{\mathrm{SL}(2,\mathbb{R})}$ Spin Chains††This paper is a
contribution to the Special Issue on Mathematics of Integrable Systems:
Classical and Quantum in honor of Leon Takhtajan. The full collection is
available at https://www.emis.de/journals/SIGMA/Takhtajan.html

Sergey É. DERKACHOV a, Karol K. KOZLOWSKI b and Alexander N. MANASHOV ca

S.É. Derkachov, K.K. Kozlowski and A.N. Manashov

a) St. Petersburg Department of Steklov Mathematical Institute of Russian
Academy of Sciences,
a) Fontanka 27, 191023 St. Petersburg, Russia<EMAIL_ADDRESS>
b) Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de
Physique,
b) F-69342 Lyon, France<EMAIL_ADDRESS>
c) Institut für Theoretische Physik, Universität Hamburg, D-22761 Hamburg,
Germany<EMAIL_ADDRESS>
Received March 08, 2021, in final form June 14, 2021; Published online June
25, 2021

This work develops a new method, based on the use of Gustafson’s integrals and
on the evaluation of singular integrals, allowing one to establish the
unitarity of the separation of variables transform for infinite-dimensional
representations of rank one quantum integrable models. We examine in detail
the case of the $\mathrm{SL}(2,\mathbb{R})$ spin chains.

spin chains; separation of variables; Gustafson’s integrals

33C70; 81R12

Dedicated to Professor Leon Armenovich Takhtajan
on the occasion of his 70th birthday

##  1 Introduction

The field of quantum integrable models takes its roots in the seminal work of
Hans Bethe [2] on the XXX Heisenberg chain who developed the so-called
coordinate Bethe ansatz method allowing one to construct the eigenvectors and
eigenvalues of the mentioned Hamilton operator by means of combinatorial
expressions involving auxiliary parameters. In order to obtain an eigenvector,
one needs to impose certain constraints on these parameters, the so-called
Bethe ansatz equations. Over the years, the method was refined and applied to
numerous other models, such as the XXZ Heisenberg chain [31], the
$\delta$-function Bose gas [26], or the Hubbard model [27], so as to name a
few. In the late 70s, the method was raised to a higher level of effectiveness
by Faddeev, Sklyanin, Takhtadjan [42], thus becoming known as the so-called
algebraic Bethe ansatz. This new approach provided an algebraic setting
allowing one to connect quantum integrability to the representation theory of
quantum groups, which had several advantages. To start with, the construction
of the eigenvectors of a given integrable model was significantly simplified,
hence allowing to address more involved problems such as the calculation of
norms [21] and scalar products [43] of Bethe vectors and, subsequently, the
one of correlation functions [17, 20]. Moreover, the method allowed one to
significantly enlarge the family of known integrable models, see, e.g., the
review [24], and in particular efficiently and systematically address the
question of constructing the eigenvectors of the higher rank integrable models
[23]. However, it soon turned out that the method had also its limitations in
that not all quantum integrable models were within its grasp, the quantum Toda
chain being a prominent example thereof. In 1985, Sklyanin pioneered a new
technique allowing one to address the calculation of the spectrum of this
model: the quantum separation of variables [40]. He developed several aspects
of the method in [39, 40], and this progress was subsequently continued by
Kharchev and Lebedev [18, 19] in the case of the Toda chain. Derkachov,
Korchemsky and Manashov [5, 6], Bytsko and Teschner [3], Silantyev [36] and,
more recently, Maillet and Niccoli [28] pushed the development of the method
in the case of other, more involved, models (see also [13, 33, 34]). Recently,
many important results have appeared in this area [4, 11, 12, 29, 30].

In fact, there are nowadays many indications that the quantum separation of
variables is a much more general technique for solving quantum integrable
models that encompasses the algebraic Bethe ansatz [5] and provides one with
the quantum analogue of the classical separation of variables technique.

In precise terms, the quantum separation of variables consists in exhibiting a
map $\mathcal{U}$ between an auxiliary Hilbert space $\mathfrak{h}_{\rm sov}$
and the original Hilbert space $\mathfrak{h}_{\rm org}$ on which a given model
is formulated. This map should be unitary so as to ensure the equivalence of
Hilbert space structure and, above all, such that it strongly simplifies the
form taken by the spectral problem associated with a given quantum integrable
Hamiltonian. More precisely, integrability of a given quantum Hamiltonian
means that there exists a commutative subalgebra in the space of operators
$\\{\mathtt{H}_{k}\\}$ containing the Hamiltonian. Thus, the spectral problem
associated with the original Hamiltonian is, in fact, a multi-parameter
spectral problem, in that each eigenvector is associated with the tower of
eigenvalues of the $\mathtt{H}_{k}$s. Now the role of the map $\mathcal{U}$ is
to realise the unitary equivalence between $\mathfrak{h}_{\rm sov}$ and
$\mathfrak{h}_{\rm org}$ in such a way that the original multi-dimensional
(because the Hamiltonians have a non-trivial structure) and multi-parametric
spectral problem on $\mathfrak{h}_{\rm org}$ is reduced into a multi-
parametric (because one has to keep track of all the eigenvalues) one-
dimensional spectral problem on $\mathfrak{h}_{\rm sov}$. This thus explains
the separation of variables terminology. In fact, this one-dimensional
spectral problem corresponds to the resolution of the so-called Baxter $T-Q$
equation associated with the model, proving in this way a remarkable bridge
between the spectrum, the $T$-part, and the eigenvectors, the $Q$ part.

Several ingredients are needed so as to implement the separation of variables
program as described above. First, one should construct a map $\mathcal{U}$
satisfying to the desired requirements and then show that it indeed
corresponds to a unitary map between the Hilbert spaces. In fact, the
construction of $\mathcal{U}$ can be dealt with by exploiting the Yang–Baxter
algebra underlying the integrability of the model. A first method was
suggested by Sklyanin in [40]. Later, an alternative construction was proposed
in [5], where, in particular, it was pointed out that $\mathcal{U}$ can be
constructed by using the Baxter $Q$ operator associated with the model. For
the first time, the idea of a connection between the Baxter $Q$ operator and
separation of variables was apparently formulated in the work [25].

This last idea was later generalised in [28] to other conserved quantities, in
the case of models having finite-dimensional local Hilbert spaces. To be more
precise about the construction of $\mathcal{U}$, we recall that the original
Hilbert space $\mathfrak{h}_{\rm org}$, where the model is formulated and
$\mathfrak{h}_{\rm sov}$, where the separation of variables takes place can be
identified with appropriate $L^{2}$ spaces $\mathfrak{h}_{\rm
org}=L^{2}(\mathcal{X},{\rm d}\nu_{\rm org})$ and $\mathfrak{h}_{\rm
sov}=L^{2}(\mathcal{Y},{\rm d}\mu_{\rm sov})$. This is a very general setting
which allows $\mathcal{X}$, $\mathcal{Y}$ to be finite, discrete or
continuous. Upon such an identification of the Hilbert spaces, the map
$\mathcal{U}$ is defined as an integral transform acting on smooth, compactly
supported functions on $\mathcal{Y}$:

$\displaystyle[\mathcal{U}\varphi](x)=\int_{\mathcal{Y}}{}\varphi(y)\Psi_{y}(x)\,{\rm
d}\mu_{\rm sov}(y).$

The functions $\Psi_{y}(x)$ describing the integral kernel of the transform
can be thought of as the analogues of the function ${\rm e}^{{\rm i}yx}$
giving the integral kernel of the Fourier transform. That case, in fact,
corresponds to $\mathcal{X}=\mathcal{Y}=\mathbb{R}$ and both ${\rm d}\nu_{\rm
org}$ and ${\rm d}\mu_{\rm sov}$ coinciding with the Lebesgue measure. In the
above case, just as $\\{x\mapsto{\rm e}^{{\rm i}yx}\\}$ corresponds to the
system of generalised eigenfunctions of translation operators,
$\\{x\mapsto\Psi_{y}(x)\\}$ corresponds to the system of generalised
eigenfunctions of a commutative operator subalgebra of the representation of
the Yang–Baxter algebra which gives rise to the original model of interest.
The construction of $\mathcal{U}$ hence boils down to the construction of this
system of eigenfunctions, which become possible since it is reduced to solving
hypergeometric like problems [40], viz. first order finite difference
equations in several variables. In fact, the very structure of the Yang–Baxter
algebra which allows one to construct $\Psi_{y}(x)$ in the first place, does
also ensure that, by construction, $\mathcal{U}$ fulfills the desired
requirement of simplifying the original spectral problem. However, unitarity
is a completely different issue. It boils down to proving the orthogonality
and completeness of the system $\Psi_{y}(x)$ which can be framed as the
following relations understood in the sense of distributions

$\displaystyle\int_{\mathcal{X}}\big{(}\Psi_{{y}^{\prime}}(x)\big{)}^{*}\Psi_{y}(x)\,{\rm
d}\nu_{\rm org}(x)=\frac{1}{{\rm d}\mu_{\rm sov}/{\rm d}y}\delta_{\rm
sov}({y}^{\prime},y)$ (1.1)

and

$\displaystyle\int_{\mathcal{Y}}\big{(}\Psi_{y}({x}^{\prime})\big{)}^{*}\Psi_{y}(x)\,{\rm
d}\mu_{\rm sov}(y)=\frac{1}{{\rm d}\nu_{\rm org}/{\rm d}x}\delta_{\rm
org}({x}^{\prime},x).$ (1.2)

Above $\delta_{\rm sov}({y}^{\prime},y)$, resp. $\delta_{\rm
org}({x}^{\prime},x)$, corresponds to the generalised function which
represents the integral kernel of the identity operator on $\mathcal{Y}$,
resp. $\mathcal{X}$. Moreover, ${\rm d}\mu_{\rm sov}/{\rm d}y$, resp. ${\rm
d}\nu_{\rm org}/{\rm d}x$, is the Radon–Nikodym derivative of $\mu_{\rm sov}$,
resp. $\nu_{\rm org}$, in respect to the canonical measure on $\mathcal{Y}$,
resp. $\mathcal{X}$.

The technique for proving unitarity of $\mathcal{U}$, viz. (1.1)–(1.2),
strongly depends on the dimension of the original Hilbert space
$\mathfrak{h}_{\rm org}$. If $\mathfrak{h}_{\rm org}$ is finite-dimensional,
checking unitarity amounts to a simple comparison of dimensions between
$\mathfrak{h}_{\rm org}$ and $\mathfrak{h}_{\rm sov}$. However, many of the
physically interesting quantum integrable models are defined on an infinite-
dimensional Hilbert space $\mathfrak{h}_{\rm org}$. There, unitarity is a much
more delicate issue. In fact, unitarity was first established for the Toda
chain case by using harmonic analysis of Lie groups techniques [35, 45].
However the methods which were used to establish this were quite sophisticated
and hardly generalisable to the more complex quantum integrable models. The
first step towards proving unitarity, in a simpler and systematic way, was
achieved in [5], where a quantum inverse scattering based technique for
proving the isometry of $\mathcal{U}$ was invented. Then, [22] developed a
technique, solely based on the use of natural objects for the quantum inverse
scattering, allowing one to prove rigorously the isometry of
$\mathcal{U}^{\dagger}$ in the case of the Toda chain. All together with the
results of [5], this construction ensures the unitarity of $\mathcal{U}$. An
even more efficient method allowing one to establish the isometry of
$\mathcal{U}^{\dagger}$ was proposed recently by the authors in [8]. Around
the same time, the work [9] has connected certain scalar products of functions
being the building blocks of $\mathcal{U}$ to Gustafson integrals [14].

In the present work, we push further this link and use the relation to the
Gustafson integrals, along with the closed formula for the latter, so as to
propose a novel and remarkably simple method for proving the unitarity of the
map realising the separation of variables for the higher spin, non-compact,
XXX chains. While focusing on this example, we are deeply convinced of the
method’s generality and hence applicability to many other quantum integrable
models possessing infinite-dimensional local Hilbert spaces and which are
solvable by the quantum separation of variables. In order to illustrate the
main features of the method without obscuring them by technicalities of the
model, as a warm up to our main result, we illustrate how it works in the case
of the Toda chain.

The paper is organised as follows. Section 2 outlines, on formal grounds, the
key ideas of our method in the case of the Toda chain. Then, Section 3
introduces the XXX non-compact spin-chain model along with the main notations.
In particular, it defines the operators $\mathcal{U}$ of interest to the
analysis and establishes their isometry. Finally, Section 4 establishes the
isometry of their adjoint, and hence completeness of the underlying system of
functions giving rise to their integral kernels.

##  2 Preliminaries

In this section we illustrate some details of our approach on the example of
the open Toda chain [15, 37]. which is a one-dimensional system of $N$
particles on the line associated with the Hamiltonian

$\displaystyle H=-\frac{1}{2}\sum_{k=1}^{N}\,\frac{\partial^{2}}{\partial
x_{k}^{2}}+\sum_{k=1}^{N-1}{\rm e}^{x_{k}-x_{k+1}}$

on the Hilbert space $L^{2}\left(\mathbb{R}^{N},{\rm d}^{N}x\right)$. The
model is integrable and can be solved by the quantum inverse scattering method
(QISM) [41, 42]. For further discussion it is important that eigenfunctions
can be constructed iteratively [18, 19],

$\displaystyle\Psi^{\lambda}_{N}(x)=\lim_{\varepsilon\rightarrow
0^{+}}\int_{\mathbb{R}^{N-1}}\prod_{k=1}^{N}\prod_{j=1}^{N-1}\Gamma({\rm
i}\lambda_{k}-{\rm i}\gamma_{j}+\varepsilon)\,{\rm e}^{{\rm
i}(\Lambda-\Gamma)x_{N}}\,\Psi_{N-1}^{\gamma}(x)\,\mu_{N-1}(\gamma)\prod_{j=1}^{N-1}{{\rm
d}\gamma_{j}},$ (2.1)

where $x=(x_{1},\dots,x_{N})$, $\lambda=(\lambda_{1},\dots,\lambda_{N})$,
$\gamma=(\gamma_{1},\dots,\gamma_{N-1})$ and
$\Lambda=\sum^{N}_{j=1}\lambda_{j}$, $\Gamma=\sum^{N-1}_{j=1}\gamma_{j}$. The
measure $\mu_{N}(\lambda)$ – the Sklyanin measure – is given by a product of
$\Gamma$-functions

$\displaystyle\mu_{N}^{-1}(\gamma)=(2\pi)^{N}{N!}\prod^{N}_{j<k}\Gamma\big{(}{\rm
i}(\gamma_{k}-\gamma_{j})\big{)}\Gamma\big{(}{\rm
i}(\gamma_{j}-\gamma_{k})\big{)}.$

Finally, the one-particle eigenfunctions are given by plane-waves
$\Psi^{\lambda}_{1}(y)={\rm e}^{{\rm i}y\lambda}$.

Note, that equation (2.1) is nothing other as the expansion of the
$N$-particle function $\Psi^{\lambda}_{N}(x_{1},\dots,x_{N})$ over the product
$\Psi^{\gamma}_{N-1}(x_{1},\dots,x_{N-1})\cdot\Psi_{1}^{\Lambda-\Gamma}(x_{N})$.
The expansion coefficients are given by products of $\Gamma$ functions. This
property – the possibility to find the expansion coefficients of $N$ particle
functions over $N-1$ particle functions – is very important since it allows
one to prove orthogonality and completeness relations for the eigenfunctions
using induction on $N$.111Although in our analysis of spin chains we use a
different, more direct, approach to establish the orthogonality of the
eigenfunctions it also can be done inductively. Indeed, for $N=1$ the
eigenfunctions obviously form an orthogonal and complete system. Let us assume
that the following identities hold for a certain $N\geq 1$

$\displaystyle\int_{\mathbb{R}^{N}}\Psi_{N}^{\lambda}(x)\big{(}\Psi_{N}^{\lambda^{\prime}}(x)\big{)}^{\dagger}\,{\rm
d}^{N}x=\mu^{-1}_{N}(\lambda)\delta^{N}(\lambda,\lambda^{\prime}),$ (2.2a)
$\displaystyle\int_{\mathbb{R}^{N}}\Psi_{N}^{\lambda}(x)\big{(}\Psi_{N}^{\lambda}(x^{\prime})\big{)}^{\dagger}\mu_{N}(\lambda)\,{\rm
d}^{N}\lambda=\delta^{N}(x-x^{\prime}),$ (2.2b)

where $\delta^{N}(x-x^{\prime})=\prod_{k=1}^{N}\delta(x_{k}-x^{\prime}_{k})$
and

$\displaystyle\delta^{N}(\lambda,\lambda^{\prime})=\frac{1}{N!}\sum_{w\in
S_{N}}\delta^{N}(\lambda^{\prime}-w\lambda),\qquad
w\lambda=\big{(}\lambda_{w_{1}},\dots,\lambda_{w_{N}}\big{)}$ (2.3)

and try to prove that these identities hold for $N+1$ as well.

Let us start with equation (2.2a). Substituting $\Psi_{N+1}^{\lambda}$ in the
form (2.1) into this equation and integrating over $x$ one gets that the
orthogonality condition is equivalent to the following equation

$\displaystyle\lim_{\varepsilon,\,\varepsilon^{\prime}\rightarrow
0^{+}}2\pi\delta(\Lambda-\Lambda^{\prime})\int_{\mathbb{R}^{N}}\prod_{k=1}^{N+1}\prod_{j=1}^{N}\Gamma({\rm
i}\lambda_{k}-{\rm i}\gamma_{j}+\varepsilon)\,\Gamma({\rm i}\gamma_{j}-{\rm
i}\lambda^{\prime}_{k}+\varepsilon^{\prime})\mu_{N}(\gamma)\,{\rm
d}^{N}\gamma$
$\displaystyle\hphantom{\lim_{\varepsilon,\,\varepsilon^{\prime}\rightarrow
0^{+}}}{}=\mu^{-1}_{N+1}(\lambda)\,\delta^{N+1}(\lambda,\lambda^{\prime}).$
(2.4)

Next, replacing $N\to N+1$ in (2.2b) and projecting both sides on the
functions $\Psi^{\gamma}_{N}(x)$ and $\Psi^{\gamma^{\prime}}_{N}(x)$ one gets
that the completeness relation is reduced to the following identity

$\displaystyle\lim_{\varepsilon,\,\varepsilon^{\prime}\rightarrow
0^{+}}\int_{\mathbb{R}^{N+1}}\prod_{k=1}^{N}\prod_{j=1}^{N+1}\Gamma({\rm
i}\gamma_{k}-{\rm i}\lambda_{j}+\varepsilon)\Gamma({\rm i}\lambda_{j}-{\rm
i}\gamma^{\prime}_{k}+\varepsilon^{\prime})\,{\rm e}^{{\rm
i}\Lambda(x^{\prime}_{N}-x_{N})}\mu_{N+1}(\lambda)\,{\rm d}^{N+1}\lambda$
$\displaystyle\hphantom{\lim_{\varepsilon,\,\varepsilon^{\prime}\rightarrow
0^{+}}}{}=\mu^{-1}_{N}(\gamma)\,\delta(x_{N}-x^{\prime}_{N})\,\delta^{N}(\gamma,\gamma^{\prime}).$
(2.5)

Of course, both of these relations should be understood in the sense of
distributions. Thus the problem of establishing the orthogonality and
completeness relations for the eigenfunctions of the open Toda chain is
equivalent to proving these two integral identities. The problem is greatly
simplified by the following remarkable result due to R.A. Gustafson [14,
Theorem 5.1]

$\displaystyle\int_{\mathbb{R}^{N}}\,\prod_{k=1}^{N+1}\prod_{j=1}^{N}\Gamma(\alpha_{k}-{\rm
i}\lambda_{j})\,\Gamma({\rm i}\lambda_{j}+\beta_{k})\,\mu_{N}(\lambda)\,{\rm
d}^{N}\lambda=\frac{\prod_{k,j=1}^{N+1}\Gamma(\alpha_{k}+\beta_{j})}{\Gamma\left(\sum_{j=1}^{N+1}(\alpha_{j}+\beta_{j})\right)},$
(2.6)

where $\text{Re}(\alpha_{k})>0$, $\text{Re}(\beta_{k})>0$ for all $k$. Note,
that the integral in the l.h.s. (2.6) is exactly the integral appearing in
(2.4) which, therefore, can be brought to the form

$\displaystyle\lim_{\varepsilon,\varepsilon^{\prime}\rightarrow
0^{+}}(2\pi)\delta(\Lambda-\Lambda^{\prime})\,\frac{\prod_{k,j=1}^{N+1}\Gamma({\rm
i}\lambda^{\prime}_{j}-{\rm
i}\lambda_{k}+\varepsilon+\varepsilon^{\prime})}{\Gamma\big{(}{\rm
i}\Lambda^{\prime}-{\rm
i}\Lambda+N(\varepsilon+\varepsilon^{\prime})\big{)}}={\mu_{N+1}^{-1}(\lambda)}\,\delta^{N+1}\left(\lambda,\lambda^{\prime}\right).$
(2.7)

The proof of this identity is already rather straightforward. Some details can
be found in Appendix A.

It takes a little more work to prove the identity (2.5). Having put
$\alpha_{N+1}=L$ and $\beta_{N+1}=tL$, $t>0$, in (2.6) and sending
$L\to+\infty$ one arrives at the reduced version of the integral (2.6) which
takes the form [9]

$\displaystyle\int_{\mathbb{R}^{N}}t^{{\rm
i}\Lambda}\prod_{k,\,j=1}^{N}\Gamma(\alpha_{k}-{\rm
i}\lambda_{j})\,\Gamma({\rm i}\lambda_{j}+\beta_{k})\,\mu_{N}(\lambda)\,{\rm
d}^{N}\lambda=\frac{t^{A}}{(1+t)^{A+B}}{\prod_{k,\,j=1}^{N}\Gamma(\alpha_{k}+\beta_{j})},$
(2.8)

where $A(B)=\sum_{k=1}^{N}\alpha_{k}(\beta_{k})$. At the same time
representing the l.h.s. of equation (2.5) in the following form (for more
details see Lemma 4.3):

$\displaystyle\lim_{L\rightarrow\infty}\lim_{\varepsilon,\,\varepsilon^{\prime}\rightarrow
0^{+}}\frac{1}{\Gamma^{{2N+2}}(L)}\int_{\mathbb{R}^{N+1}}\prod_{k,j=1}^{N+1}\Gamma({\rm
i}\gamma_{k}-{\rm i}\lambda_{j}+\varepsilon)\,\Gamma({\rm i}\lambda_{j}-{\rm
i}\gamma^{\prime}_{k}+\varepsilon^{\prime})$
$\displaystyle\hphantom{\lim_{L\rightarrow\infty}\lim_{\varepsilon,\,\varepsilon^{\prime}\rightarrow
0^{+}}\frac{1}{\Gamma^{{2N+2}}(L)}\int_{\mathbb{R}^{N+1}}\prod_{k,j=1}^{N+1}}{}\times{\rm
e}^{{\rm i}\Lambda(x^{\prime}_{N}-x_{N})}\mu_{N+1}(\lambda)\,{\rm
d}^{N+1}\lambda,$

where ${\rm i}\gamma_{N+1}=-{\rm i}\gamma^{\prime}_{N+1}=L$, one can evaluate
the integral using equation (2.8). Then, after some algebra, one can show that
(2.5) is equivalent to the following identity

$\displaystyle\lim_{L\rightarrow\infty}\lim_{\epsilon\rightarrow 0^{+}}L^{{\rm
i}(\Gamma^{\prime}-\Gamma)}\prod_{k,j=1}^{N}\Gamma\big{(}{\rm
i}(\gamma^{\prime}_{k}-\gamma_{j})+\varepsilon\big{)}\sqrt{\frac{L}{4\pi}}{\cosh^{-2L}\bigg{(}\frac{x_{N}-x^{\prime}_{N}}{2}\bigg{)}}$
$\displaystyle\hphantom{\lim_{L\rightarrow\infty}\lim_{\epsilon\rightarrow
0^{+}}}{}={\mu^{-1}_{N}(\gamma)}\,\delta(x_{N}-x^{\prime}_{N})\,\delta^{N}(\gamma,\gamma^{\prime}).$
(2.9)

We recall here that equations (2.7) and (2.9) have to be understood in the
sense of distributions and relegate further details to Appendix A.

##  3 Spin chains: operators and eigenfunctions

In this section we construct the representation of separated variables for
generic spin chain models and recall some elements of the quantum inverse
scattering method (QISM) relevant for our purposes.

The spin chain is a quantum system of interacting spins $S_{k}^{\pm}$,
$S^{0}_{k}$, $k=1,\dots,N$, where the index $k$ enumerates the nodes of the
chain. The spin operators are the symmetry generators of the
$\mathrm{SL}(2,\mathbb{R})$ group [10] which are determined by a real number
(spin) $\boldsymbol{s}_{k}>1/2$,

$\displaystyle S_{k}^{-}=-\partial_{z_{k}},\qquad
S_{k}^{0}=z_{k}\partial_{z_{k}}+\boldsymbol{s}_{k},\qquad
S_{k}^{+}=z^{2}_{k}\partial_{z_{k}}+2\boldsymbol{s}_{k}z_{k}.$

Operators with the index $k$ act in a Hilbert space associated with the $k$-th
site, $\mathcal{H}_{k}$, which is the Hilbert space of functions holomorphic
in the upper complex half-plane, $\mathbb{H}^{+}$. The scalar product in the
Hilbert space $\mathcal{H}_{k}=\big{\\{}f\in
L^{2}(\mathbb{H}^{+},\mathrm{d}\mu_{s_{k}})\colon f\text{ is holomorphic on
}\mathbb{H}^{+}\big{\\}}$ is defined as follows

$\displaystyle(f,\psi)=\int(f(z_{k}))^{\dagger}\psi(z_{k})\,\mu_{\boldsymbol{s}_{k}}(z_{k})\,{\rm
d}^{2}z_{k}.$ (3.1)

The measure takes the form

$\displaystyle\mu_{s}(z_{k})=\frac{2s-1}{\pi}\,\theta(\mathop{\rm
Im}z_{k})(2\mathop{\rm Im}z_{k})^{2s-2},$ (3.2)

where $\theta(x)$ is the Heaviside step function. The operators
$S_{k}^{\alpha}$ are anti-hermitian with respect to the scalar product (3.1).

Function in $\mathcal{H}_{k}$ can be represented by Fourier integrals where
the integration runs only over positive momenta

$\displaystyle f(z)=\int_{0}^{\infty}{\rm e}^{{\rm
i}pz}\mathcal{F}[f](p)\,{\rm d}p.$

Then, in Fourier space, viz. in the momentum representation, the scalar
product takes the form

$\displaystyle(f,\psi)=\Gamma(2s)\int_{0}^{\infty}\big{(}\mathcal{F}[f](p)\big{)}^{*}\mathcal{F}[\psi](p)p^{1-2s}\,{\rm
d}p.$

One of the main objects of QISM is the monodromy matrix. For the closed/open
spin chain of our interest, the monodromy matrix is given by a product of
$L$-operators [42] which are two by two matrices,

$\displaystyle L_{k}(u)=u+{\rm i}\begin{pmatrix}S^{0}_{k}&S^{-}_{k}\\\
S^{+}_{k}&-S_{k}^{0}\end{pmatrix}\\!,$

where $u\in\mathbb{C}$ is the spectral parameter. The monodromy matrix for the
closed chain of length $N$ has the form [42]

$\displaystyle T_{N}(u)=L_{1}(u+\xi_{1})\cdots
L_{N}(u+\xi_{N})=\begin{pmatrix}A_{N}(u)&B_{N}(u)\\\
C_{N}(u)&D_{N}(u)\end{pmatrix}\\!,$

while, for the open spin chain, it is given by the following expression [38]

$\displaystyle\mathbb{T}_{N}(u)=T_{N}(-u)\sigma_{2}T_{N}^{t}(u)\sigma_{2}=\begin{pmatrix}\mathbb{A}_{N}(u)&\mathbb{B}_{N}(u)\\\
\mathbb{C}_{N}(u)&\mathbb{D}_{N}(u)\end{pmatrix}\\!,$ (3.3)

where $\sigma_{2}$ is the Pauli matrix. The entries of the monodromy matrices
form commuting polynomial operator families [38, 42]
$\left\\{A_{N}(u)\right\\}_{u\in\mathbb{C}}$,
$\left\\{B_{N}(u)\right\\}_{u\in\mathbb{C}}$,
$\left\\{\mathbb{B}_{N}(u)\right\\}_{u\in\mathbb{C}}$:

$\displaystyle\left[A_{N}(u),A_{N}(v)\right]=\left[B_{N}(u),B_{N}(v)\right]=\left[\mathbb{B}_{N}(u),\mathbb{B}_{N}(v)\right]=0,$

which act on the Hilbert space of the model,
$\mathbb{H}_{N}=\bigotimes_{k=1}^{N}\mathcal{H}_{k}$. Operators in each of the
commuting families share the same eigenfunctions. These systems of functions
have proven to be very useful for analysing the properties of spin chains.
They determine the so-called Sklyanin representation of separated variables
[40]. For the homogeneous chains, viz. $\xi_{a}=0$, the corresponding systems
for $B_{N}$, $\mathbb{B}_{N}$ and $A_{N}$ operators were constructed in [1, 6,
7], respectively. Below, we recall these constructions and, on the occasion,
extend them to the general case of inhomogeneous spin chains where the
$\xi_{a}$’s are generic. Since the technical details are essentially the same
in all three cases, we consider in some detail the $B_{N}$-system and only
quote the results for the other two.

All three families of eigenfunctions can be represented as a convolution of
functions of a special type. Namely, let us define a function of two complex
variables, $z,w\in\mathbb{H}^{+}$, and the variable $\alpha\in\mathbb{C}$
which is called index,

$\displaystyle D_{\alpha}(z,w)=\bigg{(}\frac{\rm
i}{z-\bar{w}}\bigg{)}^{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{0}^{\infty}{\rm
e}^{{\rm i}p\,(z-\bar{w})}p^{\alpha-1}\,{\rm d}p.$

This is a single valued function222In many cases it is quite helpful to
visualize all further constructions as Feynmann diagrams with the function
$D_{\alpha}$ playing the role of a propagator. of $z$, $w$ which is fixed by
the condition $\arg\big{(}{\rm i}/(x+{\rm i}y)\big{)}\to 0$ for $x\to 0$. Some
properties of the function $D_{\alpha}$ can be found in [9].

###  3.1 $\boldsymbol{B_{N}}$ operator

####  3.1.1 Layer operators

The eigenfunctions can be most conveniently written down in term of the so-
called layer operators. Let $\boldsymbol{\Lambda}_{n+1}(\gamma,x)$ be an
operator which maps functions of $n$ complex variables to functions of $n+1$
variables. It depends on the spectral parameter $x\in\mathbb{C}$ and the
complex vector
$\gamma=(\alpha_{1},\dots,\alpha_{n},\beta_{1},\dots,\beta_{n})\in\mathbb{C}^{2n}$.
Its action takes the form

$\displaystyle\left[\boldsymbol{\Lambda}_{n+1}(\gamma,x)f\right](z_{1},\dots,z_{n+1})=\idotsint\prod_{k=1}^{n}D_{\alpha_{k}-{\rm
i}x}(z_{k},w_{k})D_{\beta_{k}+{\rm i}x}(z_{k+1},w_{k})$
$\displaystyle\hphantom{\left[\boldsymbol{\Lambda}_{n+1}(\gamma,x)f\right](z_{1},\dots,z_{n+1})=\idotsint}{}\times
f(w_{1},\dots,w_{n})\prod_{a=1}^{n}\mu_{{(\alpha_{a}+\beta_{a})}/2}(w_{a})\,{\rm
d}^{2}w_{1}\cdots{\rm d}^{2}w_{n}.$

The weight function $\mu_{s}(w_{j})$ has been defined in equation (3.2). The
integral is well defined provided $\mathop{\rm Re}(\alpha_{k}-{\rm i}x)>0$,
$\mathop{\rm Re}(\beta_{k}+{\rm i}x)>0$ for $k=1,\dots,n$.

In the momentum representation obtained by taking the Fourier transform

$\displaystyle
f(z_{1},\dots,z_{n})=\int_{0}^{\infty}\dots\int_{0}^{\infty}\mathcal{F}[f](p_{1},\dots,p_{n})\,{\rm
e}^{{\rm i}\sum_{k=1}^{n}p_{k}z_{k}}\,{\rm d}p_{1}\cdots{\rm d}p_{n}$

the action of the layer operator can be expressed as

$\displaystyle\mathcal{F}\left[\boldsymbol{\Lambda}_{n+1}(\gamma,x)g\right](q_{1},\dots,q_{n+1})=\lambda_{n}(\gamma,x)\int_{0}^{q_{2}}\frac{{\rm
d}\ell_{1}}{p_{1}}\int_{0}^{q_{3}}\frac{{\rm
d}\ell_{2}}{p_{2}}\cdots\int_{0}^{q_{n}}\frac{{\rm d}\ell_{n-1}}{p_{n-1}}$
$\displaystyle\hphantom{\mathcal{F}\left[\boldsymbol{\Lambda}_{n+1}(\gamma,x)g\right]=\qquad}{}\times\mathcal{F}[g](p_{1},p_{2},\dots,p_{n})\prod_{k=1}^{n}\bigg{(}\frac{q_{k}-\ell_{k-1}}{p_{k}}\bigg{)}^{\alpha_{k}-{\rm
i}x-1}\bigg{(}\frac{\ell_{k}}{p_{k}}\bigg{)}^{\beta_{k}+{\rm i}x-1},$ (3.4)

where $\ell_{k}$ are the “loop” momenta, $p_{k}=q_{k}+\ell_{k}-\ell_{k-1}$ and
$\ell_{0}\equiv 0$, $\ell_{n}\equiv q_{n+1}$ and the factor $\lambda_{n}$
reads

$\displaystyle\lambda_{n}(\gamma,x)=\prod_{k=1}^{n}\frac{\Gamma(\alpha_{k}+\beta_{k})}{\Gamma(\alpha_{k}-{\rm
i}x)\,\Gamma(\beta_{k}+{\rm i}x)}.$

Note also that all momenta in (3.4) are positive and
$\sum_{k=1}^{n+1}q_{k}=\sum_{k=1}^{n}p_{k}$.

The layer operators possess two important properties. Let us define a map
$t\colon\mathbb{C}^{2n}\mapsto\mathbb{C}^{2(n-1)}$ as follows

$\displaystyle{t}\gamma=t(\alpha_{1},\dots,\alpha_{n},\beta_{1},\dots,\beta_{n})=(\alpha_{1},\dots,\alpha_{n-1},\beta_{2},\dots,\beta_{n}).$

It can be checked that the operators $\boldsymbol{\Lambda}_{n}$ satisfy the
permutation identity

$\displaystyle\boldsymbol{\Lambda}_{n+1}(\gamma,x)\,\boldsymbol{\Lambda}_{n}(t\gamma,x^{\prime})=\boldsymbol{\Lambda}_{n+1}(\gamma,x^{\prime})\,\boldsymbol{\Lambda}_{n}(t\gamma,x).$
(3.5)

The derivation is based on integral identities for the functions $D_{\alpha}$
which can be found in [7].

Next, for the spin chain of length $N$, we introduce the following
combinations of spins and impurities

$\displaystyle s_{k}=\boldsymbol{s}_{k}-{\rm
i}\xi_{k},\qquad\bar{s}_{k}=s_{k}^{*}=\boldsymbol{s}_{k}+{\rm i}\xi_{k},\qquad
k=1,\dots,N,$

and define the vector $\gamma_{N}\in\mathbb{C}^{2N-2}$:

$\displaystyle\gamma_{N}=(s_{1},\dots,s_{N-1},\bar{s}_{2},\dots,\bar{s}_{N}).$
(3.6)

It can be shown, see [7], that the operator
$\boldsymbol{\Lambda}_{N}(\gamma_{N},x)$ is nullified by $B_{N}(x)$,

$\displaystyle B_{N}(x)\boldsymbol{\Lambda}_{N}(\gamma_{N},x)=0.$ (3.7)

These two properties of the layer operators are crucial for constructing the
eigenfunctions of the operator $B_{N}(u)$.

####  3.1.2 Eigenfunctions

Let us define a function of $N$ complex variables

$\displaystyle\Psi^{N}_{p,{x}}(z)=\varkappa_{N}p^{\boldsymbol{S}-1/2}\big{[}\boldsymbol{\Lambda}_{N}\left(\gamma_{N},x_{1}\right)\boldsymbol{\Lambda}_{N-1}\left(t\gamma_{N},x_{2}\right)\cdots\boldsymbol{\Lambda}_{2}\left(t^{N-2}\gamma_{N},x_{N-1}\right)\,E_{p}\big{]}(z).$
(3.8)

Here $x=(x_{1},\dots,x_{N-1})$, $z=(z_{1},\dots,z_{N})$, $E_{p}$ is the
exponential function, $E_{p}(w)={\rm e}^{{\rm i}pw}$, $p>0$,
$\boldsymbol{S}\equiv\sum_{k=1}^{N}\boldsymbol{s}_{k}$ and the normalisation
constant $\varkappa_{N}$ reads

$\displaystyle\varkappa_{N}^{-1}=\Bigg{(}\prod_{k=1}^{N}\Gamma(s_{k}+\bar{s}_{k})\Bigg{)}^{1/2}\prod_{1\leq
i<j\leq N}\Gamma(s_{i}+\bar{s}_{j}).$

Following the lines of [22] one can show that the integrals arising from the
action of the $\boldsymbol{\Lambda}_{k}$’s in (3.8) converge absolutely and
that integrations can be performed in an arbitrary order. Due to the
properties of the layer operators, equations (3.5) and (3.7), the function
$\Psi^{N}_{p,{x}}$ is a symmetric function of $x_{1},\dots,x_{N-1}$ which
satisfies the equation $B_{N}(x_{k})\Psi^{N}_{p,{x}}({z})=0$ for all $k$.
Taking into account that the operator $B_{N}(u)$ is a polynomial of degree
$N-1$ in $u$, $B_{N}(u)=u^{N-1}\sum_{k=1}^{N}S^{-}_{k}$ $+\cdots$, and that
$\sum_{k=1}^{N}S^{-}_{k}\cdot\boldsymbol{\Lambda}_{N}(\gamma_{N},x_{1})=\boldsymbol{\Lambda}_{N}(\gamma_{N},x_{1})\cdot\sum_{k=1}^{N-1}S^{-}_{k}$
one gets

$\displaystyle
B_{N}(u)\Psi^{N}_{p,{x}}({z})=p(u-x_{1})\cdots(u-x_{N-1})\,\Psi^{N}_{p,{x}}({z}).$

In the momentum representation, the function
$\mathcal{F}\big{[}\Psi^{N}_{p,{x}}\big{]}(q_{1},\dots,q_{N})$ is given by the
convolution of the layer operators in the momentum representation, equation
(3.4), acting on the function $\delta(p-q)$.

Our aim is to show that the functions
$\big{\\{}\Psi^{N}_{p,{x}},\,\,p>0,{x}\in\mathbb{R}^{N-1}\big{\\}}$ form a
complete orthogonal set in the Hilbert space $\mathbb{H}_{N}$. It is
straightforward to check this statement for $N=1,2$. The proof for general $N$
is more involved and presents the main task of this paper.

For real $x$ and $p$ the functions $\Psi^{N}_{p,x}({z})$ do not belong to the
Hilbert space $\mathbb{H}_{N}$. However, they allow one to define a linear
transform from the Hilbert space $\mathbb{H}_{N}^{B}$ defined below into
$\mathbb{H}_{N}$:

$\displaystyle\mathbb{H}_{N}^{B}=L^{2}(\mathbb{R}^{+})\otimes L^{2}_{\rm
sym}\left(\mathbb{R}^{N-1},{\rm d}\mu^{B}_{N-1}(x)\right),\qquad{\rm
d}\mu^{B}_{N-1}(x)=\mu^{B}_{N-1}(x)\,{\rm d}^{N-1}x,$

where we agree upon

$\displaystyle\mu_{N-1}^{B}(x)=\frac{1}{(2\pi)^{N-1}(N-1)!}\frac{\prod_{k=1}^{N-1}\prod_{j=1}^{N}\Gamma(s_{j}-{\rm
i}x_{k})\Gamma(\bar{s}_{j}+{\rm i}x_{k})}{\prod_{j<k}\Gamma({\rm
i}(x_{k}-x_{j}))\,\Gamma({\rm i}(x_{j}-x_{k}))}.$ (3.9)

To start with, given a smooth, compactly supported function $\varphi$ on
$\mathbb{R}^{+}\times\mathbb{R}^{N-1}$, one introduces the transform

$\displaystyle\left[\mathrm{T}_{N}^{B}\varphi\right](z)=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{N-1}}\varphi(p,x)\Psi^{N}_{p,{x}}({z})\,\mu_{N-1}^{B}(x)\,{\rm
d}p\,{\rm d}^{N-1}x.$ (3.10)

######  Theorem 3.1.

For any smooth, compactly supported function $\varphi$ on
$\mathbb{R}^{+}\times\mathbb{R}^{N-1}$,
$\mathrm{T}_{N}^{B}\varphi\in\mathbb{H}_{N}$ and the following relation holds

$\displaystyle\left\|\mathrm{T}_{N}^{B}\varphi\right\|_{\mathbb{H}_{N}}^{2}=\|{\varphi}\|_{\mathbb{H}_{N}^{B}}^{2}\equiv\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{N-1}}|\varphi(p,x)|^{2}\,\mu_{N-1}^{B}(x)\,{\rm
d}p\,{\rm d}^{N-1}x.$ (3.11)

As such, $T_{N}^{B}$ extends to a linear isometry
$T_{N}^{B}\colon\mathbb{H}_{N}^{B}\mapsto\mathbb{H}_{N}$ satisfying

$\displaystyle\left\|\mathrm{T}_{N}^{B}\varphi\right\|_{\mathbb{H}_{N}}^{2}=\|{\varphi}\|_{\mathbb{H}_{N}^{B}}^{2}.$

One may already draw several consequences from this theorem. First of all,
(3.11) ensures that the system of functions
$\left\\{\Psi^{N}_{p,{x}},\,p\in\mathbb{R}^{+},\,x\in\mathbb{R}^{N-1}\right\\}$
forms an orthogonal system in $\mathbb{H}_{N}$, viz. that

$\displaystyle\left(\Psi^{N}_{p^{\prime},x^{\prime}},\Psi^{N}_{p,{x}}\right)_{\mathbb{H}_{N}}=\delta(p-p^{\prime})\,\delta^{N-1}(x,x^{\prime})\,\left(\mu_{N-1}^{B}(x)\right)^{-1}\\!,$

where the multi-dimensional Dirac delta-function has been introduced in (2.3)
while $\mu_{N}^{B}(x)$ has been defined in (3.9). This corresponds to the
orthogonality relation for the system $\left\\{\Psi^{N}_{p,{x}}(z)\right\\}$.

Next, the equality (3.11) implies that $\|\mathrm{T}_{N}^{B}\|=1$ and that the
image of $\mathbb{H}_{N}^{B}$ is a closed subspace of $\mathbb{H}_{N}$. Now,
if one is able to show that $\mathrm{T}_{N}^{B}$ is a unitary operator, what
amounts to showing $\mathrm{T}_{N}^{B}\mathbb{H}_{N}^{B}=\mathbb{H}_{N}$, then
this will also ensure that the system of functions
$\big{\\{}(p,x)\mapsto\Psi^{N}_{p,{x}}(z)$,
$z\in(\mathbb{H}^{+})^{N}\big{\\}}$ forms an complete system in
$\mathbb{H}_{N}$, viz. that given $z_{a}=x_{a}+{\rm i}y_{a}$

$\displaystyle\big{(}\Psi^{N}_{*,*}(z^{\prime}),\Psi^{N}_{*,*}(z)\big{)}_{\mathbb{H}_{N}^{B}}=\mathcal{I}(z,z^{\prime}).$

The r.h.s. of this equation, $\mathcal{I}(z,z^{\prime})$, is the kernel of the
unit operator in $\mathbb{H}_{N}$ – the so-called reproducing kernel, see,
e.g., [16] – which takes the form
$\mathcal{I}(z,z^{\prime})=\prod_{k=1}^{N}\mathcal{I}_{k}(z_{k},z^{\prime}_{k})$,
where

$\displaystyle\mathcal{I}_{k}(z_{k},z^{\prime}_{k})=D_{2s_{k}}(z_{k},z^{\prime}_{k})=\bigg{(}\frac{\rm
i}{z_{k}-\bar{z}^{\prime}_{k}}\bigg{)}^{2\boldsymbol{s}_{k}}\\!.$

It means that for any $\Psi(z)\in\mathbb{H}_{N}$ the following identity holds

$\displaystyle\Psi(z)=\int
I(z,z^{\prime})\Psi(z^{\prime})\prod_{k=1}^{N}\mu_{\boldsymbol{s}_{k}}(z_{k})\,{\rm
d}^{2}z_{k}.$

The unitarity of $\mathrm{T}_{N}^{B}$, viz. that the map has dense range, will
be established in Section 4, hence leading to the main result of the paper.

###### Proof.

In order to prove the theorem, we first establish that (3.11) holds for
smooth, compactly supported functions $\varphi$ on
$\mathbb{R}^{+}\times\mathbb{R}^{N-1}$. For that purpose, let us introduce the
regularised function $\Psi^{N,\epsilon}_{p,{x}}({z})$ which is obtained from
$\Psi^{N}_{p,{x}}({z})$ by giving small positive imaginary parts to all
variables $x_{k}$, $x_{k}\to x_{k}+{\rm i}\epsilon_{k}$, $\epsilon_{k}>0$ and
by changing
$\bar{s}_{N}\to\bar{s}^{\epsilon}_{N}=\bar{s}_{N}+\sum_{k=1}^{N}\epsilon_{k}$
in the definition of the vector $\gamma_{N}$, equation (3.6). Further, let

$\displaystyle\big{[}\mathrm{T}_{N}^{B,\epsilon}\varphi(z)\big{]}=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{N-1}}\varphi(p,x)\,\Psi^{N,\epsilon}_{p,{x}}({z})\,\mu_{N-1}^{B}(x)\,{\rm
d}p\,{\rm d}^{N-1}x.$

One may readily check that
$\big{[}\mathrm{T}_{N}^{B,\epsilon}\varphi\big{]}\rightarrow\big{[}\mathrm{T}_{N}^{B}\varphi\big{]}(z)$
pointwise as $\epsilon=(\epsilon_{1},\dots,\epsilon_{N})\rightarrow 0^{+}$. We
want to show that

$\displaystyle\lim_{\epsilon,\epsilon^{\prime}\rightarrow
0^{+}}\big{(}\mathrm{T}_{N}^{B,\epsilon^{\prime}}\varphi,\mathrm{T}_{N}^{B,\epsilon}\varphi\big{)}_{\mathbb{H}_{N}}=\big{\|}\mathrm{T}_{N}^{B}\varphi\big{\|}_{\mathbb{H}_{N}}^{2}.$

Let us demonstrate that one can invoke Fubini’s theorem to get

$\displaystyle\big{\|}\mathrm{T}_{N}^{B,\epsilon}\varphi\big{\|}_{\mathbb{H}_{N}}^{2}=\int_{(\mathbb{R}^{+})^{2}}\\!\\!{\rm
d}p\,{\rm d}p^{\prime}\int_{\mathbb{R}^{2(N-1)}}{\rm d}^{N-1}x\,{\rm
d}^{N-1}{x}^{\prime}\varphi(p,x)\,\varphi^{*}(p^{\prime},{x}^{\prime})\mu_{N-1}^{B}(x)$
$\displaystyle\hphantom{\big{\|}\mathrm{T}_{N}^{B,\epsilon}\varphi\big{\|}_{\mathbb{H}_{N}}^{2}=\int_{(\mathbb{R}^{+})^{2}}\\!\\!{\rm
d}p\,{\rm
d}p^{\prime}\int\quad}{}\times\mu_{N-1}^{B}({x}^{\prime})\big{(}\Psi^{N,\epsilon^{\prime}}_{p^{\prime},{x}^{\prime}},\Psi^{N,\epsilon}_{p,{x}}\big{)}_{\mathbb{H}_{N}}\,.$

The scalar product of the regularised functions $\Psi^{N,\epsilon}_{p,{x}}$
may be computed in closed form as [7]

$\displaystyle\big{(}\Psi^{N,\epsilon^{\prime}}_{p^{\prime},{x}^{\prime}},\Psi^{N,\epsilon}_{p,{x}}\big{)}_{\mathbb{H}_{N}}=\delta(p-p^{\prime})\,C_{N}^{(\epsilon,\epsilon^{\prime})}(p,x,x^{\prime}),$

where

$\displaystyle
C_{N}^{(\epsilon,\epsilon^{\prime})}(p,x,x^{\prime})=\bigg{(}\frac{\varkappa_{N}^{2}}{\varkappa^{\epsilon}_{N}\varkappa^{\epsilon^{\prime}}_{N}}\bigg{)}p^{\sum_{k=1}^{N}(\epsilon_{k}+\epsilon^{\prime}_{k})}\,\frac{\Gamma\Big{(}\epsilon_{N}+\epsilon^{\prime}_{N}+{\rm
i}\sum_{k=1}^{N-1}(x_{k}-x^{\prime}_{k})\Big{)}}{\Gamma\Big{(}\sum_{k=1}^{N}\epsilon_{k}+\epsilon^{\prime}_{k}\Big{)}}$
$\displaystyle\qquad\ {}\times\frac{\prod_{k,j=1}^{N-1}\Gamma\big{(}{\rm
i}(x^{\prime}_{k}-x_{j})+\epsilon_{kj}\big{)}}{\prod_{k=1}^{N-1}\Gamma\big{(}\bar{s}^{\epsilon}_{N}\\!+\\!{\rm
i}x_{k}\\!-\\!\epsilon_{k}\big{)}\Gamma\big{(}s_{N}^{\epsilon^{\prime}}\\!-\\!{\rm
i}x^{\prime}_{k}\\!-\\!\epsilon_{k}\big{)}\prod_{j=1}^{N\\!-\\!1}\Gamma\big{(}\bar{s}_{j}\\!+\\!{\rm
i}x^{\prime}_{k}\\!+\\!\epsilon^{\prime}_{k}\big{)}\Gamma(s_{j}\\!-\\!{\rm
i}x_{k}\\!+\\!\epsilon_{k})},$ (3.12)

and $\epsilon_{kj}=\epsilon^{\prime}_{k}+\epsilon_{j}$ and
$\varkappa^{\epsilon}_{N}=\varkappa_{N}(s^{\epsilon})$.

In order to obtain this result it is convenient to perform calculation in the
momentum space representation. Using the momentum representation for the layer
operators (3.4) one can obtain an expression for
$C_{N}^{(\epsilon,\epsilon^{\prime})}(p,x,x^{\prime})$ in the form of a
multidimensional momentum integral (which can be thought of as a Feynman
diagram)

$\displaystyle
C_{N}^{(\epsilon,\epsilon^{\prime})}(p,x,x^{\prime})=\int_{X}f\left(\epsilon,p,x,x^{\prime},\\{\ell_{ij}\\}\right)\prod_{ij}{\rm
d}\ell_{ij}.$ (3.13)

Here the function $f$ is a product of linear combinations of momenta
$\ell_{ij}$ and $p$ raised to some powers. It is important to note that all
these combinations are positive and that the integration runs over a compact
region $X$. Performing integrations in a special order using the integral
identities for the product of the propagators, see, e.g., [6, 7, 9], gives the
expression (3.12). Of course one has to justify that the order of integrations
does not influence the answer. To this end we note that the integral of $|f|$
can be written in the form

$\displaystyle\int\left|f\left(\epsilon,p,x,x^{\prime},\\{\ell_{ij}\\}\right)\right|\prod_{ij}{\rm
d}\ell_{ij}=R(x,x^{\prime},\epsilon)\int
f\left(\epsilon,p,0,0,\\{\ell_{ij}\\}\right)\big{|}_{\xi_{1}=\dots=\xi_{N}=0}\prod_{ij}{\rm
d}\ell_{ij},$

where $R(x,x^{\prime},\epsilon)$ is some nonsingular factor given by a product
of $\Gamma$ functions. The function
$f\left(\epsilon,p,0,0,\\{\ell_{ij}\\}\right)\big{|}_{\xi_{1}=\dots=\xi_{N}=0}$
is positive and the integral is a particular case of equation (3.13). Thus
this integral can be evaluated, as was discussed before, in a closed form, see
equation (3.12). Then, by Fubini theorem, the integral (3.13) exists, the
integrations can be performed in an arbitrary order, what thus justifies the
result (3.12).

Equation (3.12) therefore leads to

$\displaystyle\big{(}\mathrm{T}_{N}^{B,\epsilon^{\prime}}\varphi,\mathrm{T}_{N}^{B,\epsilon}\varphi\big{)}_{\mathbb{H}_{N}}=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}}\cdots\int_{\mathbb{R}}\varphi(p,x)(\varphi(p,x^{\prime}))^{\star}\,C_{N}^{(\epsilon,\epsilon^{\prime})}(p,x,x^{\prime})$
$\displaystyle\hphantom{\big{(}\mathrm{T}_{N}^{B,\epsilon^{\prime}}\varphi,\mathrm{T}_{N}^{B,\epsilon}\varphi\big{)}_{\mathbb{H}_{N}}=\int_{\mathbb{R}^{+}}\int_{\mathbb{R}}\cdots\int_{\mathbb{R}}}{}\times\mu_{N-1}^{B}(x)\,\mu_{N-1}^{B}(x^{\prime})\,{\rm
d}p\,{\rm d}^{N-1}x\,{\rm d}^{N-1}x^{\prime},$

where we recall that $\varphi$ is a smooth function with a compact support.
For $\epsilon,\epsilon^{\prime}\to 0^{+}$ the integral in the r.h.s. can be
easily estimated, see Appendix A for the details, resulting in

$\displaystyle\big{(}\mathrm{T}_{N}^{B,\epsilon^{\prime}}\varphi,\mathrm{T}_{N}^{B,\epsilon}\varphi\big{)}_{\mathbb{H}_{N}}=K+o(1),$

where

$\displaystyle
K=\int_{\mathbb{R}_{+}}\int_{\mathbb{R}^{N-1}}|\varphi(p,x)|^{2}\mu_{N-1}^{B}(x)\,{\rm
d}p\,{\rm d}x^{N-1}$

and $\mu_{N-1}^{B}$ is as introduced in (3.9).

Since at $\epsilon\to 0^{+}$, one has that
$\big{[}\mathrm{T}_{N}^{B,\epsilon}\varphi\big{]}\rightarrow\big{[}\mathrm{T}_{N}^{B}\varphi\big{]}(z)$
almost everywhere, it follows from Fatou’s theorem that

$\big{\|}\mathrm{T}_{N}^{B}\varphi\big{\|}^{2}_{\mathbb{H}_{N}}\leq\liminf_{\epsilon\to
0^{+}}\big{\|}\mathrm{T}_{N}^{B,\epsilon}\varphi\big{\|}^{2}_{\mathbb{H}_{N}}=K.$

Thus, the function $\mathrm{T}_{N}^{B}\varphi$ belongs to the Hilbert space
$\mathbb{H}_{N}$. Finally, taking into account that
$\big{(}\mathrm{T}_{N}^{B}\varphi,\mathrm{T}_{N}^{B,\epsilon}\varphi\big{)}_{\mathbb{H}_{N}}=K+o(1)$
one derives from $\big{\|}\mathrm{T}_{N}^{B}\varphi-
T_{N}^{B,\epsilon}\varphi\big{\|}^{2}_{\mathbb{H}_{N}}\geq 0$ that
$K\leq\big{\|}\mathrm{T}_{N}^{B}\varphi\big{\|}^{2}$. Thus one gets for the
norm of $\mathrm{T}_{N}^{B}\varphi$,
$\big{\|}\mathrm{T}_{N}^{B}\varphi\big{\|}^{2}_{\mathbb{H}_{N}}=K.$

Finally, the remaining follows from the fact that the set of smooth functions
with a compact support is dense in the Hilbert space $\mathbb{H}_{N}^{B}$. ∎

###  3.2 $\boldsymbol{A_{N}}$ operator

In this section we give a brief description of the eigenfunctions of the
operator $A_{N}$. We start with defining of a layer operator suitable for this
case. Let $\eta$ be a complex vector and

$\displaystyle\eta=(\alpha_{1},\dots,\alpha_{n+1},\beta_{1},\dots,\beta_{n})\in\mathbb{C}^{2n+1}.$
(3.14)

The layer operator $\boldsymbol{\Lambda}_{n+1}^{(\sigma)}(\eta,x)$, which
depends on the vector $\eta$ and two complex parameters, $x$ and $\sigma$,
$\textrm{Im}\,\sigma\geq 0$, maps a function of $n$-complex variables to a
function of $n+1$ variables as follows

$\displaystyle\big{[}\boldsymbol{\Lambda}^{(\sigma)}_{n+1}(\eta,x)f\big{]}(z_{1},\dots,z_{n+1})\\!=\\!D_{\alpha_{n+1}-{\rm
i}x}(z_{n+1},\sigma)\\!\int\\!\cdots\\!\int\\!\\!\prod_{k=1}^{n}\\!D_{\alpha_{k}-{\rm
i}x}(z_{k},w_{k})D_{\beta_{k}+{\rm i}x}(z_{k+1},w_{k})$
$\displaystyle\hphantom{\big{[}\boldsymbol{\Lambda}^{(\sigma)}_{n+1}(\eta,x)f\big{]}(z_{1},\dots,z_{n+1})\\!=}{}\times
f(w_{1},\dots,w_{n})\prod_{a=1}^{n}\mu_{{(\alpha_{a}+\beta_{a})}/2}(w_{a})\,{\rm
d}^{2}w_{1}\cdots{\rm d}^{2}w_{n}.$ (3.15)

The integrals converge provided $\mathop{\rm Re}(\alpha_{k}-{\rm i}x)>0$,
$\mathop{\rm Re}(\beta_{k}+{\rm i}x)>0$ for $k=1,\dots,n$.

Let $\varrho$ be a map: $\mathbb{C}^{2n+1}\mapsto\mathbb{C}^{2n-1}$,

$\displaystyle\varrho\eta=\varrho(\alpha_{1},\dots,\alpha_{n+1},\beta_{1},\dots,\beta_{n})=(\alpha_{1},\dots,\alpha_{n},\beta_{2},\dots,\beta_{n}).$

The layer operators satisfy the following permutation relation [1],

$\displaystyle\boldsymbol{\Lambda}^{(\sigma)}_{n+1}(\eta,x)\,\boldsymbol{\Lambda}^{(\sigma)}_{n}(\varrho\eta,x^{\prime})=\boldsymbol{\Lambda}^{(\sigma)}_{n+1}(\eta,x^{\prime})\,\boldsymbol{\Lambda}^{(\sigma)}_{n}(\varrho\eta,x).$
(3.16)

Let us put

$\displaystyle\eta_{N}\equiv\big{(}s_{1},\dots,s_{N},\bar{s}_{2},\dots,\bar{s}_{N}\big{)}$
(3.17)

and define the function $\Phi^{(\sigma)}_{{x}}(z_{1},\dots,z_{N})$ as

$\displaystyle\Phi^{N}_{\sigma,{x}}(z)=\varkappa_{N}\Big{(}\boldsymbol{\Lambda}^{(\sigma)}_{N}(\eta_{N},x_{1})\boldsymbol{\Lambda}^{(\sigma)}_{N-1}(\varrho\eta_{N},x_{2})\cdots\boldsymbol{\Lambda}^{(\sigma)}_{1}\big{(}\varrho^{N-1}\eta_{N},x_{N}\big{)}\Big{)}.$

By virtue of equation (3.16) $\Phi^{N}_{\sigma,{x}}$ is a symmetric function
of $x_{1},\dots,x_{N}$. It can be shown, see, e.g., [7], that the operator
$A_{N}(x_{1})+\sigma B_{N}(x_{1})$ annihilates the layer operator
$\boldsymbol{\Lambda}^{(\sigma)}_{N}(\eta_{N},x_{1})$,

$\displaystyle\big{(}A_{N}(x_{1})+\sigma
B_{N}(x_{1})\big{)}\boldsymbol{\Lambda}^{(\sigma)}_{N}(\gamma_{N},x_{1})=0$

and, hence, the function $\Phi^{N}_{\sigma,{x}}$ satisfies the equation

$\displaystyle\big{(}A_{N}(u)+\sigma\,B_{N}(u)\big{)}\Phi^{N}_{\sigma,{x}}(z)=(u-x_{1})\cdots(u-x_{N})\Phi^{N}_{\sigma,{x}}(z).$

Thus the function $\Phi^{N}_{{x}}\equiv\Phi^{N}_{\sigma=0,{x}}$ diagonalizes
the operator $A_{N}(u)$. For the separated variables $x$ with small positive
imaginary parts the function $\Phi^{N}_{\sigma,x}$ has a finite norm. Indeed
one can find for the scalar product of two $\Phi^{N}$ functions [1]

$\displaystyle\big{(}\Phi^{N}_{\sigma,y},\Phi^{N}_{\upsilon,x}\big{)}_{\mathbb{H}_{N}}=\bigg{(}\frac{\rm
i}{\sigma-\bar{\upsilon}}\bigg{)}^{{\rm
i}(\bar{Y}-X)}\frac{\prod_{k,j=1}^{N}\Gamma\left({\rm
i}(\bar{y}_{k}-x_{j})\right)}{\prod_{k=1}^{N}\prod_{j=1}^{N}\Gamma(s_{j}-{\rm
i}x_{k})\,\Gamma(\bar{s}_{j}+{\rm i}\bar{y}_{k})}.$ (3.18)

Here $X=\sum_{k=1}^{N}x_{k}$, $Y=\sum_{k=1}^{N}y_{k}$.

For real $x$ the functions $\Phi^{N}_{x}$ are orthogonal to each other (see
Appendix A for more details)

$\displaystyle\big{(}\Phi^{N}_{x^{\prime}},\Phi^{N}_{x}\big{)}$
$\displaystyle=\lim_{\sigma\to 0}\lim_{\epsilon\to
0^{+}}\big{(}\Phi^{N}_{\sigma,x^{\prime}+{\rm
i}\epsilon},\Phi^{N}_{x}\big{)}=\delta^{N}(x,x^{\prime})\,\big{(}\mu_{N}^{A}(x)\big{)}^{-1},$
(3.19)

where

$\displaystyle\mu_{N}^{A}(x)=\frac{1}{(2\pi)^{N}N!}\frac{\prod_{k=1}^{N}\prod_{j=1}^{N}\big{[}\Gamma(s_{j}-{\rm
i}x_{k})\,\Gamma(\bar{s}_{j}+{\rm i}x_{k})\big{]}}{\prod_{j<k}\Gamma({\rm
i}(x_{k}-x_{j}))\,\Gamma({\rm i}(x_{j}-x_{k}))}.$

Upon repeating the argument given in the previous subsection, one can prove
the following statement:

######  Theorem 3.2.

Let $\mathbb{H}_{N}^{A}$ be the Hilbert space of symmetric functions

$\displaystyle\mathbb{H}_{N}^{A}=L^{2}_{\rm sym}\big{(}\mathbb{R}^{N},{\rm
d}\mu^{A}_{N}(x)\big{)},\qquad{\rm d}\mu^{A}_{N}(x)=\mu^{A}_{N}(x)\,{\rm
d}^{N}x.$

The transformation $\mathrm{T}_{N}^{A}$ defined for smooth, compactly
supported functions $\chi$ on $\mathbb{R}^{N}{:}$

$\displaystyle\Psi_{\chi}(z)=\big{[}\mathrm{T}_{N}^{A}\chi\big{]}(z)=\int_{\mathbb{R}^{N}}\chi(x)\,\Phi^{N}_{{x}}(z)\,\mu_{N}^{A}(x)\,{\rm
d}^{N}x$ (3.20)

extends into a linear isometry from $\mathbb{H}_{N}^{A}$ into
$\mathbb{H}_{N}$. In particular it has unit operator norm
$\big{\|}\mathrm{T}_{N}^{A}\big{\|}=1$ and satisfies

$\displaystyle\|\mathrm{T}_{N}^{A}\chi\|^{2}_{\mathbb{H}_{N}^{\phantom{A}}}=\|\chi\|^{2}_{\mathbb{H}_{N}^{A}}=\int_{\mathbb{R}^{N}}|\chi(x)|^{2}\,\mu_{N}^{A}(x)\,{\rm
d}^{N}x.$

It will be show in Section 4 that $\mathrm{T}_{N}^{A}$ is, in fact, an unitary
map between the corresponding Hilbert spaces.

###  3.3 $\boldsymbol{\mathbb{B}_{N}}$ operator

Let us construct eigenfunctions of the operator $\mathbb{B}_{N}(u)$, see
equation (3.3). It can be shown [6] that $\mathbb{B}_{N}(u)=(2u+{\rm
i})\widehat{\mathbb{B}}_{N}(u)$, where
$\widehat{\mathbb{B}}_{N}(u)=\widehat{\mathbb{B}}_{N}(-u)$. In order to write
down eigenfunctions of $\widehat{\mathbb{B}}_{N}(u)$ we define the
corresponding layer operator,
$\widetilde{\boldsymbol{\Lambda}}_{n+1}(\eta,x)$, where
$\eta\in\mathbb{C}^{2n+1}$, see equation (3.14) and $x\in\mathbb{C}$ is a
spectral parameter, maps function of $n$-complex variables to a function of
$n+1$ variables. The layer operator is written in terms of the operators
$\boldsymbol{\Lambda}_{n}^{(\sigma)}$, defined in the previous section,
equation (3.15), as follows

$\displaystyle\big{[}\widetilde{\boldsymbol{\Lambda}}_{n+1}(\eta,x)f\big{]}(z)=\\!\int\big{[}\boldsymbol{\Lambda}^{(\sigma)}_{n+1}(\eta,x)\boldsymbol{\Lambda}^{(\sigma)}_{n}(\varrho\eta,-x)f\big{]}(z_{1},\dots,z_{n+1})\mu_{{(\alpha_{n}+\alpha_{n+1})}/2}(\sigma)\,{\rm
d}^{2}\sigma.$ (3.21)

Above, the product of two layer operators,
$\boldsymbol{\Lambda}^{(\sigma)}_{n+1}(\eta,x)\boldsymbol{\Lambda}^{(\sigma)}_{n}(\varrho\eta,-x)$,
maps a function of $n-1$ variables $(w_{1},\dots,w_{n-1})$,
$f(w_{1},\dots,w_{n-1},\sigma)$, to a function of $n+1$ variables,
$z=(z_{1},\dots,z_{n+1})$, as indicated in the above formula. By virtue of
(3.16) the layer operator $\widetilde{\boldsymbol{\Lambda}}_{n+1}$ is an even
function of $x$,

$\displaystyle\widetilde{\boldsymbol{\Lambda}}_{n+1}(\eta,x)=\widetilde{\boldsymbol{\Lambda}}_{n+1}(\eta,-x).$

Let $\omega$ be a map: $\mathbb{C}^{2n+1}\mapsto\mathbb{C}^{2n-1}$, defined as

$\displaystyle\omega\,\gamma=\omega(\alpha_{1},\dots,\alpha_{n+1},\beta_{1},\dots,\beta_{n})=(\alpha_{1},\dots,\alpha_{n},\beta_{3},\dots,\beta_{n},\alpha_{n+1}).$

The layer operators (3.21) satisfy the permutation relation [6]

$\displaystyle\widetilde{\boldsymbol{\Lambda}}_{n+1}(\gamma,x)\widetilde{\boldsymbol{\Lambda}}_{n}(\omega\gamma,x^{\prime})=\widetilde{\boldsymbol{\Lambda}}_{n+1}(\gamma,x^{\prime})\widetilde{\boldsymbol{\Lambda}}_{n}(\omega\gamma,x)$

and is nullified by the operator $\widehat{\mathbb{B}}_{N}(x)$

$\displaystyle\widehat{\mathbb{B}}_{N}(x)\widetilde{\boldsymbol{\Lambda}}_{N}(\eta_{N},x)=0,$
(3.22)

where the vector $\eta_{N}$ is given by equation (3.17).

Given $z=(z_{1},\dots,z_{N})$, define the function

$\displaystyle\Upsilon^{N}_{p,{x}}(z)=\kappa_{N}p^{\boldsymbol{S}-\frac{1}{2}}\Big{[}\widetilde{\boldsymbol{\Lambda}}_{N}(\gamma_{N},x_{1})\widetilde{\boldsymbol{\Lambda}}_{N-1}(\omega\gamma_{N},x_{2})\cdots\widetilde{\boldsymbol{\Lambda}}_{2}(\omega^{N-2}\gamma_{N},x_{N-1})\cdot
E_{p}\Big{]}(z),$

where $\boldsymbol{S}=\sum_{k=1}^{N}\boldsymbol{s}_{k}$,
$x=(x_{1},\dots,x_{N-1})$ and the normalisation factor is

$\displaystyle\kappa_{N}^{-1}=\bigg{(}\prod_{k=1}^{N}\Gamma(s_{k}+\bar{s}_{k})\bigg{)}^{1/2}\prod_{1\leq
i<j\leq
N}\Gamma(s_{i}+s_{j})\,\Gamma(s_{i}+\bar{s}_{j})=\varkappa_{N}\prod_{1\leq
i<j\leq N}\Gamma(s_{i}+s_{j}).$

The function $\Upsilon^{N}_{p,x}$ is a symmetric even function of
$x_{1},\dots,x_{N-1}$ which is well defined for $|\mathop{\rm
Im}(x_{k})|\allowbreak<\min_{k}s_{k}$. By virtue of equation (3.22), the
function $\Upsilon^{N}_{p,{x}}$ diagonalizes the operator
$\widehat{\mathbb{B}}_{N}$

$\displaystyle\widehat{\mathbb{B}}_{N}(u)\Upsilon^{N}_{p,{x}}(z)=p\big{(}u^{2}-x_{1}^{2}\big{)}\cdots\big{(}u^{2}-x_{N-1}^{2}\big{)}\,\Upsilon^{N}_{p,{x}}(z).$

The functions $\Upsilon^{N}_{p,x}$ are orthogonal to each other for real
separated variables, $x\in(\mathbb{R}^{+})^{N-1}$. Namely,

$\displaystyle\big{(}\Upsilon^{N}_{q,{y}},\Upsilon^{N}_{p,{x}}\big{)}_{\mathbb{H}_{N}}=\delta(p-q)\delta^{N-1}(x,y)\big{(}\mu_{N-1}^{\mathbb{B}}(x)\big{)}^{-1},$

where

$\displaystyle\mu_{N-1}^{\mathbb{B}}(x)=\frac{1}{(2\pi)^{N-1}(N-1)!}\frac{\prod_{j=1}^{N}\prod_{k=1}^{N-1}|\Gamma(s_{j}+{\rm
i}x_{k})\,\Gamma(s_{j}-{\rm i}x_{k})|^{2}}{\prod_{n=1}^{N-1}|\Gamma(2{\rm
i}x_{n})|^{2}\prod_{j<k}|\Gamma({\rm i}(x_{k}+x_{j}))\,\Gamma({\rm
i}(x_{k}-x_{j}))|^{2}}.$

We are now in position to formulate the theorem:

######  Theorem 3.3.

Let $\mathbb{H}_{N}^{\mathbb{B}}$ be the Hilbert space

$\displaystyle\mathbb{H}_{N}^{\mathbb{B}}=L^{2}(\mathbb{R}^{+})\otimes
L^{2}_{\rm sym}\big{(}(\mathbb{R}^{+})^{N-1},{\rm
d}\mu^{\mathbb{B}}_{N-1}(x)\big{)},\qquad{\rm
d}\mu^{\mathbb{B}}_{N-1}(x)=\mu^{\mathbb{B}}_{N-1}(x){\rm d}^{N-1}x.$

The transformation $\mathrm{T}_{N}^{\mathbb{B}}$ defined for smooth, compactly
supported functions $\phi$ on $\mathbb{R}^{+}\times(\mathbb{R}^{+})^{N-1}$
that are symmetric in respect to the last $N-1$ variables as

$\displaystyle\big{[}\mathrm{T}_{N}^{\mathbb{B}}\phi\big{]}(z)=\int_{(\mathbb{R}^{+})^{N}}\phi(p,x)\,\Upsilon^{N}_{p,{x}}(z)\,\mu_{N-1}^{\mathbb{B}}(x)\,{\rm
d}p\,{\rm d}^{N-1}x$

extends to a linear isometry from $\mathbb{H}_{N}^{\mathbb{B}}$ into
$\mathbb{H}_{N}$. In particular, it has unit operator norm
$\big{\|}\mathrm{T}_{N}^{\mathbb{B}}\big{\|}=1$ and satisfies

$\displaystyle\big{\|}\mathrm{T}_{N}^{\mathbb{B}}\phi\big{\|}^{2}_{\mathbb{H}_{N}}=\|\phi\|^{2}_{\mathbb{H}_{N}^{\mathbb{B}}}=\int_{\mathbb{R}_{+}^{N}}|\phi(p,x)|^{2}\,\mu_{N-1}^{\mathbb{B}}(x)\,{\rm
d}p\,{\rm d}^{N-1}x.$

We will show in Section 4 that $\mathrm{T}_{N}^{\mathbb{B}}$ is an unitary
operator.

##  4 Completeness

In the previous section we constructed three systems of functions,
$\Psi_{x,p}$, $\Phi_{x}$ and $\Upsilon_{x,p}$. They allow one to define the
linear operators, $\mathrm{T}_{N}^{\alpha}$, $\alpha=\\{B,A,{\mathbb{B}}\\}$,
which map the Hilbert spaces $\mathbb{H}_{N}^{\alpha}$ to the Hilbert space
$\mathbb{H}_{N}$. For $N=1$ the transformations $\mathrm{T}_{1}^{B}$ and
$\mathrm{T}_{1}^{\mathbb{B}}$ are the Fourier transform and
$\mathrm{T}_{1}^{A}$ is the Mellin transform. Thus, these transformations are
unitary maps and, in particular,
$\mathrm{T}_{1}^{\alpha}\mathbb{H}_{1}^{\alpha}=\mathbb{H}_{1}$.

In order to prove the unitarity of the maps $\mathrm{T}^{\alpha}_{N}$ for
arbitrary $N$ we use induction on $N$. Namely, we will show that if the map
$\mathrm{T}_{N}^{A}$ is unitary then the maps $\mathrm{T}_{N}^{B}$,
$\mathrm{T}_{N}^{\mathbb{B}}$ and $\mathrm{T}_{N+1}^{B}$ are also unitary. We
also show that the unitarity of the map $\mathrm{T}_{N}^{B}$ implies the one
for $\mathrm{T}_{N}^{A}$. Schematically it is shown on the diagram below

${B_{N}}$${A_{N}}$${B_{N+1}}$${A_{N+1}}$${\mathbb{B}_{N}}$${\mathbb{B}_{N+1}.}$

The backward arrow is dispensable here, but we consider it first because its
proof is most transparent and all other proofs follow the same scheme.

It was shown in the previous section that
$\mathcal{R}\big{(}\mathrm{T}_{N}^{B}\big{)}$ is a closed subspace of the
Hilbert space $\mathbb{H}_{N}$. If this subspace coincides with the whole
$\mathbb{H}_{N}$ then the orthogonal complement is trivial,
$\mathcal{R}\big{(}\mathrm{T}_{N}^{B}\big{)}^{\perp}=0$. Since
$\mathcal{R}\big{(}\mathrm{T}_{N}^{B}\big{)}^{\perp}=\ker\big{(}\mathrm{T}_{N}^{B}\big{)}^{\star}$
it is enough to prove that the kernel of the adjoint operator
$\big{(}\mathrm{T}_{N}^{B}\big{)}^{\star}$ is empty. In order to do it let us
consider a linear map from $\mathbb{H}_{N}^{A}$ to $\mathbb{H}_{N}^{B}$
defined by

$\displaystyle\mathrm{S}_{BA}=\big{(}\mathrm{T}_{N}^{B}\big{)}^{\star}\,\mathrm{T}_{N}^{A}.$
(4.1)

Since the map $\mathrm{T}_{N}^{A}$ is an isometry, by assumption, it maps
$\ker(\mathrm{S}_{BA})\mapsto\ker\big{(}\mathrm{T}_{N}^{B}\big{)}^{\star}$.
Our immediate aim is to show that $\ker(\mathrm{S}_{BA})=0$.

We prove the following statement:

######  Lemma 4.1.

Let $\mathrm{S}_{BA}$ be the operator from $\mathbb{H}_{N}^{A}$ to
$\mathbb{H}_{N}^{B}$ defined in equation (4.1). Then, for any
$\chi\in\mathbb{H}_{N}^{A}$ the following holds

$\displaystyle\|\mathrm{S}_{BA}\chi\|^{2}_{\mathbb{H}_{N}^{B}}=\|\chi\|^{2}_{\mathbb{H}_{N}^{A}}.$
(4.2)

###### Proof.

First, we calculate the action of the operator $\mathrm{S}_{BA}$ on the space
of smooth functions with a compact support, $\chi(x)$. The action of
$\mathrm{T}_{N}^{A}$ on a function $\chi(x)$ is given by equation (3.20). In
full similarity with the construction in Section 3.1.2, we define the
regularized function $\mathrm{T}_{N}^{A,\sigma,\epsilon}\chi$ obtained by
replacing $\Phi^{N}_{x}$ in (3.20) by $\Phi^{N}_{\sigma,x+{\rm i}\epsilon}$,
where $x=(x_{1},\dots,x_{N})$ and
$\epsilon=(\epsilon_{1},\dots,\epsilon_{N})$, all $\epsilon_{k}>0$,

$\displaystyle\mathrm{T}_{N}^{A,\sigma,\epsilon}\chi(z)=\int_{\mathbb{R}^{N}}\chi(x)\,\Phi^{N}_{{\sigma,x+{\rm
i}\epsilon}}(z)\,{\rm d}\mu_{N}^{A}(x).$ (4.3)

As $\sigma,\epsilon\to 0$ one has that
$\big{\|}\mathrm{T}_{N}^{A,\sigma,\epsilon}\chi-\mathrm{T}_{N}^{A}\chi\big{\|}_{\mathbb{H}_{N}}\to
0$. Since $\|\mathrm{T}_{N}^{B}\|=1$, the adjoint to $\mathrm{T}_{N}^{B}$ is a
bounded operator which acts on a vector $\Psi$ by projecting it on the
eigenfunction $\Psi^{N}_{p,y}$, see equation (3.10),

$\displaystyle\big{(}\mathrm{T}_{N}^{B}\big{)}^{\star}\Psi=\big{(}\Psi^{N}_{p,y},\Psi\big{)}_{\mathbb{H}_{N}}\equiv\varphi(p,y).$

Thus we write

$\displaystyle\varphi(p,y)$
$\displaystyle\equiv[\mathrm{S}_{BA}\chi](p,y)=\big{[}\big{(}\mathrm{T}_{N}^{B}\big{)}^{\star}T_{N}^{A}\chi\big{]}(p,y)=\lim_{\sigma,\epsilon\to
0}\big{[}\big{(}\mathrm{T}_{N}^{B}\big{)}^{\star}\mathrm{T}_{N}^{A,\sigma,\epsilon}\chi\big{]}(p,y)$
$\displaystyle=\lim_{\sigma,\epsilon\to
0}\big{(}\Psi^{N}_{p,y},T_{N}^{A,\sigma,\epsilon}\chi\big{)}_{\mathbb{H}_{N}}.$

Moreover, one has

$\displaystyle\|\varphi\|^{2}_{\mathbb{H}_{N}^{B}}=\lim_{\sigma\to
0}\lim_{\epsilon\to
0^{+}}\|\varphi_{\sigma,\epsilon}\|^{2}_{\mathbb{H}_{N}^{B}},\qquad\varphi_{\sigma,\epsilon}(p,y)=\big{(}\Psi^{N}_{p,y},T_{N}^{A,\sigma,\epsilon}\chi\big{)}_{\mathbb{H}_{N}}.$
(4.4)

The further analysis depends on the remarkable fact that the scalar product of
the functions $\Psi^{N}_{p,y}$ and $\Phi^{N}_{\sigma,x+{\rm i}\epsilon}$ can
be obtained in a closed form [1]:

$\displaystyle\big{(}\Psi^{N}_{p,y},\Phi^{N}_{\sigma,x+{\rm
i}\epsilon}\big{)}_{\mathbb{H}_{N}}$ $\displaystyle\qquad{}=p^{-1/2-{\rm
i}\Xi-{\rm i}X+\mathcal{E}}{\rm e}^{-{\rm
i}p\bar{\sigma}}\frac{\prod_{k=1}^{N}\prod_{j=1}^{N-1}\Gamma({\rm
i}(y_{j}-x_{k})+\epsilon_{k}))}{\prod_{j=1}^{N}\left(\prod_{k=1}^{N-1}\Gamma(\bar{s}_{j}+{\rm
i}y_{k})\prod_{k=1}^{N}\Gamma(s_{j}-{\rm i}x_{k}+\epsilon_{k})\right)},$ (4.5)

where $X=\sum_{k=1}^{N}x_{k}$ and $\Xi=\sum_{k=1}^{N}\xi_{k}$, and
$\mathcal{E}=\sum_{k=1}^{N}\epsilon_{k}$. That is

$\displaystyle\varphi_{\sigma,\epsilon}(p,y)=\frac{p^{-1/2-{\rm
i}\Xi+\mathcal{E}}{\rm e}^{-{\rm
i}p\bar{\sigma}}}{\prod_{j=1}^{N}\prod_{k=1}^{N-1}\Gamma(\bar{s}_{j}+{\rm
i}y_{k})}$
$\displaystyle\hphantom{\varphi_{\sigma,\epsilon}(p,y)=}{}\times\int_{\mathbb{R}^{N}}\frac{\prod_{k=1}^{N}\prod_{j=1}^{N-1}\Gamma({\rm
i}(y_{j}-x_{k})+\epsilon_{k})}{\prod_{k,j=1}^{N}\Gamma(s_{j}-{\rm
i}x_{k}+\epsilon_{k})}p^{-{\rm i}X}\chi(x)\,{\rm d}\mu_{N}^{A}(x).$ (4.6)

By assumption the function $\chi$ is nonzero only in a compact region.
Therefore the function $\varphi_{\sigma,\epsilon}(p,y)$ grows no faster that
some power of $y$ for large $y$ while at large $p$ it decays exponentially
fast $\sim\exp\\{-\mathop{\rm Im}(\sigma)p\\}$). Taking into account that the
measure $\mu_{N-1}^{B}(y)$ decays exponentially fast for large $y$

$\displaystyle\mu^{B}_{N-1}(y)\simeq\frac{(4\pi)^{\frac{N(N-1)}{2}}}{2^{N-1}(N-1)!}\prod_{1\leq
i<j\leq N-1}{y_{ij}}\sinh\pi
y_{ij}\,\prod_{j=1}^{N}\prod_{k=1}^{N-1}y_{k}^{2\boldsymbol{s}_{j}-1}{\rm
e}^{-\pi|y_{k}+\xi_{j}|},$

one concludes that the normalisation integral for $\varphi_{\sigma,\epsilon}$
converges

$\displaystyle\|\varphi_{\sigma,\epsilon}\|^{2}_{\mathbb{H}_{N}^{B}}=\int_{\mathbb{R}_{+}}\int_{\mathbb{R}^{N-1}}|\varphi_{\sigma,\epsilon}(p,y)|^{2}\,{\rm
d}p\,{\rm d}\mu_{N-1}^{B}(y)<\infty.$

Moreover, substituting the expression for $\varphi_{\sigma,\epsilon}$,
equation (4.6), one can change the order of integration and integrate first
over $p$ and $y$. The momentum integral is trivial and produces the factor
$\Gamma({\rm i}(X^{\prime}-X)+2\mathcal{E})(2\mathop{\rm Im}\sigma)^{{\rm
i}(X-X^{\prime})-2\mathcal{E}}$, while the integral over $y$ can be calculated
in closed form [14, Theorem 5.1], see also equation (2.6). Namely,

$\displaystyle\frac{1}{(N-1)!}\int_{\mathbb{R}^{N}}\frac{\prod_{k=1}^{N}\prod_{j=1}^{N-1}\Gamma({\rm
i}(y_{j}-x_{k})+\epsilon_{k})\,\Gamma({\rm
i}(x^{\prime}_{k}-y_{j})+\epsilon_{k})}{\prod_{j<k}\Gamma({\rm
i}(y_{k}-y_{j}))\,\Gamma({\rm i}(y_{j}-y_{k}))}\prod_{m=1}^{N-1}\frac{{\rm
d}y_{m}}{2\pi}$
$\displaystyle\hphantom{\frac{1}{(N-1)!}}{}=\frac{\prod_{k,j=1}^{N}\Gamma({\rm
i}(x^{\prime}_{k}-x_{j})+\epsilon_{k}+\epsilon_{j})}{\Gamma({\rm
i}(X^{\prime}-X)+2\mathcal{E})}.$ (4.7)

Thus we get for the norm of $\varphi_{\sigma,\epsilon}$

$\displaystyle\|\varphi_{\sigma,\epsilon}\|^{2}_{\mathbb{H}_{N}^{B}}=\int_{\mathbb{R}^{2N}}\left((2\mathop{\rm
Im}\sigma)^{{\rm
i}(X-X^{\prime})-2\mathcal{E}}\frac{\prod_{k,j=1}^{N}\Gamma({\rm
i}(x^{\prime}_{k}-x_{j})+\epsilon_{k}+\epsilon_{j})}{\prod_{k,j=1}^{N}\Gamma\left(\bar{s}_{j}+{\rm
i}x^{\prime}_{k}+\epsilon_{k}\right)\,\Gamma(s_{j}-{\rm
i}x_{k}+\epsilon_{k})}\right)$
$\displaystyle\hphantom{\|\varphi_{\sigma,\epsilon}\|^{2}_{\mathbb{H}_{N}^{B}}=}{}\times\chi(x)(\chi(x^{\prime}))^{*}\,{\rm
d}\mu_{N}^{A}(x)\,{\rm d}\mu_{N}^{A}(x^{\prime}).$ (4.8)

Note that the expression in the bracket is nothing else as the scalar product,
$\big{(}\Phi^{N}_{\sigma,x^{\prime}+{\rm i}\epsilon},\Phi^{N}_{\sigma,x+{\rm
i}\epsilon}\big{)}_{\mathbb{H}_{N}}$, see equation (3.18). Finally, taking
into account (3.19), see also Appendix A, we obtain that for any smooth
function $\chi$ with a compact support

$\displaystyle\|\varphi\|^{2}_{\mathbb{H}_{N}^{B}}=\lim_{\sigma\to
0}\lim_{\epsilon\to
0^{+}}\|\varphi_{\sigma,\epsilon}\|^{2}_{\mathbb{H}_{N}^{B}}=\|\chi\|^{2}_{\mathbb{H}_{N}^{A}}=\int_{\mathbb{R}^{N}}\,|\chi(x)|^{2}\,{\rm
d}\mu_{N}^{A}(x)$

or

$\displaystyle\|\mathrm{S}_{BA}\chi\|^{2}_{\mathbb{H}_{N}^{B}}=\|\chi\|^{2}_{\mathbb{H}_{N}^{A}}.$

Since the space of smooth, compactly supported functions is dense in
$\mathbb{H}_{N}^{A}$ this equation holds on the whole Hilbert space. ∎

The identity (4.2) implies that $\ker\mathrm{S}_{BA}=0$ and hence
$\mathcal{R}\big{(}\mathrm{T}_{N}^{B}\big{)}=\mathbb{H}_{N}$, which guarantees
the unitarity of the map $\mathrm{T}_{N}^{B}$.

The proof of the unitarity of the maps $\mathrm{T}_{N}^{\mathbb{B}}$ and
$\mathrm{T}_{N+1}^{B}$ follows the same lines and is based on the following
result:

######  Lemma 4.2.

Let $\mathrm{S}_{\mathbb{B}A}$ and $\mathrm{S}_{B}$ be maps from
$\mathbb{H}_{N}^{A}\mapsto\mathbb{H}^{\mathbb{B}}_{N}$ and
$\mathbb{H}^{A}_{N}\otimes\mathbb{H}_{1}^{A}\mapsto\mathbb{H}^{B}_{N+1}$
defined as follows

$\displaystyle\mathrm{S}_{\mathbb{B}A}=\big{(}\mathrm{T}_{N}^{\mathbb{B}}\big{)}^{\star}\mathrm{T}_{N}^{A}\qquad\text{and}\qquad\mathrm{S}_{B}=\big{(}\mathrm{T}_{N+1}^{B}\big{)}^{\star}\,\big{(}\mathrm{T}_{N}^{A}\otimes\mathrm{T}_{1}^{A}\big{)}.$

Provided the map
$\mathrm{T}_{N}^{A}\colon\mathbb{H}^{A}_{N}\mapsto\mathbb{H}_{N}$ is unitary
the following identities,

$\displaystyle\|\mathrm{S}_{\mathbb{B}A}\chi\|^{2}_{\mathbb{H}_{N}^{\mathbb{B}}}=\|\chi\|^{2}_{\mathbb{H}_{N}^{A}}\qquad\text{and}\qquad\|\mathrm{S}_{B}\chi^{\prime}\|^{2}_{\mathbb{H}_{N+1}^{B}}=\|\chi^{\prime}\|^{2}_{\mathbb{H}_{N}^{A}\otimes\mathbb{H}_{1}^{A}},$

hold for any $\chi\in\mathbb{H}_{N}^{A}$,
$\chi^{\prime}\in\mathbb{H}_{N}^{A}\otimes\mathbb{H}_{1}^{A}$.

###### Proof.

In the proof of these assertions, the main difference from the proof of Lemma
4.1 lies in the type of the $\Gamma$-integrals arising in the process. We
briefly discuss these differences below.

For the $S_{\mathbb{B}A}$ operator the problem is reduced to calculating the
norm of the function

$\displaystyle\phi_{\sigma,\epsilon}(p,y)=\big{(}\Upsilon^{N}_{p,y},T_{N}^{A,\sigma,\epsilon}\chi\big{)}_{\mathbb{H}_{N}},$

which is an analogue of the function $\varphi_{\sigma,\epsilon}$, see equation
(4.4). The relevant scalar product takes the form333For the homogeneous chain
this scalar product was calculated in [9] and its extension to the general
case is straightforward.

$\displaystyle(\Upsilon^{N}_{p,y},\Phi^{N}_{\sigma,x+{\rm
i}\epsilon})_{\mathbb{H}_{N}}=p^{-1/2-{\rm i}\Xi-{\rm i}X+\mathcal{E}}{\rm
e}^{-{\rm i}p\bar{\sigma}}\frac{1}{\prod_{1\leq k<j\leq N}\Gamma(-{\rm
i}(x_{k}+x_{j})+\epsilon_{k}+\epsilon_{j})}$
$\displaystyle\hphantom{(\Upsilon^{N}_{p,y},\Phi^{N}_{\sigma,x+{\rm
i}\epsilon})_{\mathbb{H}_{N}}=}{}\times\frac{\prod_{k=1}^{N}\prod_{j=1}^{N-1}\Gamma(-{\rm
i}(x_{k}\pm
y_{j})+\epsilon_{k})}{\prod_{k=1}^{N}\left(\prod_{j=1}^{N-1}\Gamma(\bar{s}_{k}\pm{\rm
i}y_{j})\right)\left(\prod_{m=1}^{N}\Gamma(s_{k}-{\rm
i}x_{m}+\epsilon_{m})\right)}.$

Above, the symbol $\pm$ stands for

$\displaystyle f(a\pm b)\equiv f(a+b)f(a-b).$

Calculating the norm

$\displaystyle\|\phi_{\sigma,\epsilon}\|^{2}_{\mathbb{H}_{N}^{\mathbb{B}}}=\int_{\mathbb{R}_{+}^{N}}|\phi_{\sigma,\epsilon}(p,y)|^{2}\,{\rm
d}p\,{\rm d}\mu_{N-1}^{\mathbb{B}}(y)$

one substitutes the function $\phi_{\sigma,\epsilon}$ in the form

$\displaystyle\phi_{\sigma,\epsilon}(p,y)=\int_{\mathbb{R}_{+}^{N}}\big{(}\Upsilon_{p,y},\Phi^{N}_{\sigma,x+{\rm
i}\epsilon}\big{)}_{\mathbb{H}_{N}}\chi(x)\,{\rm d}p\,{\rm d}\mu_{N}^{A}(x).$

One can change the order of integrations and take the integral over $p$ and
$y$ first. The integral over $p$ is exactly the same while the other integral
takes the form of second Gustafson integral [14, Theorem 9.3],

$\displaystyle\frac{1}{(N-1)!}\int_{\mathbb{R}^{N}}\frac{\prod_{k=1}^{N}\prod_{j=1}^{N-1}\Gamma({\rm
i}(\pm y_{j}-x_{k})+\epsilon_{k})\,\Gamma({\rm i}(x^{\prime}_{k}\pm
y_{j})+\epsilon_{k})}{\prod_{m=1}^{N-1}\Gamma(\pm 2{\rm
i}y_{m})\prod_{j<k}\Gamma({\rm i}(y_{k}\pm y_{j}))\,\Gamma(-{\rm i}(y_{k}\pm
y_{j}))}\prod_{m=1}^{N-1}\frac{{\rm d}y_{m}}{4\pi}$
$\displaystyle\qquad{}=\frac{\prod_{k,j=1}^{N}\Gamma({\rm
i}(x^{\prime}_{k}-x_{j})+\epsilon_{kj})\prod_{1\leq m<n\leq N}\Gamma({\rm
i}(x^{\prime}_{n}+x^{\prime}_{m})+\epsilon_{nm})\Gamma(-{\rm
i}(x_{n}+x_{m})+\epsilon_{nm})}{\Gamma({\rm i}(X^{\prime}-X)+2\mathcal{E})},$

where $\epsilon_{kj}=\epsilon_{j}+\epsilon_{k}$. We also extended the integral
over $y_{k}$ from the positive half-axis to the real line using the symmetry
of the integrand with respect to the reflection $y_{k}\to-y_{k}$. Finally,
collecting all factors, one finds that the norm
$\|\phi_{\sigma,\epsilon}\|^{2}_{\mathbb{H}_{N}^{\mathbb{B}}}$ is given by the
expression on the r.h.s. of equation (4.8). Repeating all the same arguments
as in the previous case we conclude that

$\displaystyle\|\mathrm{S}_{\mathbb{B}A}\chi\|^{2}_{\mathbb{H}_{N}^{\mathbb{B}}}=\|\chi\|^{2}_{\mathbb{H}_{N}^{A}}$

for any $\chi\in\mathbb{H}_{N}^{A}$.

Now let us show that the map $\mathrm{T}_{N+1}^{B}$ is unitary. In this case
we consider the map

$\displaystyle\mathrm{S}_{B}=\big{(}\mathrm{T}_{N+1}^{B}\big{)}^{\star}\big{(}\mathrm{T}_{N}^{A}\otimes\mathrm{T}_{1}^{A}\big{)}.$

The last factor in the above equation is the unitary map from
$\mathbb{H}_{N}^{A}\otimes\mathbb{H}_{1}^{A}$ to
$\mathbb{H}_{N+1}=\mathbb{H}_{N}\otimes\mathcal{H}_{N+1}$, where
$\mathcal{H}_{N+1}$ is the Hilbert space of holomorphic functions in the upper
complex half-plane discussed around equation (3.1). Namely, similar to
equation (4.3) we define

$\displaystyle\Psi^{\sigma,\underline{\epsilon}}_{\chi}(\underline{z})=\int_{\mathbb{R}^{N+1}}\chi(\underline{x})\,\Phi^{N}_{{\sigma,x+{\rm
i}\epsilon}}(z)\Phi^{1}_{{\sigma,x_{N+1}+{\rm
i}\epsilon_{N+1}}}(z_{N+1})\,{\rm d}\mu_{N}^{A}(x)\,{\rm
d}\mu_{1}^{A}(x_{N+1}).$

Here $z$, $x$, $\epsilon$ and $\underline{z}$, $\underline{x}$,
$\underline{\epsilon}$ are $N$ and ($N+1)$-dimensional vectors, respectively,
e.g., $x=(x_{1},\dots,x_{N})$,
$\underline{x}=(\underline{x}_{1},\dots,\underline{x}_{N+1})$, etc. Note also
that the parameter $\sigma$ is the same for the functions $\Phi^{N}$ and
$\Phi^{1}$. Completely similar to the previous consideration one can show that

$\displaystyle\lim_{\sigma\to 0}\lim_{\underline{\epsilon}\to
0^{+}}\Psi^{\sigma,\underline{\epsilon}}_{\chi}=\Psi_{\chi}\equiv\big{(}\mathrm{T}_{N}^{A}\otimes\mathrm{T}_{1}^{A}\big{)}\chi.$

Again we define the function

$\displaystyle\varphi_{\sigma,\underline{\epsilon}}(p,\underline{y})=\big{(}\Psi^{N}_{p,\underline{y}},\Psi^{\sigma,\underline{\epsilon}}_{\chi}\big{)}\equiv\big{(}\mathrm{T}_{N+1}^{B}\big{)}^{\star}\Psi^{\sigma,\underline{\epsilon}}_{\chi},$

where $\underline{y}=(y_{1},\dots,y_{N})$. The scalar product of the function
$\Psi^{N}_{p,\underline{y}}$ and $\Phi^{N}_{{\sigma,x+{\rm
i}\epsilon}}\otimes\Phi^{1}_{\sigma,x_{N+1}+{\rm i}\epsilon_{N+1}}$ takes the
form (see, e.g., [9])

$\displaystyle\big{(}\Psi^{N}_{p,\underline{y}},\Phi^{N}_{{\sigma,x}}\otimes\Phi^{1}_{\sigma,x_{N+1}}\big{)}_{\mathbb{H}_{N}}=\frac{1}{\sqrt{p}}p^{-{\rm
i}\underline{X}-{\rm i}\Xi+{\rm i}\xi_{N+1}}{\rm e}^{-{\rm
i}p\bar{\sigma}}\prod_{k=1}^{N}\frac{\Gamma(\bar{s}_{k}+s_{N+1})}{\Gamma(s_{k}+\bar{s}_{N+1})}\frac{1}{\Gamma(s_{N+1}-{\rm
i}x_{N+1})}$
$\displaystyle\hphantom{\big{(}\Psi^{N}_{p,\underline{y}},\Phi^{N}_{{\sigma,x}}\otimes\Phi^{1}_{\sigma,x_{N+1}}\big{)}_{\mathbb{H}_{N}}=}{}\times\prod_{k,j=1}^{N}\frac{\Gamma({\rm
i}(y_{j}-x_{k}))}{\Gamma(\bar{s}_{k}+{\rm i}y_{j})\,\Gamma(s_{j}-{\rm
i}x_{k})}$
$\displaystyle\hphantom{\big{(}\Psi^{N}_{p,\underline{y}},\Phi^{N}_{{\sigma,x}}\otimes\Phi^{1}_{\sigma,x_{N+1}}\big{)}_{\mathbb{H}_{N}}=}{}\times\prod_{k=1}^{N}\frac{1}{\Gamma(s_{N+1}-{\rm
i}y_{k})}\frac{\Gamma(-{\rm i}(y_{k}+x_{N+1}))}{\Gamma(-{\rm
i}(x_{k}+x_{N+1}))}.$

Calculating the norm of $\varphi_{\sigma,\underline{\epsilon}}$ we change the
order of integration and first take the integral over $y$. It takes the form
of $N$-fold Gustafson’s integral [14, Theorem 5.1] that we encountered
earlier, see equation (4.7). After some algebra we obtain

$\displaystyle\|\varphi_{\sigma,\underline{\epsilon}}\|^{2}_{\mathbb{H}_{N+1}^{B}}$
$\displaystyle=\int_{\mathbb{R}^{2N+2}}\biggl{(}(2\mathop{\rm Im}\sigma)^{{\rm
i}(\underline{X}-\underline{X}^{\prime})-2\underline{\mathcal{E}}}\frac{\prod_{k,j=1}^{N}\Gamma\big{(}{\rm
i}(x^{\prime}_{k}-x_{j})+\epsilon_{k}+\epsilon_{j}\big{)}}{\prod_{k,j=1}^{N}\Gamma(\bar{s}_{j}+{\rm
i}x^{\prime}_{k}+\epsilon_{k})\,\Gamma(s_{j}-{\rm i}x_{k}+\epsilon_{k})}$
$\displaystyle\quad\times\frac{\Gamma\big{(}{\rm
i}(x^{\prime}_{N+1}-x_{N+1})+2\epsilon_{N+1}\big{)}}{\Gamma\big{(}\bar{s}_{N+1}+{\rm
i}x^{\prime}_{N+1}+\epsilon_{N+1}\big{)}\,\Gamma(s_{N+1}-{\rm
i}x_{N+1}+\epsilon_{N+1})}\,\biggr{)}\chi(\underline{x})\,\big{(}\chi(\underline{x}^{\prime})\big{)}^{*}$
$\displaystyle\quad\times{\rm d}\mu_{N}^{A}(x)\,{\rm
d}\mu_{N}^{A}(x^{\prime})\,{\rm d}\mu_{1}^{A}(x_{N+1})\,{\rm
d}\mu_{1}^{A}(x^{\prime}_{N+1}).$

The analysis of the above expression in the limit $\underline{\epsilon}\to
0^{+}$ and $\sigma\to 0$ is exactly the same as before, see Appendix A. It
results in the following expression for the norm

$\displaystyle\|\varphi\|^{2}_{\mathbb{H}_{N+1}^{B}}=\|\mathrm{S}_{B}\chi\|^{2}_{\mathbb{H}_{N+1}^{B}}=\|\chi\|^{2}_{\mathbb{H}_{N}^{A}\otimes\mathbb{H}_{1}^{A}}=\int_{\mathbb{R}^{N+1}}\,|\chi(\underline{x})|^{2}\,{\rm
d}\mu_{N}^{A}(x)\,{\rm d}\mu_{1}^{A}(x_{N+1}),$

that completes the proof of the lemma. ∎

It follows from Lemma 4.2 $\ker\mathrm{S}_{\mathbb{B}A}=0$ and
$\ker\mathrm{S}_{B}=0$ and, hence, that the operators
$\mathrm{T}_{N}^{\mathbb{B}}\colon\mathbb{H}_{N}^{\mathbb{B}}\mapsto\mathbb{H}_{N}$
and $\mathrm{T}_{N+1}^{B}\colon\mathbb{H}_{N+1}^{B}\mapsto\mathbb{H}_{N+1}$
are unitary provided that $\mathrm{T}_{N}^{A}$ is.

The final step required to complete the induction on $N$ is to show that the
unitarity of the map $\mathrm{T}_{N}^{A}$ follows from that of the map
$\mathrm{T}_{N}^{B}$. As it was argued earlier in this section in order to
prove this statement it is enough to show that the kernel of the operator

$\displaystyle\mathrm{S}_{AB}=\big{(}\mathrm{T}_{N}^{A}\big{)}^{\star}\,\mathrm{T}_{N}^{B},\qquad\mathrm{S}_{AB}\colon\
\mathbb{H}_{N}^{B}\mapsto\mathbb{H}_{N}^{A}$ (4.9)

is trivial.

######  Lemma 4.3.

Let $\mathrm{S}_{AB}$ be the linear operator defined in equation (4.9). If the
map $\mathrm{T}_{N}^{B}\colon\mathbb{H}_{N}^{B}\mapsto\mathbb{H}_{N}$ is
unitary then for any $\varphi\in\mathbb{H}_{N}^{B}$ the following identity
holds:
$\|\mathrm{S}_{AB}\varphi\|^{2}_{\mathbb{H}_{N}^{A}}=\|\varphi\|^{2}_{\mathbb{H}_{N}^{B}}$.

###### Proof.

Let $\varphi(p,x)$ be a smooth function with compact support having the
factorised form

$\displaystyle\varphi(p,x)=f(p)\tilde{\varphi}(x).$ (4.10)

The linear span of these functions is dense in $\mathbb{H}_{N}^{B}$. The
action of the operator $\mathrm{S}_{AB}$ on a function $\varphi$ can be
represented as follows

$\displaystyle\chi(y)$
$\displaystyle=[\mathrm{S}_{AB}\varphi](y)=\big{(}\Phi_{y},\mathrm{T}_{N}^{B}\varphi\big{)}_{\mathbb{H}_{N}}=\lim_{\epsilon\to
0^{+}}\big{(}\Phi_{y+{\rm
i}\epsilon},\mathrm{T}_{N}^{B}\varphi\big{)}_{\mathbb{H}_{N}}$
$\displaystyle=\lim_{\epsilon\to
0^{+}}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}^{N-1}}\big{(}\Phi_{y+{\rm
i}\epsilon},\Psi^{N}_{p,{x}}\big{)}_{\mathbb{H}_{N}}\,\varphi(p,x)\,\,{\rm
d}p\,{\rm d}\mu_{N-1}^{B}(x),$ (4.11)

where $y+{\rm i}\epsilon=(y_{1}+{\rm i}\epsilon,\dots,y_{N}+{\rm i}\epsilon)$
and the scalar product of two eigenfunctions is given by equation (4.5). We
also denote the function given by the integral in the above equation by
$\chi^{\epsilon}(y)$, i.e.,

$\displaystyle\chi(y)=\lim_{\epsilon\to 0^{+}}\chi^{\epsilon}(y).$

It can be shown, see [22, Lemma 3.1], that the function
$\chi(y)\prod_{i<k}y_{ik}$ is a smooth function. Our final aim is to show that
$\chi\in\mathbb{H}_{N}^{B}$ and that the
$\|\chi\|^{2}_{\mathbb{H}_{N}^{A}}=|\varphi\|^{2}_{\mathbb{H}_{N}^{B}}$.

In contrast to the previous cases, the function $\chi(y)$ does not decrease
fast enough for large $y_{k}$ to justify changing the order of integration
over $x$, $x^{\prime}$ and $y$ in the norm integral. To overcome this
difficulty we proceed as follows. Let us define a regularized function
$\chi_{L}(y)$ as

$\displaystyle\chi_{L}(y)=\chi(y)g_{L}(y),\qquad\text{where}\quad
g_{L}(y)=\prod_{k=1}^{N}\left|\frac{\Gamma(L+{\rm
i}y_{k})}{\Gamma(L)}\right|.$

The factor $g_{L}(y)$ has the following properties:

  1. ($i$)

$g_{L}(y)<1$ for all $y$,

  2. ($ii$)

$g_{L}(y)\to 1$ monotonically as $L\to\infty$ for fixed $y$,

  3. ($iii$)

$g_{L}(y)\sim\exp\big{\\{}-\pi/2\sum_{k}|y_{k}|\big{\\}}$ for fixed $L$ and
$|y_{k}|\to\infty$.

It follows from $(ii)$ that for any bounded region $D\in\mathbb{R}^{N}$

$\displaystyle\int_{D}|\chi(y)-\chi_{L}(y)|^{2}\,{\rm d}\mu_{N}^{A}(y)\to
0\qquad\text{as}\quad L\to\infty.$

Due to $(iii)$ one concludes that, for finite $L$, the integral of
$|\chi_{L}(y)|^{2}$ over $\mathbb{R}^{N}$ converges

$\displaystyle I_{L}=\int_{\mathbb{R}^{N}}|\chi_{L}(y)|^{2}\,{\rm
d}\mu_{N}^{A}(y)<\infty.$

Then one derives the following inequality

$\displaystyle\int_{D}|\chi(y)|^{2}\,{\rm
d}\mu_{N}^{A}(y)\leq\int_{D}|\chi(y)-\chi_{L}(y)|^{2}\,{\rm
d}\mu_{N}^{A}(y)+\int_{D}|\chi_{L}(y)|^{2}\,{\rm d}\mu_{N}^{A}(y)\leq
2^{-M}+I_{L},$

which holds for any $L$ greater than some $L_{M}$. Since $M$ is arbitrary we
get the following inequality

$\displaystyle\int_{D}|\chi(y)|^{2}\,{\rm
d}\mu_{N}^{A}(y)\leq\lim_{L\to\infty}I_{L}\equiv I,$

which holds for an arbitrary bounded region $D$. Therefore
$\int_{\mathbb{R}^{N}}|\chi(y)|^{2}{\rm d}\mu_{N}^{A}(y)\leq I$. Since due to
$(i)$ $I\leq\int_{\mathbb{R}^{N}}|\chi(y)|^{2}{\rm d}\mu_{N}^{A}(y)$ we
conclude that

$\displaystyle\|\chi\|^{2}_{\mathbb{H}_{N}^{A}}=\int_{\mathbb{R}^{N}}|\chi(y)|^{2}\,{\rm
d}\mu_{N}^{A}(y)=I.$

Thus one has to find the limit of $I_{L}$ at $L\to\infty$. First we note that
$I_{L}$ can be written in the form

$\displaystyle I_{L}=\int_{\mathbb{R}^{N}}|\chi_{L}(y)|\,{\rm
d}\mu_{N}^{A}(y)=\int_{\mathbb{R}^{N}}\lim_{\epsilon\to
0^{+}}|\chi_{L}^{\epsilon}(y)|^{2}\,{\rm d}\mu_{N}^{A}(y)=\lim_{\epsilon\to
0^{+}}\int_{\mathbb{R}^{N}}|\chi_{L}^{\epsilon}(y)|^{2}\,{\rm
d}\mu_{N}^{A}(y),$

where $\chi_{L}^{\epsilon}(y)=g_{L}(y)\chi^{\epsilon}(y)$. At the last step,
the limit $\epsilon\to 0^{+}$ is taken after the integration. It is possible
to do so since the function $\chi_{L}^{\epsilon}(y)$ is bounded,
$|\chi_{L}^{\epsilon}(y)|<C_{L}$ for all $y$, and the measure $\mu_{N}^{A}(y)$
decays exponentially fast at large $y$,

$\displaystyle\mu^{A}_{N}(y)\simeq\frac{(4\pi)^{\frac{N(N-1)}{2}}}{N!}\prod_{1\leq
i<j\leq N}y_{ij}\sinh\pi
y_{ij}\,\prod_{k=1}^{N}\prod_{j=1}^{N}y_{k}^{2\boldsymbol{s}_{j}-1}{\rm
e}^{-\pi|y_{k}+\xi_{j}|},$

so that the measure of the whole space is finite

$\displaystyle\int_{\mathbb{R}^{N}}{\rm
d}\mu_{N}^{A}(y)=2^{-\sum_{i=1}^{N}(s_{i}+\bar{s}_{i})}\prod_{k,j=1}^{N}\Gamma(s_{k}+\bar{s}_{j}).$

The calculation of the integral of $|\chi_{L}^{\epsilon}(y)|^{2}$ follows the
familiar pattern: one substitutes the function $\chi^{\epsilon}_{L}(y)$ using
(4) and then perform first the integral over $y$. This integral is the reduced
version of Gustafson’s integral, equation (2.8). Making use of this result one
can write $I_{L}$ in the form

$\displaystyle I_{L}=\lim_{\epsilon\to
0^{+}}\int_{0}^{\infty}\int_{0}^{\infty}\int_{\mathbb{R}^{2N-2}}\widehat{\varphi}(p,x)\big{(}\widehat{\varphi}(p^{\prime},x^{\prime})\big{)}^{*}M_{\epsilon}(L,p,x,p^{\prime},x^{\prime})\,{\rm
d}p\,{\rm d}p^{\prime}\,{\rm d}\mu_{N-1}^{B}(x)\,{\rm
d}\mu_{N-1}^{B}(x^{\prime}),$

where

$\displaystyle\widehat{\varphi}(p,x)=p^{-1/2+{\rm
i}\Xi}\,{\varphi(p,x)}\Bigg{(}\prod_{j=1}^{N}\prod_{k=1}^{N-1}\Gamma(s_{j}-{\rm
i}x_{k})\Bigg{)}^{-1}$

and

$\displaystyle M_{\epsilon}(L,p,x,p^{\prime},x^{\prime})$
$\displaystyle=\prod_{k,j=1}^{N-1}\Gamma\big{(}{\rm
i}(x^{\prime}_{j}-x_{k})+2\epsilon\big{)}\,\frac{\Gamma(2L)}{\Gamma^{2}(L)}\,\prod_{k=1}^{N-1}\frac{\Gamma(L-{\rm
i}x_{k}+\epsilon)\Gamma(L+{\rm i}x^{\prime}_{k}+\epsilon)}{\Gamma^{2}(L)}$
$\displaystyle\quad\times\bigg{(}1+\frac{p}{p^{\prime}}\bigg{)}^{-L-(N-1)\epsilon+{\rm
i}X}\bigg{(}1+\frac{p^{\prime}}{p}\bigg{)}^{-L-(N-1)\epsilon-{\rm
i}X^{\prime}}.$ (4.12)

We recall that $X(X^{\prime})=\sum_{k=1}^{N-1}x_{k}(x^{\prime}_{k})$. Due to
our assumptions on the function $\varphi$ the integral in (4.12) is restricted
to a finite region hence we can expand the function
$M_{\epsilon}(L,p,x,p^{\prime},x^{\prime})$ in series in $L^{-1}$

$\displaystyle M_{\epsilon}(L,p,x,p^{\prime},x^{\prime})$
$\displaystyle=L^{{\rm i}(X^{\prime}-X)}\prod_{k,j=1}^{N-1}\Gamma({\rm
i}(x^{\prime}_{k}-x_{j})+2\epsilon)$ $\displaystyle\quad\times
2^{2L-1}\sqrt{\frac{L}{\pi}}\bigg{(}1+\frac{p}{p^{\prime}}\bigg{)}^{-L+{\rm
i}X}\,\bigg{(}1+\frac{p^{\prime}}{p}\bigg{)}^{-L-{\rm
i}X^{\prime}}\,\bigg{(}1+O\bigg{(}\frac{1}{L}\bigg{)}\bigg{)},$ (4.13)

where we put $\epsilon\to 0$ in non-singular terms. At large $L$, the dominant
contribution to the integral over $p$, $p^{\prime}$ comes from the region
$p=p^{\prime}$ and can be easily estimated as

$\displaystyle 2^{{\rm i}(X-X^{\prime})}\int_{0}^{\infty}{\rm
d}p\,|f(p)|^{2}(1+O(1/L)),$

see equation (4.10). The factor in the first line of equation (4) has the form
we encountered earlier, see, e.g., (4.8), and can be handled in the same way
as before, see Appendix A for more details. Collecting all factors we obtain

$\displaystyle
I=\lim_{L\to\infty}I_{L}=\int_{\mathbb{R}_{+}}\int_{\mathbb{R}^{N-1}}|\varphi(p,x)|^{2}\,{\rm
d}p\,{\rm d}\mu_{N-1}^{B}(x)\equiv\|\varphi\|^{2}_{\mathbb{H}_{N}^{B}}.$

Thus one concludes that $\mathrm{S}_{AB}$ is a norm preserving map,
$\|S_{AB}\varphi\|^{2}_{\mathbb{H}_{N}^{A}}\equiv\|\chi\|^{2}_{\mathbb{H}_{N}^{A}}=\|\varphi\|^{2}_{\mathbb{H}_{N}^{B}}$
and, hence, $\ker\mathrm{S}_{AB}=0$. Therefore one concludes that
$\mathrm{T}_{N}^{A}$ is a unitary operator between the Hilbert spaces
$\mathbb{H}_{N}^{A}$ and $\mathbb{H}_{N}$. ∎

Lemma 4.3 completes the inductive proof that the maps $\mathrm{T}_{N}^{B}$,
$\mathrm{T}_{N}^{A}$ and $\mathbb{T}_{N}^{\mathbb{B}}$ are unitary for all
$N$.

##  5 Summary

This work devised a very effective inductive scheme allowing one to prove the
completeness of Sklyanin’s separated variables which arise in the analysis of
the closed and open $\mathrm{SL}(2,\mathbb{R})$ spin chain magnets. The method
we proposed heavily relies on the use of multidimensional Mellin–Barnes
integrals which were calculated in closed form by R.A. Gustafson [14]. The
attractive feature of our approach is that it does not depends on the details
of the spin chain – spins, $s_{k}$, and inhomogeneity parameters, $\xi_{k}$.
Moreover, the core identities which are to be used for the closed spin chain
or for Toda chain are exactly the same, what stressed a certain generality of
our method. Since the Gustafson integrals can be viewed as a special case of
the elliptic hypergeometric integrals, see, e.g., [32, 44], we believe that
our method can be adapted to such models as spin chains with the trigonometric
and elliptic $R$-matrices or non-compact magnets with the
$\mathrm{SL}(2,\mathbb{C})$ symmetry group.

##  Appendix A Some representations for multi-dimensional Dirac
$\boldsymbol{\delta}$-functions

I. Define

$\displaystyle\widetilde{C}_{N}^{(\epsilon,\epsilon^{\prime})}(p,x,x^{\prime})=\frac{\Gamma\big{(}\epsilon_{N}+\epsilon^{\prime}_{N}+{\rm
i}\sum_{a=1}^{N-1}(x_{a}-x^{\prime}_{a})\big{)}}{\Gamma\big{(}\sum_{a=1}^{N}\epsilon_{a}+\epsilon^{\prime}_{a}\big{)}}\frac{\prod_{a,b}^{N-1}\Gamma({\rm
i}(x^{\prime}_{b}-x_{a})+\epsilon^{\prime}_{b}+\epsilon_{a})}{\prod_{a\not=b}^{N-1}\Gamma({\rm
i}(x_{a}^{\prime}-x_{b}^{\prime}))\Gamma({\rm i}(x_{a}-x_{b}))}.$

In the following we show that, in the sense of distributions, it holds

$\displaystyle\lim_{\epsilon,\epsilon^{\prime}\rightarrow
0^{+}}\big{\\{}\widetilde{C}_{N}^{(\epsilon;\epsilon^{\prime})}\big{(}p,x,x^{\prime}\big{)}\big{\\}}=W_{N-1}(x)\cdot\delta^{N-1}\big{(}x,x^{\prime}\big{)},$

where

$\displaystyle
W_{N-1}(x)=(2\pi)^{N-1}(N-1)!\prod\limits_{a\not=b}^{N-1}\frac{1}{\Gamma\big{(}{\rm
i}(x_{a}-x_{b})\big{)}}.$ (A.1)

In other words, given

$\displaystyle\mathscr{I}_{N}^{(\epsilon;\epsilon^{\prime})}=\int\limits_{\mathbb{R}^{+}}{}{\rm
d}p\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x^{\prime}\widetilde{C}_{N}^{(\epsilon;\epsilon^{\prime})}\big{(}p,x,x^{\prime}\big{)}\varphi\big{(}p,x\big{)}\varphi^{*}\big{(}p,x^{\prime}\big{)}$
(A.2)

it holds that

$\displaystyle\lim_{\epsilon,\epsilon^{\prime}\rightarrow
0^{+}}\mathscr{I}_{N}^{(\epsilon;\epsilon^{\prime})}=\int\limits_{\mathbb{R}^{+}}{}{\rm
d}p\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x\,W(x)\big{|}\,\varphi\big{(}p,x\big{)}\big{|}^{2}.$

In order to establish the result, one starts by reorganising the integral in
(A.2) as

$\displaystyle\mathscr{I}_{N}^{(\epsilon;\epsilon^{\prime})}=\int\limits_{\mathbb{R}^{+}}{}{\rm
d}p\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x^{\prime}\mathscr{U}_{N}^{(\epsilon;\epsilon^{\prime})}\big{(}p,x,x^{\prime}\big{)}$
$\displaystyle\hphantom{\mathscr{I}_{N}^{(\epsilon;\epsilon^{\prime})}=}\times\det_{N-1}\bigg{[}\frac{1}{x^{\prime}_{k}-x_{j}-{\rm
i}(\epsilon_{j}+\epsilon_{k}^{\prime})}\bigg{]}\,\frac{-{\rm
i}\sum_{k=1}^{N}(\epsilon_{k}+\epsilon_{k}^{\prime})}{-{\rm
i}(\epsilon_{N}+\epsilon_{N}^{\prime})+\sum_{k=1}^{N-1}(x_{k}-x_{k}^{\prime})},$

where

$\displaystyle\mathscr{U}_{N}^{(\epsilon;\epsilon^{\prime})}\big{(}p,x,x^{\prime}\big{)}=\prod\limits_{a<b}^{N-1}\big{\\{}(x_{a}-x_{b})\,(x_{b}^{\prime}-x_{a}^{\prime})\big{\\}}
\,\widehat{C}_{N}^{(\epsilon;\epsilon^{\prime})}\big{(}p,x,x^{\prime}\big{)}\,\varphi\big{(}p,x\big{)}\,\varphi^{*}\big{(}p,x^{\prime}\big{)},$

and

$\displaystyle\widehat{C}_{N}^{(\epsilon;\epsilon^{\prime})}\big{(}p,x,x^{\prime}\big{)}\\!=\\!(-{\rm
i})^{(N-1)^{2}}\frac{\Gamma\big{(}\\!1\\!+\\!\epsilon_{N}\\!+\\!\epsilon^{\prime}_{N}\\!+\\!\sum_{k=1}^{N-1}(x_{k}\\!-\\!x_{k}^{\prime})\\!\big{)}\\!\prod_{a,b=1}^{N-1}\\!\Gamma\big{(}1\\!+\\!{\rm
i}(x_{a}^{\prime}\\!-\\!x_{b})\\!+\\!\epsilon^{\prime}_{a}\\!+\\!\epsilon_{b}\big{)}}{\Gamma\big{(}1\\!+\\!\sum_{k=1}^{N}(\epsilon_{k}\\!+\\!\epsilon_{k}^{\prime})\big{)}\prod_{a\not=b}^{N-1}\Gamma\big{(}1\\!+\\!{\rm
i}(x_{a}^{\prime}\\!-\\!x_{b}^{\prime})\big{)}\Gamma\big{(}1\\!+\\!{\rm
i}(x_{a}\\!-\\!x_{b})\big{)}}.$

Thus, $\mathscr{U}_{N}^{(\epsilon;\epsilon^{\prime})}$ is antisymmetric in
$x$, $x^{\prime}$ taken singly, and it is smooth and compactly supported in
$x,x^{\prime}\in\mathbb{R}^{N-1}$ and smooth in a small neighbourhood of zero
in respect to $\epsilon$, $\epsilon^{\prime}$.

Expanding the determinant as a sum over the permutation group and using the
antisymmetry in $x$, $x^{\prime}$ and the smoothness in $\epsilon$,
$\epsilon^{\prime}$ of $\mathscr{U}_{N}^{(\epsilon;\epsilon^{\prime})}$ as
well as the Sokhotsky–Plemejl formulae for the limits
$\epsilon,\epsilon^{\prime}\rightarrow 0^{+}$ of the singular factors, one
gets that

$\displaystyle\lim_{\epsilon,\epsilon^{\prime}\rightarrow
0^{+}}\mathscr{I}_{N}^{(\epsilon;\epsilon^{\prime})}=(N-1)!\lim_{\varepsilon\rightarrow
0^{+}}\lim_{\epsilon\rightarrow 0^{+}}\int\limits_{\mathbb{R}^{+}}{}{\rm
d}p\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x^{\prime}\mathscr{U}_{N}^{(0;0)}\big{(}p,x,x^{\prime}\big{)}$
$\displaystyle\hphantom{\lim_{\epsilon,\epsilon^{\prime}\rightarrow
0^{+}}\mathscr{I}_{N}^{(\epsilon;\epsilon^{\prime})}=}\times\prod\limits_{a=1}^{N-1}\frac{1}{x^{\prime}_{a}-x_{a}-{\rm
i}\epsilon_{a}}\,\frac{-{\rm i}\varepsilon}{-{\rm
i}\varepsilon+\sum_{k=1}^{N-1}(x_{k}-x_{k}^{\prime})}.$

It is thus enough to study the $\varepsilon\rightarrow 0^{+}$,
$\epsilon\rightarrow 0^{+}$ limit of the model integral

$\displaystyle\mathscr{J}_{N}^{(\varepsilon;\epsilon)}=\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x^{\prime}\chi\big{(}x,x^{\prime}\big{)}\prod\limits_{a=1}^{N-1}\frac{1}{x^{\prime}_{a}-x_{a}-{\rm
i}\epsilon_{a}}\,\frac{-{\rm i}\varepsilon}{-{\rm
i}\varepsilon+\sum_{k=1}^{N-1}(x_{k}-x_{k}^{\prime})},$

in which $\chi$ is antisymmetric in $x$, $x^{\prime}$ taken singly. Observe
that, for fixed $x$, by the Stone–Weierstrass theorem, there exists a sequence
of smooth, compactly supported functions $\varphi_{k,a}$ on $\mathbb{R}$ such
that

$\displaystyle\chi\big{(}x,x^{\prime}\big{)}\;=\;\sum_{k\geq
1}{}\prod\limits_{a=1}^{N-1}\varphi_{k,a}(x_{a}^{\prime}).$

Next, one observes that

$\displaystyle\prod\limits_{a=1}^{N-1}\frac{\varphi_{k,a}(x_{a}^{\prime})}{x^{\prime}_{a}-x_{a}-{\rm
i}\epsilon_{a}}=\sum_{s=0}^{N-1}\sum\limits_{\begin{subarray}{c}\alpha_{+}\cup\alpha_{-}=[\\![1;N-1]\\!]\\\
|\alpha_{+}|=s\end{subarray}}{}\prod\limits_{a\in\alpha_{+}}{}\Delta_{x_{a},x_{a}^{\prime};\epsilon_{a}}\big{[}\varphi_{k,a}\big{]}\prod\limits_{a\in\alpha_{-}}{}\frac{\varphi_{k,a}(x_{a})}{x^{\prime}_{a}-x_{a}-{\rm
i}\epsilon_{a}},$

where

$\displaystyle\Delta_{x_{a},x_{a}^{\prime};\epsilon_{a}}\big{[}f\big{]}=\frac{f(x_{a}^{\prime})-f(x_{a})}{x^{\prime}_{a}-x_{a}-{\rm
i}\epsilon_{a}}.$

Thus, inserting the expansion in the integral, summing up, setting
$\epsilon_{a}=0$ in the regular part of the integrand and using the
antisymmetry in $x$, $x^{\prime}$ of $\chi$, one gets that

$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\lim_{\epsilon\rightarrow
0^{+}}\mathscr{J}_{N}^{(\varepsilon;\epsilon)}=\lim_{\varepsilon\rightarrow
0^{+}}\lim_{\epsilon\rightarrow
0^{+}}\sum_{s=0}^{N-1}C^{s}_{N-1}\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x^{\prime}\Delta^{(s)}_{x,x^{\prime}}\,\chi\big{(}x,(\boldsymbol{x}_{s}^{\prime},x^{(s+1)})\big{)}$
$\displaystyle\hphantom{\lim_{\varepsilon\rightarrow
0^{+}}\lim_{\epsilon\rightarrow
0^{+}}\mathscr{J}_{N}^{(\varepsilon;\epsilon)}=}\times\prod\limits_{a=s+1}^{N-1}\frac{1}{x^{\prime}_{a}-x_{a}-{\rm
i}\varepsilon_{a}}\,\frac{-{\rm i}\varepsilon}{-{\rm
i}\varepsilon+\sum_{k=1}^{N-1}(x_{k}-x_{k}^{\prime})},$ (A.3)

where $\Delta^{(s)}_{x,x^{\prime}}$ is a composite of operators acting on the
variables $x_{1},\dots,x_{s},x_{1}^{\prime},\dots,x_{s}^{\prime}$

$\displaystyle\Delta^{(s)}_{x,x^{\prime}}=\prod\limits_{a=1}^{s}\Delta_{x_{a},x_{a}^{\prime}}^{(0)}.$

Also, establishing (A.3), we took the freedom to relabeling the variables

$\displaystyle
x^{(k)}=(x_{k},\dots,x_{N-1})\qquad\text{and}\qquad\boldsymbol{x}_{s}^{\prime}=(x_{1}^{\prime},\dots,x_{s}^{\prime}).$

One may now readily take the integrals in respect to
$x^{\prime}_{s+1},\dots,x^{\prime}_{N-1}$ in (A.3), what yields

$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\lim_{\epsilon\rightarrow
0^{+}}\mathscr{J}_{N}^{(\varepsilon;\epsilon)}=\lim_{\varepsilon\rightarrow
0^{+}}\sum_{s=0}^{N-1}C^{s}_{N-1}(2{\rm
i}\pi)^{N-1-s}\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x\int\limits_{\mathbb{R}^{s}}{}{\rm d}^{s}x^{\prime}$
$\displaystyle\hphantom{\lim_{\varepsilon\rightarrow
0^{+}}\lim_{\epsilon\rightarrow
0^{+}}\mathscr{J}_{N}^{(\varepsilon;\epsilon)}=}\times\Delta^{(s)}_{x,x^{\prime}}\,\chi\big{(}x,\big{(}\boldsymbol{x}_{s}^{\prime},x^{(s+1)}\big{)}\big{)}\,\frac{-{\rm
i}\varepsilon}{-{\rm i}\varepsilon+\sum_{k=1}^{s}(x_{k}-x_{k}^{\prime})}.$

Apart from the term arising in the last line, the integrand is a smooth
function. Thus, by changing the variables

$\displaystyle x\hookrightarrow
y=\bigg{(}\sum_{k=1}^{s}x_{k},x_{2},\dots,x_{N-1}\bigg{)}\qquad\text{and}\qquad
x^{\prime}\hookrightarrow
y^{\prime}=\bigg{(}\sum_{k=1}^{s}x_{k}^{\prime},x_{2}^{\prime},\dots,x_{s}^{\prime}\bigg{)}$

one may apply the Sokhotsky–Plemejl formula for the remaining singular factor
what ensures that solely the $s=0$ term contributes to the integral. Hence,

$\displaystyle\lim_{\varepsilon\rightarrow 0^{+}}\lim_{\epsilon\rightarrow
0^{+}}\mathscr{J}_{N}^{(\varepsilon;\epsilon)}=(2{\rm
i}\pi)^{N-1}\int\limits_{\mathbb{R}^{N-1}}{}{\rm
d}^{N-1}x\chi\big{(}x,x\big{)}.$

This entails the claim.

II. Let us define

$\displaystyle S_{N}^{L,\varepsilon}(x,x^{\prime})=L^{{\rm
i}\sum_{a=1}^{N}(x^{\prime}_{a}-x_{a})}\frac{\prod_{a,b=1}^{N}\Gamma({\rm
i}(x^{\prime}_{a}-x_{b})+\epsilon)}{\prod_{a\not=b}^{N}\Gamma\big{(}{\rm
i}(x_{a}^{\prime}-x_{b}^{\prime})\big{)}\Gamma\big{(}{\rm
i}(x_{a}-x_{b})\big{)}}.$

We will show that in the sense of distributions the following identity holds

$\displaystyle\lim_{L\rightarrow\infty}\lim_{\epsilon\rightarrow
0^{+}}S_{N}^{L,\epsilon}(x,x^{\prime})=W_{N}(x)\,\delta^{N}\big{(}x,x^{\prime}\big{)},$

where $W_{N}$ is defined in equation (A.1). Namely, given

$\displaystyle\mathscr{T}_{N}^{L,\epsilon}=\int\limits_{\mathbb{R}^{N}}{\rm
d}^{N}x\int\limits_{\mathbb{R}^{N}}{}{\rm
d}^{N}x^{\prime}{S}_{N}^{L,\epsilon}\big{(}x,x^{\prime}\big{)}\varphi\big{(}x\big{)}\varphi^{*}\big{(}x^{\prime}\big{)}$
(A.4)

it holds that

$\displaystyle\lim_{L\rightarrow\infty}\lim_{\epsilon\rightarrow
0^{+}}\mathscr{T}_{N}^{L,\epsilon} =\int\limits_{\mathbb{R}^{N}}{}{\rm
d}^{N}xW_{N}(x)\big{|}\varphi\big{(}x\big{)}\big{|}^{2}.$

Repeating the same arguments as above one can rewrite (A.4) in the form

$\displaystyle\lim_{L\rightarrow\infty}\lim_{\epsilon\rightarrow
0^{+}}\mathscr{T}_{N}^{L,\epsilon}=N!\lim_{L\rightarrow\infty}\lim_{\epsilon\rightarrow
0^{+}}\int\limits_{\mathbb{R}^{N}}{}{\rm
d}^{N}x\int\limits_{\mathbb{R}^{N}}{}{\rm
d}^{N}x^{\prime}\mathscr{V}_{N}\big{(}x,x^{\prime}\big{)}\prod\limits_{a=1}^{N}\frac{L^{{\rm
i}(x^{\prime}_{a}-x_{a})}}{x^{\prime}_{a}-x_{a}-{\rm i}\epsilon},$

where

$\displaystyle\mathscr{V}_{N}\big{(}x,x^{\prime}\big{)}=(-{\rm
i})^{N^{2}}\frac{\prod_{a<b}^{N}(x_{a}-x_{b})\,(x_{b}^{\prime}-x_{a}^{\prime})
\prod_{a,b=1}^{N}\Gamma\big{(}1+{\rm
i}(x^{\prime}_{a}-x_{b})\big{)}}{\prod_{a\not=b}^{N}\Gamma\big{(}1+{\rm
i}(x_{a}^{\prime}-x_{b}^{\prime})\big{)}\Gamma\big{(}1+{\rm
i}(x_{a}-x_{b})\big{)}}\,\varphi\big{(}p,x\big{)}\,\varphi^{*}\big{(}p,x^{\prime}\big{)}.$

Finally, taking into account that

$\displaystyle\lim_{\epsilon\rightarrow 0^{+}}\int_{\mathbb{R}}{\rm
d}x\,L^{{\rm i}x}\frac{\varphi(x)}{x-{\rm i}\epsilon}$
$\displaystyle=\lim_{\epsilon\rightarrow 0^{+}}\varphi(0)\int_{\mathbb{R}}{\rm
d}x\frac{L^{{\rm i}x}}{x-{\rm i}\epsilon}+\int_{\mathbb{R}}{\rm d}x\,L^{{\rm
i}x}\frac{\varphi(x)-\varphi(0)}{x}$ $\displaystyle=2\pi\varphi(0)+O(1/\ln L)$

and using the Stone–Weierstrass theorem one gets the necessary result.

### Acknowledgements

This work was supported by the Russian Science Foundation project No
19-11-00131 and by the DFG grants MO 1801/4-1, KN 365/13-1 (A.M.). The work of
K.K.K. is supported by CNRS.

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