Completeness of SoV Representation for SL(2, ℝ) Spin Chains
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...
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| citation_txt | Completeness of SoV Representation for SL(2, ℝ) Spin Chains. Sergey É. Derkachov, Karol K. Kozlowski and Alexander N. Manashov. SIGMA 17 (2021), 063, 26 pages |
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| description | 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 SL(2, ℝ) spin chains.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 063, 26 pages
Completeness of SoV Representation
for SL(2,R) Spin Chains
Sergey É. DERKACHOV a, Karol K. KOZLOWSKI b and Alexander N. MANASHOV ca
a) St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences,
Fontanka 27, 191023 St. Petersburg, Russia
E-mail: derkach@pdmi.ras.ru
b) Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique,
F-69342 Lyon, France
E-mail: karol.kozlowski@ens-lyon.fr
c) Institut für Theoretische Physik, Universität Hamburg, D-22761 Hamburg, Germany
E-mail: alexander.manashov@desy.de
Received March 08, 2021, in final form June 14, 2021; Published online June 25, 2021
https://doi.org/10.3842/SIGMA.2021.063
Abstract. 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 sepa-
ration of variables transform for infinite-dimensional representations of rank one quantum
integrable models. We examine in detail the case of the SL(2,R) spin chains.
Key words: spin chains; separation of variables; Gustafson’s integrals
2020 Mathematics Subject Classification: 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 opera-
tor 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 δ-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 inte-
grability 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 simpli-
fied, 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
This paper is a contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quan-
tum in honor of Leon Takhtajan.
The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html
mailto:derkach@pdmi.ras.ru
mailto:email@address
mailto:alexander.manashov@desy.de
https://doi.org/10.3842/SIGMA.2021.063
https://www.emis.de/journals/SIGMA/Takhtajan.html
2 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
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 U bet-
ween an auxiliary Hilbert space hsov and the original Hilbert space horg 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 prob-
lem 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 {Hk} 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 Hks. Now the role of the map U is to realise
the unitary equivalence between hsov and horg in such a way that the original multi-dimensional
(because the Hamiltonians have a non-trivial structure) and multi-parametric spectral problem
on horg is reduced into a multi-parametric (because one has to keep track of all the eigenvalues)
one-dimensional spectral problem on hsov. This thus explains the separation of variables ter-
minology. 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 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 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 alter-
native construction was proposed in [5], where, in particular, it was pointed out that 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 U ,
we recall that the original Hilbert space horg, where the model is formulated and hsov, where
the separation of variables takes place can be identified with appropriate L2 spaces horg =
L2(X ,dνorg) and hsov = L2(Y,dµsov). This is a very general setting which allows X , Y to be
finite, discrete or continuous. Upon such an identification of the Hilbert spaces, the map U is
defined as an integral transform acting on smooth, compactly supported functions on Y:
[Uϕ](x) =
∫
Y
ϕ(y)Ψy(x) dµsov(y).
The functions Ψy(x) describing the integral kernel of the transform can be thought of as the ana-
Completeness of SoV Representation for SL(2,R) Spin Chains 3
logues of the function eiyx giving the integral kernel of the Fourier transform. That case, in fact,
corresponds to X = Y = R and both dνorg and dµsov coinciding with the Lebesgue measure.
In the above case, just as {x 7→ eiyx} corresponds to the system of generalised eigenfunctions
of translation operators, {x 7→ Ψy(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 U hence boils down to the con-
struction 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 Ψy(x) in
the first place, does also ensure that, by construction, U fulfills the desired requirement of sim-
plifying the original spectral problem. However, unitarity is a completely different issue. It boils
down to proving the orthogonality and completeness of the system Ψy(x) which can be framed
as the following relations understood in the sense of distributions∫
X
(
Ψy′(x)
)∗
Ψy(x) dνorg(x) =
1
dµsov/dy
δsov(y′, y) (1.1)
and ∫
Y
(
Ψy(x
′)
)∗
Ψy(x) dµsov(y) =
1
dνorg/dx
δorg(x′, x). (1.2)
Above δsov(y′, y), resp. δorg(x′, x), corresponds to the generalised function which represents the
integral kernel of the identity operator on Y, resp. X . Moreover, dµsov/dy, resp. dνorg/dx, is
the Radon–Nikodym derivative of µsov, resp. νorg, in respect to the canonical measure on Y,
resp. X .
The technique for proving unitarity of U , viz. (1.1)–(1.2), strongly depends on the dimension
of the original Hilbert space horg. If horg is finite-dimensional, checking unitarity amounts to
a simple comparison of dimensions between horg and hsov. However, many of the physically
interesting quantum integrable models are defined on an infinite-dimensional Hilbert space horg.
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 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 U† in the case of the Toda chain. All together with the results of [5],
this construction ensures the unitarity of U . An even more efficient method allowing one to
establish the isometry of U† 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 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.
4 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
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 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
H = −1
2
N∑
k=1
∂2
∂x2
k
+
N−1∑
k=1
exk−xk+1
on the Hilbert space L2
(
RN , dNx
)
. The model is integrable and can be solved by the quan-
tum inverse scattering method (QISM) [41, 42]. For further discussion it is important that
eigenfunctions can be constructed iteratively [18, 19],
Ψλ
N (x) = lim
ε→0+
∫
RN−1
N∏
k=1
N−1∏
j=1
Γ(iλk − iγj + ε) ei(Λ−Γ)xN Ψγ
N−1(x)µN−1(γ)
N−1∏
j=1
dγj , (2.1)
where x = (x1, . . . , xN ), λ = (λ1, . . . , λN ), γ = (γ1, . . . , γN−1) and Λ =
∑N
j=1 λj , Γ =
∑N−1
j=1 γj .
The measure µN (λ) – the Sklyanin measure – is given by a product of Γ-functions
µ−1
N (γ) = (2π)NN !
N∏
j<k
Γ
(
i(γk − γj)
)
Γ
(
i(γj − γk)
)
.
Finally, the one-particle eigenfunctions are given by plane-waves Ψλ
1(y) = eiyλ.
Note, that equation (2.1) is nothing other as the expansion of the N -particle function
Ψλ
N (x1, . . . , xN ) over the product Ψγ
N−1(x1, . . . , xN−1) · ΨΛ−Γ
1 (xN ). The expansion coefficients
are given by products of Γ functions. This property – the possibility to find the expansion coeffi-
cients 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 .1
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 ≥ 1∫
RN
Ψλ
N (x)
(
Ψλ′
N (x)
)†
dNx = µ−1
N (λ)δN (λ, λ′), (2.2a)∫
RN
Ψλ
N (x)
(
Ψλ
N (x′)
)†
µN (λ) dNλ = δN (x− x′), (2.2b)
where δN (x− x′) =
∏N
k=1 δ(xk − x′k) and
δN (λ, λ′) =
1
N !
∑
w∈SN
δN (λ′ − wλ), wλ =
(
λw1 , . . . , λwN
)
(2.3)
and try to prove that these identities hold for N + 1 as well.
1Although 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.
Completeness of SoV Representation for SL(2,R) Spin Chains 5
Let us start with equation (2.2a). Substituting Ψλ
N+1 in the form (2.1) into this equation
and integrating over x one gets that the orthogonality condition is equivalent to the following
equation
lim
ε, ε′→0+
2πδ(Λ− Λ′)
∫
RN
N+1∏
k=1
N∏
j=1
Γ(iλk − iγj + ε) Γ(iγj − iλ′k + ε′)µN (γ) dNγ
= µ−1
N+1(λ) δN+1(λ, λ′). (2.4)
Next, replacingN → N+1 in (2.2b) and projecting both sides on the functions Ψγ
N (x) and Ψγ′
N (x)
one gets that the completeness relation is reduced to the following identity
lim
ε, ε′→0+
∫
RN+1
N∏
k=1
N+1∏
j=1
Γ(iγk − iλj + ε)Γ(iλj − iγ′k + ε′) eiΛ(x′N−xN )µN+1(λ) dN+1λ
= µ−1
N (γ) δ(xN − x′N ) δN (γ, γ′). (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]∫
RN
N+1∏
k=1
N∏
j=1
Γ(αk − iλj) Γ(iλj + βk)µN (λ) dNλ =
∏N+1
k,j=1 Γ(αk + βj)
Γ
(∑N+1
j=1 (αj + βj)
) , (2.6)
where Re(αk) > 0, Re(βk) > 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
lim
ε,ε′→0+
(2π)δ(Λ− Λ′)
∏N+1
k,j=1 Γ(iλ′j − iλk + ε+ ε′)
Γ
(
iΛ′ − iΛ +N(ε+ ε′)
) = µ−1
N+1(λ) δN+1
(
λ, λ′
)
. (2.7)
The proof of this identity is already rather straightforward. Some details can be found in Appen-
dix A.
It takes a little more work to prove the identity (2.5). Having put αN+1 = L and βN+1 = tL,
t > 0, in (2.6) and sending L → +∞ one arrives at the reduced version of the integral (2.6)
which takes the form [9]∫
RN
tiΛ
N∏
k, j=1
Γ(αk − iλj) Γ(iλj + βk)µN (λ) dNλ =
tA
(1 + t)A+B
N∏
k, j=1
Γ(αk + βj), (2.8)
where A(B) =
∑N
k=1 αk(βk). At the same time representing the l.h.s. of equation (2.5) in the
following form (for more details see Lemma 4.3):
lim
L→∞
lim
ε, ε′→0+
1
Γ2N+2(L)
∫
RN+1
N+1∏
k,j=1
Γ(iγk − iλj + ε) Γ(iλj − iγ′k + ε′)
× eiΛ(x′N−xN )µN+1(λ) dN+1λ,
where iγN+1 = −iγ′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
lim
L→∞
lim
ε→0+
Li(Γ′−Γ)
N∏
k,j=1
Γ
(
i(γ′k − γj) + ε
)√ L
4π
cosh−2L
(
xN − x′N
2
)
= µ−1
N (γ) δ(xN − x′N ) δN (γ, γ′). (2.9)
6 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
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 , S0
k , k = 1, . . . , N , where the
index k enumerates the nodes of the chain. The spin operators are the symmetry generators of
the SL(2,R) group [10] which are determined by a real number (spin) sk > 1/2,
S−k = −∂zk , S0
k = zk∂zk + sk, S+
k = z2
k∂zk + 2skzk.
Operators with the index k act in a Hilbert space associated with the k-th site, Hk, which is
the Hilbert space of functions holomorphic in the upper complex half-plane, H+. The scalar
product in the Hilbert space Hk =
{
f ∈ L2(H+, dµsk) : f is holomorphic on H+
}
is defined as
follows
(f, ψ) =
∫
(f(zk))
†ψ(zk)µsk(zk) d2zk. (3.1)
The measure takes the form
µs(zk) =
2s− 1
π
θ(Im zk)(2 Im zk)
2s−2, (3.2)
where θ(x) is the Heaviside step function. The operators Sαk are anti-hermitian with respect to
the scalar product (3.1).
Function in Hk can be represented by Fourier integrals where the integration runs only over
positive momenta
f(z) =
∫ ∞
0
eipzF [f ](p) dp.
Then, in Fourier space, viz. in the momentum representation, the scalar product takes the form
(f, ψ) = Γ(2s)
∫ ∞
0
(
F [f ](p)
)∗F [ψ](p)p1−2s dp.
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,
Lk(u) = u+ i
(
S0
k S−k
S+
k −S0
k
)
,
where u ∈ C is the spectral parameter. The monodromy matrix for the closed chain of length N
has the form [42]
TN (u) = L1(u+ ξ1) · · ·LN (u+ ξN ) =
(
AN (u) BN (u)
CN (u) DN (u)
)
,
while, for the open spin chain, it is given by the following expression [38]
TN (u) = TN (−u)σ2T
t
N (u)σ2 =
(
AN (u) BN (u)
CN (u) DN (u)
)
, (3.3)
Completeness of SoV Representation for SL(2,R) Spin Chains 7
where σ2 is the Pauli matrix. The entries of the monodromy matrices form commuting polyno-
mial operator families [38, 42] {AN (u)}u∈C, {BN (u)}u∈C, {BN (u)}u∈C:
[AN (u), AN (v)] = [BN (u), BN (v)] = [BN (u),BN (v)] = 0,
which act on the Hilbert space of the model, HN =
⊗N
k=1Hk. Operators in each of the commut-
ing 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 repre-
sentation of separated variables [40]. For the homogeneous chains, viz. ξa = 0, the corresponding
systems for BN , BN and AN operators were constructed in [1, 6, 7], respectively. Below, we
recall these constructions and, on the occasion, extend them to the general case of inhomoge-
neous spin chains where the ξa’s are generic. Since the technical details are essentially the same
in all three cases, we consider in some detail the BN -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 ∈ H+, and the variable
α ∈ C which is called index,
Dα(z, w) =
(
i
z − w̄
)α
=
1
Γ(α)
∫ ∞
0
eip (z−w̄)pα−1 dp.
This is a single valued function2 of z, w which is fixed by the condition arg
(
i/(x + iy)
)
→ 0
for x→ 0. Some properties of the function Dα can be found in [9].
3.1 BN operator
3.1.1 Layer operators
The eigenfunctions can be most conveniently written down in term of the so-called layer opera-
tors. Let Λn+1(γ, x) be an operator which maps functions of n complex variables to functions
of n + 1 variables. It depends on the spectral parameter x ∈ C and the complex vector γ =
(α1, . . . , αn, β1, . . . , βn) ∈ C2n. Its action takes the form
[Λn+1(γ, x)f ] (z1, . . . , zn+1) =
∫
· · ·
∫ n∏
k=1
Dαk−ix(zk, wk)Dβk+ix(zk+1, wk)
× f(w1, . . . , wn)
n∏
a=1
µ(αa+βa)/2(wa) d2w1 · · · d2wn.
The weight function µs(wj) has been defined in equation (3.2). The integral is well defined
provided Re(αk − ix) > 0, Re(βk + ix) > 0 for k = 1, . . . , n.
In the momentum representation obtained by taking the Fourier transform
f(z1, . . . , zn) =
∫ ∞
0
· · ·
∫ ∞
0
F [f ](p1, . . . , pn) ei
∑n
k=1 pkzk dp1 · · · dpn
the action of the layer operator can be expressed as
F [Λn+1(γ, x)g] (q1, . . . , qn+1) = λn(γ, x)
∫ q2
0
d`1
p1
∫ q3
0
d`2
p2
· · ·
∫ qn
0
d`n−1
pn−1
×F [g](p1, p2, . . . , pn)
n∏
k=1
(
qk − `k−1
pk
)αk−ix−1( `k
pk
)βk+ix−1
, (3.4)
2In many cases it is quite helpful to visualize all further constructions as Feynmann diagrams with the func-
tion Dα playing the role of a propagator.
8 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
where `k are the “loop” momenta, pk = qk + `k − `k−1 and `0 ≡ 0, `n ≡ qn+1 and the factor λn
reads
λn(γ, x) =
n∏
k=1
Γ(αk + βk)
Γ(αk − ix) Γ(βk + ix)
.
Note also that all momenta in (3.4) are positive and
∑n+1
k=1 qk =
∑n
k=1 pk.
The layer operators possess two important properties. Let us define a map t : C2n 7→ C2(n−1)
as follows
tγ = t(α1, . . . , αn, β1, . . . , βn) = (α1, . . . , αn−1, β2, . . . , βn).
It can be checked that the operators Λn satisfy the permutation identity
Λn+1(γ, x) Λn(tγ, x′) = Λn+1(γ, x′) Λn(tγ, x). (3.5)
The derivation is based on integral identities for the functions Dα which can be found in [7].
Next, for the spin chain of length N , we introduce the following combinations of spins and
impurities
sk = sk − iξk, s̄k = s∗k = sk + iξk, k = 1, . . . , N,
and define the vector γN ∈ C2N−2:
γN = (s1, . . . , sN−1, s̄2, . . . , s̄N ). (3.6)
It can be shown, see [7], that the operator ΛN (γN , x) is nullified by BN (x),
BN (x)ΛN (γN , x) = 0. (3.7)
These two properties of the layer operators are crucial for constructing the eigenfunctions of the
operator BN (u).
3.1.2 Eigenfunctions
Let us define a function of N complex variables
ΨN
p,x(z) = κNpS−1/2
[
ΛN (γN , x1) ΛN−1 (tγN , x2) · · ·Λ2
(
tN−2γN , xN−1
)
Ep
]
(z). (3.8)
Here x = (x1, . . . , xN−1), z = (z1, . . . , zN ), Ep is the exponential function, Ep(w) = eipw, p > 0,
S ≡
∑N
k=1 sk and the normalisation constant κN reads
κ−1
N =
(
N∏
k=1
Γ(sk + s̄k)
)1/2 ∏
1≤i<j≤N
Γ(si + s̄j).
Following the lines of [22] one can show that the integrals arising from the action of the Λk’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 ΨN
p,x is a symmetric
function of x1, . . . , xN−1 which satisfies the equation BN (xk)Ψ
N
p,x(z) = 0 for all k. Taking into
account that the operator BN (u) is a polynomial of degree N − 1 in u, BN (u) = uN−1
∑N
k=1 S
−
k
+ · · · , and that
∑N
k=1 S
−
k ·ΛN (γN , x1) = ΛN (γN , x1) ·
∑N−1
k=1 S−k one gets
BN (u)ΨN
p,x(z) = p(u− x1) · · · (u− xN−1) ΨN
p,x(z).
Completeness of SoV Representation for SL(2,R) Spin Chains 9
In the momentum representation, the function F
[
ΨN
p,x
]
(q1, . . . , qN ) is given by the convolution
of the layer operators in the momentum representation, equation (3.4), acting on the func-
tion δ(p− q).
Our aim is to show that the functions
{
ΨN
p,x, p > 0, x ∈ RN−1
}
form a complete orthogonal
set in the Hilbert space HN . 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 ΨN
p,x(z) do not belong to the Hilbert space HN . However, they
allow one to define a linear transform from the Hilbert space HB
N defined below into HN :
HB
N = L2(R+)⊗ L2
sym
(
RN−1,dµBN−1(x)
)
, dµBN−1(x) = µBN−1(x) dN−1x,
where we agree upon
µBN−1(x) =
1
(2π)N−1(N − 1)!
∏N−1
k=1
∏N
j=1 Γ(sj − ixk)Γ(s̄j + ixk)∏
j<k Γ(i(xk − xj)) Γ(i(xj − xk))
. (3.9)
To start with, given a smooth, compactly supported function ϕ on R+ × RN−1, one introduces
the transform[
TB
Nϕ
]
(z) =
∫
R+
∫
RN−1
ϕ(p, x)ΨN
p,x(z)µBN−1(x) dp dN−1x. (3.10)
Theorem 3.1. For any smooth, compactly supported function ϕ on R+ × RN−1, TB
Nϕ ∈ HN
and the following relation holds∥∥TB
Nϕ
∥∥2
HN
= ‖ϕ‖2HBN ≡
∫
R+
∫
RN−1
|ϕ(p, x)|2 µBN−1(x) dp dN−1x. (3.11)
As such, TBN extends to a linear isometry TBN : HB
N 7→ HN satisfying∥∥TB
Nϕ
∥∥2
HN
= ‖ϕ‖2HBN .
One may already draw several consequences from this theorem. First of all, (3.11) ensures
that the system of functions
{
ΨN
p,x, p ∈ R+, x ∈ RN−1
}
forms an orthogonal system in HN ,
viz. that(
ΨN
p′,x′ ,Ψ
N
p,x
)
HN
= δ(p− p′) δN−1(x, x′)
(
µBN−1(x)
)−1
,
where the multi-dimensional Dirac delta-function has been introduced in (2.3) while µBN (x) has
been defined in (3.9). This corresponds to the orthogonality relation for the system
{
ΨN
p,x(z)
}
.
Next, the equality (3.11) implies that ‖TB
N‖ = 1 and that the image of HB
N is a closed
subspace of HN . Now, if one is able to show that TB
N is a unitary operator, what amounts to
showing TB
NHB
N = HN , then this will also ensure that the system of functions
{
(p, x) 7→ ΨN
p,x(z),
z ∈ (H+)N
}
forms an complete system in HN , viz. that given za = xa + iya(
ΨN
∗,∗(z
′),ΨN
∗,∗(z)
)
HBN
= I(z, z′).
The r.h.s. of this equation, I(z, z′), is the kernel of the unit operator in HN – the so-called
reproducing kernel, see, e.g., [16] – which takes the form I(z, z′) =
∏N
k=1 Ik(zk, z′k), where
Ik(zk, z′k) = D2sk(zk, z
′
k) =
(
i
zk − z̄′k
)2sk
.
10 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
It means that for any Ψ(z) ∈ HN the following identity holds
Ψ(z) =
∫
I(z, z′)Ψ(z′)
N∏
k=1
µsk(zk) d2zk.
The unitarity of TB
N , 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, com-
pactly supported functions ϕ on R+ ×RN−1. For that purpose, let us introduce the regularised
function ΨN,ε
p,x (z) which is obtained from ΨN
p,x(z) by giving small positive imaginary parts to all
variables xk, xk → xk + iεk, εk > 0 and by changing s̄N → s̄εN = s̄N +
∑N
k=1 εk in the definition
of the vector γN , equation (3.6). Further, let[
TB,ε
N ϕ(z)
]
=
∫
R+
∫
RN−1
ϕ(p, x) ΨN,ε
p,x (z)µBN−1(x) dp dN−1x.
One may readily check that
[
TB,ε
N ϕ
]
→
[
TB
Nϕ
]
(z) pointwise as ε = (ε1, . . . , εN )→ 0+. We want
to show that
lim
ε,ε′→0+
(
TB,ε′
N ϕ,TB,ε
N ϕ
)
HN
=
∥∥TB
Nϕ
∥∥2
HN
.
Let us demonstrate that one can invoke Fubini’s theorem to get∥∥TB,ε
N ϕ
∥∥2
HN
=
∫
(R+)2
dp dp′
∫
R2(N−1)
dN−1x dN−1x′ϕ(p, x)ϕ∗(p′, x′)µBN−1(x)
× µBN−1(x′)
(
ΨN,ε′
p′,x′ ,Ψ
N,ε
p,x
)
HN
.
The scalar product of the regularised functions ΨN,ε
p,x may be computed in closed form as [7](
ΨN,ε′
p′,x′ ,Ψ
N,ε
p,x
)
HN
= δ(p− p′)C(ε,ε′)
N (p, x, x′),
where
C
(ε,ε′)
N (p, x, x′) =
(
κ2
N
κεNκε
′
N
)
p
∑N
k=1(εk+ε′k)
Γ
(
εN + ε′N + i
∑N−1
k=1 (xk − x′k)
)
Γ
(∑N
k=1 εk + ε′k
)
×
∏N−1
k,j=1 Γ
(
i(x′k − xj) + εkj
)∏N−1
k=1 Γ
(
s̄εN+ixk−εk
)
Γ
(
sε
′
N−ix′k−εk
)∏N−1
j=1 Γ
(
s̄j+ix′k+ε′k
)
Γ(sj−ixk+εk)
, (3.12)
and εkj = ε′k + εj and κεN = κN (sε).
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
(ε,ε′)
N (p, x, x′) in the form of a multidimensional momentum integral (which
can be thought of as a Feynman diagram)
C
(ε,ε′)
N (p, x, x′) =
∫
X
f
(
ε, p, x, x′, {`ij}
)∏
ij
d`ij . (3.13)
Here the function f is a product of linear combinations of momenta `ij and p raised to some
powers. It is important to note that all these combinations are positive and that the integration
Completeness of SoV Representation for SL(2,R) Spin Chains 11
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∫ ∣∣f (ε, p, x, x′, {`ij})∣∣∏
ij
d`ij = R(x, x′, ε)
∫
f (ε, p, 0, 0, {`ij})
∣∣
ξ1=···=ξN=0
∏
ij
d`ij ,
where R(x, x′, ε) is some nonsingular factor given by a product of Γ functions. The function
f (ε, p, 0, 0, {`ij})
∣∣
ξ1=···=ξN=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 arbi-
trary order, what thus justifies the result (3.12).
Equation (3.12) therefore leads to
(
TB,ε′
N ϕ,TB,ε
N ϕ
)
HN
=
∫
R+
∫
R
· · ·
∫
R
ϕ(p, x)(ϕ(p, x′))?C
(ε,ε′)
N (p, x, x′)
× µBN−1(x)µBN−1(x′) dp dN−1x dN−1x′,
where we recall that ϕ is a smooth function with a compact support. For ε, ε′ → 0+ the integral
in the r.h.s. can be easily estimated, see Appendix A for the details, resulting in(
TB,ε′
N ϕ,TB,ε
N ϕ
)
HN
= K + o(1),
where
K =
∫
R+
∫
RN−1
|ϕ(p, x)|2µBN−1(x) dp dxN−1
and µBN−1 is as introduced in (3.9).
Since at ε → 0+, one has that
[
TB,ε
N ϕ
]
→
[
TB
Nϕ
]
(z) almost everywhere, it follows from
Fatou’s theorem that∥∥TB
Nϕ
∥∥2
HN
≤ lim inf
ε→0+
∥∥TB,ε
N ϕ
∥∥2
HN
= K.
Thus, the function TB
Nϕ belongs to the Hilbert space HN . Finally, taking into account that(
TB
Nϕ,T
B,ε
N ϕ
)
HN
= K + o(1) one derives from
∥∥TB
Nϕ− T
B,ε
N ϕ
∥∥2
HN
≥ 0 that K ≤
∥∥TB
Nϕ
∥∥2
. Thus
one gets for the norm of TB
Nϕ,
∥∥TB
Nϕ
∥∥2
HN
= K.
Finally, the remaining follows from the fact that the set of smooth functions with a compact
support is dense in the Hilbert space HB
N . �
3.2 AN operator
In this section we give a brief description of the eigenfunctions of the operator AN . We start
with defining of a layer operator suitable for this case. Let η be a complex vector and
η = (α1, . . . , αn+1, β1, . . . , βn) ∈ C2n+1. (3.14)
12 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
The layer operator Λ
(σ)
n+1(η, x), which depends on the vector η and two complex parameters,
x and σ, Imσ ≥ 0, maps a function of n-complex variables to a function of n + 1 variables as
follows[
Λ
(σ)
n+1(η, x)f
]
(z1, . . . , zn+1)=Dαn+1−ix(zn+1, σ)
∫
· · ·
∫ n∏
k=1
Dαk−ix(zk, wk)Dβk+ix(zk+1, wk)
× f(w1, . . . , wn)
n∏
a=1
µ(αa+βa)/2(wa) d2w1 · · · d2wn. (3.15)
The integrals converge provided Re(αk − ix) > 0, Re(βk + ix) > 0 for k = 1, . . . , n.
Let % be a map: C2n+1 7→ C2n−1,
%η = %(α1, . . . , αn+1, β1, . . . , βn) = (α1, . . . , αn, β2, . . . , βn).
The layer operators satisfy the following permutation relation [1],
Λ
(σ)
n+1(η, x) Λ(σ)
n (%η, x′) = Λ
(σ)
n+1(η, x′) Λ(σ)
n (%η, x). (3.16)
Let us put
ηN ≡
(
s1, . . . , sN , s̄2, . . . , s̄N
)
(3.17)
and define the function Φ
(σ)
x (z1, . . . , zN ) as
ΦN
σ,x(z) = κN
(
Λ
(σ)
N (ηN , x1)Λ
(σ)
N−1(%ηN , x2) · · ·Λ(σ)
1
(
%N−1ηN , xN
))
.
By virtue of equation (3.16) ΦN
σ,x is a symmetric function of x1, . . . , xN . It can be shown, see,
e.g., [7], that the operator AN (x1) + σBN (x1) annihilates the layer operator Λ
(σ)
N (ηN , x1),(
AN (x1) + σBN (x1)
)
Λ
(σ)
N (γN , x1) = 0
and, hence, the function ΦN
σ,x satisfies the equation(
AN (u) + σ BN (u)
)
ΦN
σ,x(z) = (u− x1) · · · (u− xN )ΦN
σ,x(z).
Thus the function ΦN
x ≡ ΦN
σ=0,x diagonalizes the operator AN (u). For the separated variables x
with small positive imaginary parts the function ΦN
σ,x has a finite norm. Indeed one can find for
the scalar product of two ΦN functions [1]
(
ΦN
σ,y,Φ
N
υ,x
)
HN
=
(
i
σ − ῡ
)i(Ȳ−X)
∏N
k,j=1 Γ (i(ȳk − xj))∏N
k=1
∏N
j=1 Γ(sj − ixk) Γ(s̄j + iȳk)
. (3.18)
Here X =
∑N
k=1 xk, Y =
∑N
k=1 yk.
For real x the functions ΦN
x are orthogonal to each other (see Appendix A for more details)(
ΦN
x′ ,Φ
N
x
)
= lim
σ→0
lim
ε→0+
(
ΦN
σ,x′+iε,Φ
N
x
)
= δN (x, x′)
(
µAN (x)
)−1
, (3.19)
where
µAN (x) =
1
(2π)NN !
∏N
k=1
∏N
j=1
[
Γ(sj − ixk) Γ(s̄j + ixk)
]∏
j<k Γ(i(xk − xj)) Γ(i(xj − xk))
.
Upon repeating the argument given in the previous subsection, one can prove the following
statement:
Completeness of SoV Representation for SL(2,R) Spin Chains 13
Theorem 3.2. Let HA
N be the Hilbert space of symmetric functions
HA
N = L2
sym
(
RN , dµAN (x)
)
, dµAN (x) = µAN (x) dNx.
The transformation TA
N defined for smooth, compactly supported functions χ on RN :
Ψχ(z) =
[
TA
Nχ
]
(z) =
∫
RN
χ(x) ΦN
x (z)µAN (x) dNx (3.20)
extends into a linear isometry from HA
N into HN . In particular it has unit operator norm∥∥TA
N
∥∥ = 1 and satisfies
‖TA
Nχ‖2HN = ‖χ‖2HAN =
∫
RN
|χ(x)|2 µAN (x) dNx.
It will be show in Section 4 that TA
N is, in fact, an unitary map between the corresponding
Hilbert spaces.
3.3 BN operator
Let us construct eigenfunctions of the operator BN (u), see equation (3.3). It can be shown [6]
that BN (u) = (2u + i)B̂N (u), where B̂N (u) = B̂N (−u). In order to write down eigenfunctions
of B̂N (u) we define the corresponding layer operator, Λ̃n+1(η, x), where η ∈ C2n+1, see equa-
tion (3.14) and x ∈ 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 Λ
(σ)
n , defined in the
previous section, equation (3.15), as follows
[
Λ̃n+1(η, x)f
]
(z) =
∫ [
Λ
(σ)
n+1(η, x)Λ(σ)
n (%η,−x)f
]
(z1, . . . , zn+1)µ(αn+αn+1)/2(σ) d2σ. (3.21)
Above, the product of two layer operators, Λ
(σ)
n+1(η, x)Λ
(σ)
n (%η,−x), maps a function of n − 1
variables (w1, . . . , wn−1), f(w1, . . . , wn−1, σ), to a function of n+ 1 variables, z = (z1, . . . , zn+1),
as indicated in the above formula. By virtue of (3.16) the layer operator Λ̃n+1 is an even function
of x,
Λ̃n+1(η, x) = Λ̃n+1(η,−x).
Let ω be a map: C2n+1 7→ C2n−1, defined as
ω γ = ω(α1, . . . , αn+1, β1, . . . , βn) = (α1, . . . , αn, β3, . . . , βn, αn+1).
The layer operators (3.21) satisfy the permutation relation [6]
Λ̃n+1(γ, x)Λ̃n(ωγ, x′) = Λ̃n+1(γ, x′)Λ̃n(ωγ, x)
and is nullified by the operator B̂N (x)
B̂N (x)Λ̃N (ηN , x) = 0, (3.22)
where the vector ηN is given by equation (3.17).
Given z = (z1, . . . , zN ), define the function
ΥN
p,x(z) = κNp
S− 1
2
[
Λ̃N (γN , x1)Λ̃N−1(ωγN , x2) · · · Λ̃2(ωN−2γN , xN−1) · Ep
]
(z),
14 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
where S =
∑N
k=1 sk, x = (x1, . . . , xN−1) and the normalisation factor is
κ−1
N =
( N∏
k=1
Γ(sk + s̄k)
)1/2 ∏
1≤i<j≤N
Γ(si + sj) Γ(si + s̄j) = κN
∏
1≤i<j≤N
Γ(si + sj).
The function ΥN
p,x is a symmetric even function of x1, . . . , xN−1 which is well defined for | Im(xk)|
< mink sk. By virtue of equation (3.22), the function ΥN
p,x diagonalizes the operator B̂N
B̂N (u)ΥN
p,x(z) = p
(
u2 − x2
1
)
· · ·
(
u2 − x2
N−1
)
ΥN
p,x(z).
The functions ΥN
p,x are orthogonal to each other for real separated variables, x ∈ (R+)N−1.
Namely,(
ΥN
q,y,Υ
N
p,x
)
HN
= δ(p− q)δN−1(x, y)
(
µBN−1(x)
)−1
,
where
µBN−1(x) =
1
(2π)N−1(N − 1)!
∏N
j=1
∏N−1
k=1 |Γ(sj + ixk) Γ(sj − ixk)|2∏N−1
n=1 |Γ(2ixn)|2
∏
j<k |Γ(i(xk + xj)) Γ(i(xk − xj))|2
.
We are now in position to formulate the theorem:
Theorem 3.3. Let HB
N be the Hilbert space
HB
N = L2(R+)⊗ L2
sym
(
(R+)N−1, dµBN−1(x)
)
, dµBN−1(x) = µBN−1(x)dN−1x.
The transformation TB
N defined for smooth, compactly supported functions φ on R+ × (R+)N−1
that are symmetric in respect to the last N − 1 variables as[
TB
Nφ
]
(z) =
∫
(R+)N
φ(p, x) ΥN
p,x(z)µBN−1(x) dp dN−1x
extends to a linear isometry from HB
N into HN . In particular, it has unit operator norm∥∥TB
N
∥∥ = 1 and satisfies∥∥TB
Nφ
∥∥2
HN
= ‖φ‖2HB
N
=
∫
RN+
|φ(p, x)|2 µBN−1(x) dp dN−1x.
We will show in Section 4 that TB
N is an unitary operator.
4 Completeness
In the previous section we constructed three systems of functions, Ψx,p, Φx and Υx,p. They
allow one to define the linear operators, Tα
N , α = {B,A,B}, which map the Hilbert spaces Hα
N
to the Hilbert space HN . For N = 1 the transformations TB
1 and TB
1 are the Fourier transform
and TA
1 is the Mellin transform. Thus, these transformations are unitary maps and, in particular,
Tα
1Hα
1 = H1.
In order to prove the unitarity of the maps Tα
N for arbitrary N we use induction on N .
Namely, we will show that if the map TA
N is unitary then the maps TB
N , TB
N and TB
N+1 are also
unitary. We also show that the unitarity of the map TB
N implies the one for TA
N . Schematically
it is shown on the diagram below
BN AN BN+1 AN+1
BN BN+1.
Completeness of SoV Representation for SL(2,R) Spin Chains 15
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 R
(
TB
N
)
is a closed subspace of the Hilbert
space HN . If this subspace coincides with the whole HN then the orthogonal complement is
trivial, R
(
TB
N
)⊥
= 0. Since R
(
TB
N
)⊥
= ker
(
TB
N
)?
it is enough to prove that the kernel of the
adjoint operator
(
TB
N
)?
is empty. In order to do it let us consider a linear map from HA
N to HB
N
defined by
SBA =
(
TB
N
)?
TA
N . (4.1)
Since the map TA
N is an isometry, by assumption, it maps ker(SBA) 7→ ker
(
TB
N
)?
. Our immediate
aim is to show that ker(SBA) = 0.
We prove the following statement:
Lemma 4.1. Let SBA be the operator from HA
N to HB
N defined in equation (4.1). Then, for any
χ ∈ HA
N the following holds
‖SBAχ‖2HBN = ‖χ‖2HAN . (4.2)
Proof. First, we calculate the action of the operator SBA on the space of smooth functions
with a compact support, χ(x). The action of TA
N on a function χ(x) is given by equation (3.20).
In full similarity with the construction in Section 3.1.2, we define the regularized function TA,σ,ε
N χ
obtained by replacing ΦN
x in (3.20) by ΦN
σ,x+iε, where x = (x1, . . . , xN ) and ε = (ε1, . . . , εN ),
all εk > 0,
TA,σ,ε
N χ(z) =
∫
RN
χ(x) ΦN
σ,x+iε(z) dµAN (x). (4.3)
As σ, ε → 0 one has that
∥∥TA,σ,ε
N χ − TA
Nχ
∥∥
HN
→ 0. Since ‖TB
N‖ = 1, the adjoint to TB
N is
a bounded operator which acts on a vector Ψ by projecting it on the eigenfunction ΨN
p,y, see
equation (3.10),(
TB
N
)?
Ψ =
(
ΨN
p,y,Ψ
)
HN
≡ ϕ(p, y).
Thus we write
ϕ(p, y) ≡ [SBAχ](p, y) =
[(
TB
N
)?
TANχ
]
(p, y) = lim
σ,ε→0
[(
TB
N
)?
TA,σ,ε
N χ
]
(p, y)
= lim
σ,ε→0
(
ΨN
p,y, T
A,σ,ε
N χ
)
HN
.
Moreover, one has
‖ϕ‖2HBN = lim
σ→0
lim
ε→0+
‖ϕσ,ε‖2HBN , ϕσ,ε(p, y) =
(
ΨN
p,y, T
A,σ,ε
N χ
)
HN
. (4.4)
The further analysis depends on the remarkable fact that the scalar product of the functi-
ons ΨN
p,y and ΦN
σ,x+iε can be obtained in a closed form [1]:(
ΨN
p,y,Φ
N
σ,x+iε
)
HN
= p−1/2−iΞ−iX+Ee−ipσ̄
∏N
k=1
∏N−1
j=1 Γ(i(yj − xk) + εk))∏N
j=1
(∏N−1
k=1 Γ(s̄j + iyk)
∏N
k=1 Γ(sj − ixk + εk)
) , (4.5)
16 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
where X =
∑N
k=1 xk and Ξ =
∑N
k=1 ξk, and E =
∑N
k=1 εk. That is
ϕσ,ε(p, y) =
p−1/2−iΞ+Ee−ipσ̄∏N
j=1
∏N−1
k=1 Γ(s̄j + iyk)
×
∫
RN
∏N
k=1
∏N−1
j=1 Γ(i(yj − xk) + εk)∏N
k,j=1 Γ(sj − ixk + εk)
p−iXχ(x) dµAN (x). (4.6)
By assumption the function χ is nonzero only in a compact region. Therefore the function
ϕσ,ε(p, y) grows no faster that some power of y for large y while at large p it decays exponentially
fast ∼ exp{− Im(σ)p}). Taking into account that the measure µBN−1(y) decays exponentially
fast for large y
µBN−1(y) ' (4π)
N(N−1)
2
2N−1(N − 1)!
∏
1≤i<j≤N−1
yij sinhπyij
N∏
j=1
N−1∏
k=1
y
2sj−1
k e−π|yk+ξj |,
one concludes that the normalisation integral for ϕσ,ε converges
‖ϕσ,ε‖2HBN =
∫
R+
∫
RN−1
|ϕσ,ε(p, y)|2 dpdµBN−1(y) <∞.
Moreover, substituting the expression for ϕσ,ε, 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 Γ(i(X ′−X) + 2E)(2 Imσ)i(X−X′)−2E , while the integral over y can be calculated in closed
form [14, Theorem 5.1], see also equation (2.6). Namely,
1
(N − 1)!
∫
RN
∏N
k=1
∏N−1
j=1 Γ(i(yj − xk) + εk) Γ(i(x′k − yj) + εk)∏
j<k Γ(i(yk − yj)) Γ(i(yj − yk))
N−1∏
m=1
dym
2π
=
∏N
k,j=1 Γ(i(x′k − xj) + εk + εj)
Γ(i(X ′ −X) + 2E)
. (4.7)
Thus we get for the norm of ϕσ,ε
‖ϕσ,ε‖2HBN =
∫
R2N
(
(2 Imσ)i(X−X′)−2E
∏N
k,j=1 Γ(i(x′k − xj) + εk + εj)∏N
k,j=1 Γ
(
s̄j + ix′k + εk
)
Γ(sj − ixk + εk)
)
× χ(x)(χ(x′))∗ dµAN (x) dµAN (x′). (4.8)
Note that the expression in the bracket is nothing else as the scalar product,
(
ΦN
σ,x′+iε,Φ
N
σ,x+iε
)
HN
,
see equation (3.18). Finally, taking into account (3.19), see also Appendix A, we obtain that for
any smooth function χ with a compact support
‖ϕ‖2HBN = lim
σ→0
lim
ε→0+
‖ϕσ,ε‖2HBN = ‖χ‖2HAN =
∫
RN
|χ(x)|2 dµAN (x)
or
‖SBAχ‖2HBN = ‖χ‖2HAN .
Since the space of smooth, compactly supported functions is dense in HA
N this equation holds
on the whole Hilbert space. �
Completeness of SoV Representation for SL(2,R) Spin Chains 17
The identity (4.2) implies that ker SBA = 0 and hence R
(
TB
N
)
= HN , which guarantees the
unitarity of the map TB
N .
The proof of the unitarity of the maps TB
N and TB
N+1 follows the same lines and is based on
the following result:
Lemma 4.2. Let SBA and SB be maps from HA
N 7→ HB
N and HA
N ⊗ HA
1 7→ HB
N+1 defined as
follows
SBA =
(
TB
N
)?
TA
N and SB =
(
TB
N+1
)? (
TA
N ⊗ TA
1
)
.
Provided the map TA
N : HA
N 7→ HN is unitary the following identities,
‖SBAχ‖2HB
N
= ‖χ‖2HAN and ‖SBχ′‖2HBN+1
= ‖χ′‖2HAN⊗HA1 ,
hold for any χ ∈ HA
N , χ′ ∈ HA
N ⊗HA
1 .
Proof. In the proof of these assertions, the main difference from the proof of Lemma 4.1 lies
in the type of the Γ-integrals arising in the process. We briefly discuss these differences below.
For the SBA operator the problem is reduced to calculating the norm of the function
φσ,ε(p, y) =
(
ΥN
p,y, T
A,σ,ε
N χ
)
HN
,
which is an analogue of the function ϕσ,ε, see equation (4.4). The relevant scalar product takes
the form3
(ΥN
p,y,Φ
N
σ,x+iε)HN = p−1/2−iΞ−iX+Ee−ipσ̄ 1∏
1≤k<j≤N Γ(−i(xk + xj) + εk + εj)
×
∏N
k=1
∏N−1
j=1 Γ(−i(xk ± yj) + εk)∏N
k=1
(∏N−1
j=1 Γ(s̄k ± iyj)
)(∏N
m=1 Γ(sk − ixm + εm)
) .
Above, the symbol ± stands for
f(a± b) ≡ f(a+ b)f(a− b).
Calculating the norm
‖φσ,ε‖2HB
N
=
∫
RN+
|φσ,ε(p, y)|2 dp dµBN−1(y)
one substitutes the function φσ,ε in the form
φσ,ε(p, y) =
∫
RN+
(
Υp,y,Φ
N
σ,x+iε
)
HN
χ(x) dpdµAN (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],
1
(N − 1)!
∫
RN
∏N
k=1
∏N−1
j=1 Γ(i(±yj − xk) + εk) Γ(i(x′k ± yj) + εk)∏N−1
m=1 Γ(±2iym)
∏
j<k Γ(i(yk ± yj)) Γ(−i(yk ± yj))
N−1∏
m=1
dym
4π
=
∏N
k,j=1 Γ(i(x′k − xj) + εkj)
∏
1≤m<n≤N Γ(i(x′n + x′m) + εnm)Γ(−i(xn + xm) + εnm)
Γ(i(X ′ −X) + 2E)
,
3For the homogeneous chain this scalar product was calculated in [9] and its extension to the general case is
straightforward.
18 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
where εkj = εj + εk. We also extended the integral over yk from the positive half-axis to the
real line using the symmetry of the integrand with respect to the reflection yk → −yk. Finally,
collecting all factors, one finds that the norm ‖φσ,ε‖2HB
N
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
‖SBAχ‖2HB
N
= ‖χ‖2HAN
for any χ ∈ HA
N .
Now let us show that the map TB
N+1 is unitary. In this case we consider the map
SB =
(
TB
N+1
)?(
TA
N ⊗ TA
1
)
.
The last factor in the above equation is the unitary map from HA
N ⊗HA
1 to HN+1 = HN ⊗HN+1,
where HN+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
Ψσ,ε
χ (z) =
∫
RN+1
χ(x) ΦN
σ,x+iε(z)Φ
1
σ,xN+1+iεN+1
(zN+1) dµAN (x) dµA1 (xN+1).
Here z, x, ε and z, x, ε areN and (N+1)-dimensional vectors, respectively, e.g., x = (x1, . . . , xN ),
x = (x1, . . . , xN+1), etc. Note also that the parameter σ is the same for the functions ΦN and Φ1.
Completely similar to the previous consideration one can show that
lim
σ→0
lim
ε→0+
Ψσ,ε
χ = Ψχ ≡
(
TA
N ⊗ TA
1
)
χ.
Again we define the function
ϕσ,ε(p, y) =
(
ΨN
p,y,Ψ
σ,ε
χ
)
≡
(
TB
N+1
)?
Ψσ,ε
χ ,
where y = (y1, . . . , yN ). The scalar product of the function ΨN
p,y and ΦN
σ,x+iε ⊗ Φ1
σ,xN+1+iεN+1
takes the form (see, e.g., [9])
(
ΨN
p,y,Φ
N
σ,x ⊗ Φ1
σ,xN+1
)
HN
=
1
√
p
p−iX−iΞ+iξN+1e−ipσ̄
N∏
k=1
Γ(s̄k + sN+1)
Γ(sk + s̄N+1)
1
Γ(sN+1 − ixN+1)
×
N∏
k,j=1
Γ(i(yj − xk))
Γ(s̄k + iyj) Γ(sj − ixk)
×
N∏
k=1
1
Γ(sN+1 − iyk)
Γ(−i(yk + xN+1))
Γ(−i(xk + xN+1))
.
Calculating the norm of ϕσ,ε 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
‖ϕσ,ε‖2HBN+1
=
∫
R2N+2
(
(2 Imσ)i(X−X′)−2E
∏N
k,j=1 Γ
(
i(x′k − xj) + εk + εj
)∏N
k,j=1 Γ(s̄j + ix′k + εk) Γ(sj − ixk + εk)
×
Γ
(
i(x′N+1 − xN+1) + 2εN+1
)
Γ
(
s̄N+1 + ix′N+1 + εN+1
)
Γ(sN+1 − ixN+1 + εN+1)
)
χ(x)
(
χ(x′)
)∗
× dµAN (x) dµAN (x′) dµA1 (xN+1) dµA1 (x′N+1).
Completeness of SoV Representation for SL(2,R) Spin Chains 19
The analysis of the above expression in the limit ε → 0+ and σ → 0 is exactly the same as
before, see Appendix A. It results in the following expression for the norm
‖ϕ‖2HBN+1
= ‖SBχ‖2HBN+1
= ‖χ‖2HAN⊗HA1 =
∫
RN+1
|χ(x)|2 dµAN (x) dµA1 (xN+1),
that completes the proof of the lemma. �
It follows from Lemma 4.2 ker SBA = 0 and ker SB = 0 and, hence, that the operators
TB
N : HB
N 7→ HN and TB
N+1 : HB
N+1 7→ HN+1 are unitary provided that TA
N is.
The final step required to complete the induction on N is to show that the unitarity of the
map TA
N follows from that of the map TB
N . 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
SAB =
(
TA
N
)?
TB
N , SAB : HB
N 7→ HA
N (4.9)
is trivial.
Lemma 4.3. Let SAB be the linear operator defined in equation (4.9). If the map TB
N : HB
N 7→ HN
is unitary then for any ϕ ∈ HB
N the following identity holds: ‖SABϕ‖2HAN = ‖ϕ‖2HBN .
Proof. Let ϕ(p, x) be a smooth function with compact support having the factorised form
ϕ(p, x) = f(p)ϕ̃(x). (4.10)
The linear span of these functions is dense in HB
N . The action of the operator SAB on a function ϕ
can be represented as follows
χ(y) = [SABϕ](y) =
(
Φy,T
B
Nϕ
)
HN
= lim
ε→0+
(
Φy+iε,T
B
Nϕ
)
HN
= lim
ε→0+
∫
R+
∫
RN−1
(
Φy+iε,Ψ
N
p,x
)
HN
ϕ(p, x) dp dµBN−1(x), (4.11)
where y+iε = (y1 +iε, . . . , yN +iε) and the scalar product of two eigenfunctions is given by equa-
tion (4.5). We also denote the function given by the integral in the above equation by χε(y),
i.e.,
χ(y) = lim
ε→0+
χε(y).
It can be shown, see [22, Lemma 3.1], that the function χ(y)
∏
i<k yik is a smooth function. Our
final aim is to show that χ ∈ HB
N and that the ‖χ‖2HAN = |ϕ‖2HBN .
In contrast to the previous cases, the function χ(y) does not decrease fast enough for large yk
to justify changing the order of integration over x, x′ and y in the norm integral. To overcome
this difficulty we proceed as follows. Let us define a regularized function χL(y) as
χL(y) = χ(y)gL(y), where gL(y) =
N∏
k=1
∣∣∣∣Γ(L+ iyk)
Γ(L)
∣∣∣∣ .
The factor gL(y) has the following properties:
(i) gL(y) < 1 for all y,
(ii) gL(y)→ 1 monotonically as L→∞ for fixed y,
(iii) gL(y) ∼ exp
{
− π/2
∑
k |yk|
}
for fixed L and |yk| → ∞.
20 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
It follows from (ii) that for any bounded region D ∈ RN∫
D
|χ(y)− χL(y)|2 dµAN (y)→ 0 as L→∞.
Due to (iii) one concludes that, for finite L, the integral of |χL(y)|2 over RN converges
IL =
∫
RN
|χL(y)|2 dµAN (y) <∞.
Then one derives the following inequality∫
D
|χ(y)|2 dµAN (y) ≤
∫
D
|χ(y)− χL(y)|2 dµAN (y) +
∫
D
|χL(y)|2 dµAN (y) ≤ 2−M + IL,
which holds for any L greater than some LM . Since M is arbitrary we get the following inequality∫
D
|χ(y)|2 dµAN (y) ≤ lim
L→∞
IL ≡ I,
which holds for an arbitrary bounded region D. Therefore
∫
RN |χ(y)|2dµAN (y) ≤ I. Since due
to (i) I ≤
∫
RN |χ(y)|2dµAN (y) we conclude that
‖χ‖2HAN =
∫
RN
|χ(y)|2 dµAN (y) = I.
Thus one has to find the limit of IL at L→∞. First we note that IL can be written in the form
IL =
∫
RN
|χL(y)|dµAN (y) =
∫
RN
lim
ε→0+
|χεL(y)|2 dµAN (y) = lim
ε→0+
∫
RN
|χεL(y)|2 dµAN (y),
where χεL(y) = gL(y)χε(y). At the last step, the limit ε → 0+ is taken after the integration.
It is possible to do so since the function χεL(y) is bounded, |χεL(y)| < CL for all y, and the
measure µAN (y) decays exponentially fast at large y,
µAN (y) ' (4π)
N(N−1)
2
N !
∏
1≤i<j≤N
yij sinhπyij
N∏
k=1
N∏
j=1
y
2sj−1
k e−π|yk+ξj |,
so that the measure of the whole space is finite∫
RN
dµAN (y) = 2−
∑N
i=1(si+s̄i)
N∏
k,j=1
Γ(sk + s̄j).
The calculation of the integral of |χεL(y)|2 follows the familiar pattern: one substitutes the
function χεL(y) using (4.11) 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 IL
in the form
IL = lim
ε→0+
∫ ∞
0
∫ ∞
0
∫
R2N−2
ϕ̂(p, x)
(
ϕ̂(p′, x′)
)∗
Mε(L, p, x, p
′, x′) dp dp′ dµBN−1(x) dµBN−1(x′),
where
ϕ̂(p, x) = p−1/2+iΞ ϕ(p, x)
(
N∏
j=1
N−1∏
k=1
Γ(sj − ixk)
)−1
Completeness of SoV Representation for SL(2,R) Spin Chains 21
and
Mε(L, p, x, p
′, x′) =
N−1∏
k,j=1
Γ
(
i(x′j − xk) + 2ε
) Γ(2L)
Γ2(L)
N−1∏
k=1
Γ(L− ixk + ε)Γ(L+ ix′k + ε)
Γ2(L)
×
(
1 +
p
p′
)−L−(N−1)ε+iX(
1 +
p′
p
)−L−(N−1)ε−iX′
. (4.12)
We recall that X(X ′) =
∑N−1
k=1 xk(x
′
k). Due to our assumptions on the function ϕ the integral
in (4.12) is restricted to a finite region hence we can expand the function Mε(L, p, x, p
′, x′)
in series in L−1
Mε(L, p, x, p
′, x′) = Li(X′−X)
N−1∏
k,j=1
Γ(i(x′k − xj) + 2ε)
× 22L−1
√
L
π
(
1 +
p
p′
)−L+iX (
1 +
p′
p
)−L−iX′ (
1 +O
(
1
L
))
, (4.13)
where we put ε→ 0 in non-singular terms. At large L, the dominant contribution to the integral
over p, p′ comes from the region p = p′ and can be easily estimated as
2i(X−X′)
∫ ∞
0
dp |f(p)|2(1 +O(1/L)),
see equation (4.10). The factor in the first line of equation (4.13) 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
I = lim
L→∞
IL =
∫
R+
∫
RN−1
|ϕ(p, x)|2 dp dµBN−1(x) ≡ ‖ϕ‖2HBN .
Thus one concludes that SAB is a norm preserving map, ‖SABϕ‖2HAN ≡ ‖χ‖
2
HAN
= ‖ϕ‖2HBN and,
hence, ker SAB = 0. Therefore one concludes that TA
N is a unitary operator between the Hilbert
spaces HA
N and HN . �
Lemma 4.3 completes the inductive proof that the maps TB
N , TA
N and TB
N 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 SL(2,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, sk,
and inhomogeneity parameters, ξk. 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 SL(2,C) symmetry group.
22 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
A Some representations for multi-dimensional Dirac δ-functions
I. Define
C̃
(ε,ε′)
N (p, x, x′) =
Γ
(
εN + ε′N + i
∑N−1
a=1 (xa − x′a)
)
Γ
(∑N
a=1 εa + ε′a
) ∏N−1
a,b Γ(i(x′b − xa) + ε′b + εa)∏N−1
a6=b Γ(i(x′a − x′b))Γ(i(xa − xb))
.
In the following we show that, in the sense of distributions, it holds
lim
ε,ε′→0+
{
C̃
(ε;ε′)
N
(
p, x, x′
)}
= WN−1(x) · δN−1
(
x, x′
)
,
where
WN−1(x) = (2π)N−1(N − 1)!
N−1∏
a6=b
1
Γ
(
i(xa − xb)
) . (A.1)
In other words, given
I
(ε;ε′)
N =
∫
R+
dp
∫
RN−1
dN−1x
∫
RN−1
dN−1x′C̃
(ε;ε′)
N
(
p, x, x′
)
ϕ
(
p, x
)
ϕ∗
(
p, x′
)
(A.2)
it holds that
lim
ε,ε′→0+
I
(ε;ε′)
N =
∫
R+
dp
∫
RN−1
dN−1xW (x)
∣∣ϕ(p, x)∣∣2.
In order to establish the result, one starts by reorganising the integral in (A.2) as
I
(ε;ε′)
N =
∫
R+
dp
∫
RN−1
dN−1x
∫
RN−1
dN−1x′U
(ε;ε′)
N
(
p, x, x′
)
× det
N−1
[
1
x′k − xj − i(εj + ε′k)
]
−i
∑N
k=1(εk + ε′k)
−i(εN + ε′N ) +
∑N−1
k=1 (xk − x′k)
,
where
U
(ε;ε′)
N
(
p, x, x′
)
=
N−1∏
a<b
{
(xa − xb) (x′b − x′a)
}
Ĉ
(ε;ε′)
N
(
p, x, x′
)
ϕ
(
p, x
)
ϕ∗
(
p, x′
)
,
and
Ĉ
(ε;ε′)
N
(
p, x, x′
)
=(−i)(N−1)2
Γ
(
1+εN+ε′N+
∑N−1
k=1 (xk−x′k)
)∏N−1
a,b=1Γ
(
1+i(x′a−xb)+ε′a+εb
)
Γ
(
1+
∑N
k=1(εk+ε′k)
)∏N−1
a6=b Γ
(
1+i(x′a−x′b)
)
Γ
(
1+i(xa−xb)
) .
Thus, U
(ε;ε′)
N is antisymmetric in x, x′ taken singly, and it is smooth and compactly supported
in x, x′ ∈ RN−1 and smooth in a small neighbourhood of zero in respect to ε, ε′.
Expanding the determinant as a sum over the permutation group and using the antisymmetry
in x, x′ and the smoothness in ε, ε′ of U
(ε;ε′)
N as well as the Sokhotsky–Plemejl formulae for the
limits ε, ε′ → 0+ of the singular factors, one gets that
lim
ε,ε′→0+
I
(ε;ε′)
N = (N − 1)! lim
ε→0+
lim
ε→0+
∫
R+
dp
∫
RN−1
dN−1x
∫
RN−1
dN−1x′U
(0;0)
N
(
p, x, x′
)
×
N−1∏
a=1
1
x′a − xa − iεa
−iε
−iε+
∑N−1
k=1 (xk − x′k)
.
Completeness of SoV Representation for SL(2,R) Spin Chains 23
It is thus enough to study the ε→ 0+, ε→ 0+ limit of the model integral
J
(ε;ε)
N =
∫
RN−1
dN−1x
∫
RN−1
dN−1x′χ
(
x, x′
)N−1∏
a=1
1
x′a − xa − iεa
−iε
−iε+
∑N−1
k=1 (xk − x′k)
,
in which χ is antisymmetric in x, x′ taken singly. Observe that, for fixed x, by the Stone–
Weierstrass theorem, there exists a sequence of smooth, compactly supported functions ϕk,a
on R such that
χ
(
x, x′
)
=
∑
k≥1
N−1∏
a=1
ϕk,a(x
′
a).
Next, one observes that
N−1∏
a=1
ϕk,a(x
′
a)
x′a − xa − iεa
=
N−1∑
s=0
∑
α+∪α−=[[1;N−1]]
|α+|=s
∏
a∈α+
∆xa,x′a;εa
[
ϕk,a
] ∏
a∈α−
ϕk,a(xa)
x′a − xa − iεa
,
where
∆xa,x′a;εa
[
f
]
=
f(x′a)− f(xa)
x′a − xa − iεa
.
Thus, inserting the expansion in the integral, summing up, setting εa = 0 in the regular part of
the integrand and using the antisymmetry in x, x′ of χ, one gets that
lim
ε→0+
lim
ε→0+
J
(ε;ε)
N = lim
ε→0+
lim
ε→0+
N−1∑
s=0
CsN−1
∫
RN−1
dN−1x
∫
RN−1
dN−1x′∆
(s)
x,x′ χ
(
x, (x′s, x
(s+1))
)
×
N−1∏
a=s+1
1
x′a − xa − iεa
−iε
−iε+
∑N−1
k=1 (xk − x′k)
, (A.3)
where ∆
(s)
x,x′ is a composite of operators acting on the variables x1, . . . , xs, x
′
1, . . . , x
′
s
∆
(s)
x,x′ =
s∏
a=1
∆
(0)
xa,x′a
.
Also, establishing (A.3), we took the freedom to relabeling the variables
x(k) = (xk, . . . , xN−1) and x′s = (x′1, . . . , x
′
s).
One may now readily take the integrals in respect to x′s+1, . . . , x
′
N−1 in (A.3), what yields
lim
ε→0+
lim
ε→0+
J
(ε;ε)
N = lim
ε→0+
N−1∑
s=0
CsN−1(2iπ)N−1−s
∫
RN−1
dN−1x
∫
Rs
dsx′
×∆
(s)
x,x′ χ
(
x,
(
x′s, x
(s+1)
)) −iε
−iε+
∑s
k=1(xk − x′k)
.
Apart from the term arising in the last line, the integrand is a smooth function. Thus, by
changing the variables
x ↪→ y =
( s∑
k=1
xk, x2, . . . , xN−1
)
and x′ ↪→ y′ =
( s∑
k=1
x′k, x
′
2, . . . , x
′
s
)
24 S.É. Derkachov, K.K. Kozlowski and A.N. Manashov
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,
lim
ε→0+
lim
ε→0+
J
(ε;ε)
N = (2iπ)N−1
∫
RN−1
dN−1xχ
(
x, x
)
.
This entails the claim.
II. Let us define
SL,εN (x, x′) = Li
∑N
a=1(x′a−xa)
∏N
a,b=1 Γ(i(x′a − xb) + ε)∏N
a6=b Γ
(
i(x′a − x′b)
)
Γ
(
i(xa − xb)
) .
We will show that in the sense of distributions the following identity holds
lim
L→∞
lim
ε→0+
SL,εN (x, x′) = WN (x) δN
(
x, x′
)
,
where WN is defined in equation (A.1). Namely, given
T L,ε
N =
∫
RN
dNx
∫
RN
dNx′SL,εN
(
x, x′
)
ϕ
(
x
)
ϕ∗
(
x′
)
(A.4)
it holds that
lim
L→∞
lim
ε→0+
T L,ε
N =
∫
RN
dNxWN (x)
∣∣ϕ(x)∣∣2.
Repeating the same arguments as above one can rewrite (A.4) in the form
lim
L→∞
lim
ε→0+
T L,ε
N = N ! lim
L→∞
lim
ε→0+
∫
RN
dNx
∫
RN
dNx′VN
(
x, x′
) N∏
a=1
Li(x′a−xa)
x′a − xa − iε
,
where
VN
(
x, x′
)
= (−i)N
2
∏N
a<b(xa − xb) (x′b − x′a)
∏N
a,b=1 Γ
(
1 + i(x′a − xb)
)∏N
a6=b Γ
(
1 + i(x′a − x′b)
)
Γ
(
1 + i(xa − xb)
) ϕ
(
p, x
)
ϕ∗
(
p, x′
)
.
Finally, taking into account that
lim
ε→0+
∫
R
dxLix ϕ(x)
x− iε
= lim
ε→0+
ϕ(0)
∫
R
dx
Lix
x− iε
+
∫
R
dxLixϕ(x)− ϕ(0)
x
= 2πϕ(0) +O(1/ lnL)
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.
Completeness of SoV Representation for SL(2,R) Spin Chains 25
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1 Introduction
2 Preliminaries
3 Spin chains: operators and eigenfunctions
3.1 BN operator
3.1.1 Layer operators
3.1.2 Eigenfunctions
3.2 AN operator
3.3 BN operator
4 Completeness
5 Summary
A Some representations for multi-dimensional Dirac delta-functions
References
|
| id | nasplib_isofts_kiev_ua-123456789-211360 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T13:28:18Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Derkachov, Sergey É. Kozlowski, Karol K. Manashov, Alexander N. 2025-12-30T15:55:51Z 2021 Completeness of SoV Representation for SL(2, ℝ) Spin Chains. Sergey É. Derkachov, Karol K. Kozlowski and Alexander N. Manashov. SIGMA 17 (2021), 063, 26 pages 1815-0659 2020 Mathematics Subject Classification: 33C70; 81R12 arXiv:2102.13570 https://nasplib.isofts.kiev.ua/handle/123456789/211360 https://doi.org/10.3842/SIGMA.2021.063 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 SL(2, ℝ) spin chains. This work was supported by the Russian Science Foundation project No 19-11-00131and by the DFG grants MO 1801/4-1, KN 365/13-1 (A.M.). The work of K.K.K. is supported by CNRS. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Completeness of SoV Representation for SL(2, ℝ) Spin Chains Article published earlier |
| spellingShingle | Completeness of SoV Representation for SL(2, ℝ) Spin Chains Derkachov, Sergey É. Kozlowski, Karol K. Manashov, Alexander N. |
| title | Completeness of SoV Representation for SL(2, ℝ) Spin Chains |
| title_full | Completeness of SoV Representation for SL(2, ℝ) Spin Chains |
| title_fullStr | Completeness of SoV Representation for SL(2, ℝ) Spin Chains |
| title_full_unstemmed | Completeness of SoV Representation for SL(2, ℝ) Spin Chains |
| title_short | Completeness of SoV Representation for SL(2, ℝ) Spin Chains |
| title_sort | completeness of sov representation for sl(2, ℝ) spin chains |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211360 |
| work_keys_str_mv | AT derkachovsergeye completenessofsovrepresentationforsl2rspinchains AT kozlowskikarolk completenessofsovrepresentationforsl2rspinchains AT manashovalexandern completenessofsovrepresentationforsl2rspinchains |