Completeness of SoV Representation for \mathrm{SL}(2,\mathbb R) Spin Chains
Sergey ?. Derkachov, Karol K. Kozlowski, Alexander N. Manashov
aa r X i v : . [ m a t h - ph ] F e b DESY–21–024
Completeness of SoV representation for
SL(2 , R ) spinchains Sergey ´E. DERKACHOV † , Karol K. KOZLOWSKI § and Alexander N. MANASHOV ‡† St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences,Fontanka 27, 191023 St. Petersburg, Russia
E-mail: [email protected] § Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique,F-69342 Lyon, France.
E-mail: [email protected] ‡ Institut f¨ur Theoretische Physik, Universit¨at Hamburg, D-22761 Hamburg, Germany,St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences,Fontanka 27, 191023 St. Petersburg, Russia
E-mail: [email protected]
Abstract.
This work develops a new method, based on the use of Gustafson’s integrals andon 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 quantumintegrable models. We examine in detail the case of the SL(2 , R ) spin chains. Key words:
Spin chains, Separation of Variables, Gustafson’s integrals
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 methodallowing one to construct the eigenvectors and eigenvalues of the mentioned Hamilton operatorby means of combinatorial expressions involving auxiliary parameters. In order to obtain aneigenvector, one needs to impose certain constraints on these parameters, the so-called BetheAnsatz equations. Over the years, the method was refined and applied to numerous other models,such as the XXZ Heisenberg chain [32], the δ -function Bose gas [27], or the Hubbard model [28],so as to name a few. In the late 70s, the method was raised to a higher level of effectivenessby Faddeev, Sklyanin, Takhtadjan [10], thus becoming known as the so-called algebraic BetheAnsatz. This new approach provided an algebraic setting allowing one to connect quantumintegrability to the representation theory of quantum groups, what had several advantages.To start with, the construction of the eigenvectors of a given integrable model was significantlysimplified, hence allowing to address more involved problems such as the calculation of norms [22]and scalar products [42] of Bethe vectors and, subsequently, the one of correlation functions[18,21]. Moreover, the method allowed one to significantly enlarge the family of known integrablemodels, see e.g. the review [25], and in particular efficiently and systematically address thequestion of constructing the eigenvectors of the higher rank integrable models [24]. However,it soon turned out that the method had also its limitations in that not all quantum integrablemodels were within its grasp, the quantum Toda chain being a prominent example thereof. In1985, Sklyanin pioneered a new technique allowing one to address the calculation of the spectrum S.´E. Derkachov, K.K. Kozlowski and A.N. Manashovof this model: the quantum separation of variables [40]. He developed several aspects of themethod in [38, 40], and this progress was subsequently continued by Kharchev and Lebedev[19, 20] relatively to the Toda chain. Derkachov, Korchemsky and Manashov [5, 6], Bytsko andTeschner [3], Silantyev [36] and, more recently, Maillet and Niccoli [29] pushed the developmentof the method in the case of other, more involved, models (see also [14, 33, 34]). Recently, manyimportant results have appeared in this area [4, 12, 13, 30, 31].In fact, there are nowadays many indications that the quantum separation of variables is amuch more general technique for solving quantum integrable models that encompasses the alge-braic Bethe Ansatz [5] and provides one with the quantum analogue of the classical separationof variables technique.In precise terms, the quantum separation of variables consists in exhibiting a map U be-tween an auxiliary Hilbert space h sov and the original Hilbert space h org on which a given modelis formulated. This map should be unitary so as to ensure the equivalence of Hilbert spacestructure 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 ofa given quantum Hamiltonian means that there exists a commutative subalgebra in the spaceof operators { H k } containing the Hamiltonian. Thus, the spectral problem associated with theoriginal Hamiltonian is, in fact, a multi-parameter spectral problem, in that each eigenvectoris associated with the tower of eigenvalues of the H k s. Now the role of the map U is to realisethe unitary equivalence between h sov and h org in such a way that the original multi-dimensional(because the Hamiltonians have a non-trivial structure) and multi-parametric spectral problemon h org is reduced into a multi-parametric (because one has to keep track of all the eigenvalues) one -dimensional spectral problem on h sov . This thus explains the separation of variables ter-minology. In fact, this one -dimensional spectral problem corresponds to the resolution of theso-called Baxter T − Q equation associated with the model, proving in this way a remarkablebridge 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 asdescribed above. First, one should construct a map U satisfying to the desired requirements andthen show that it indeed corresponds to a unitary map between the Hilbert spaces. In fact, theconstruction of U can be dealt with by making the best of the Yang-Baxter algebra underlyingthe integrability of the model. A first method was suggested by Sklyanin in [40]. Later, analternative construction was proposed in [5] where, in particular, it was pointed out that U canbe constructed by using the Baxter Q operator associated with the model. For the first time, theidea of a connection between the Baxter Q operator and separation of variables was apparentlyformulated in the work [26].This last idea was later generalised in [29] to other conserved quantities, in the case of modelshaving finite dimensional local Hilbert spaces. To be more precise about the construction of U ,we recall that the original Hilbert space h org where the model is formulated and h sov wherethe separation of variables takes place can be identified with appropriate L spaces h org = L ( X , dν org ) and h sov = L ( Y , dµ sov ). This is a very general setting which allows X , Y to befinite, discrete or continuous. Upon such an identification of the Hilbert spaces, the map U isdefined as an integral transform acting on smooth, compactly supported functions on Y :[ U ϕ ]( x ) = Z Y ϕ ( y )Ψ y ( x ) dµ sov ( y ) . (1.1)The functions Ψ y ( x ) describing the integral kernel of the transform can be thought of as theanalogues of the function e iyx giving the integral kernel of the Fourier transform. That case, infact, corresponds to X = Y = R and both dν org and dµ sov coinciding with the Lebesgue measure.In the above case, just as { x e iyx } corresponds to the system of generalised eigenfunctionsof translation operators, { x Ψ y ( x ) } corresponds to the system of generalised eigenfunctionsompleteness of SoV representations 3of a commutative operator subalgebra of the representation of the Yang-Baxter algebra whichgives rise to the original model of interest. The construction of U hence boils down to theconstruction of this system of eigenfunctions, what become possible since it is reduced to solvinghypergeometric 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 Ψ y ( x )in the first place, does also ensure that, by construction, U fulfills the desired requirement ofsimplifying the original spectral problem. However, unitarity is a completely different issue. Itboils down to proving the orthogonality and completeness of the system Ψ y ( x ) which can beframed as the following relations understood in the sense of distributions Z X (cid:0) Ψ y ′ ( x ) (cid:1) ∗ Ψ y ( x ) dν org ( x ) = 1 dµ sov /dy δ sov ( y ′ , y ) (1.2)and Z Y (cid:0) Ψ y ( x ′ ) (cid:1) ∗ Ψ y ( x ) dµ sov ( y ) = 1 dν org /dx δ org ( x ′ , x ) . (1.3)Above δ sov ( y ′ , y ), resp. δ org ( x ′ , x ), corresponds to the generalised function which represents theintegral kernel of the identity operator on Y , resp. X . Moreover, dµ sov /dy , resp. dν org /dx , isthe Radon-Nikodym derivative of µ sov , resp. ν org , in respect to the canonical measure on Y ,resp. X .The technique for proving unitarity of U , viz . (1.2)-(1.3), strongly depends on the dimensionof the original Hilbert space h org . If h org is finite dimensional, checking unitarity amounts toa simple comparison of dimensions between h org and h sov . However, many of the physicallyinteresting quantum integrable models are defined on an infinite dimensional Hilbert space h org .There, unitarity is a much more delicate issue. In fact, unitarity was first established for theToda chain case by using harmonic analysis of Lie groups techniques [35, 43]. However themethods which were used to establish this were quite sophisticated and hardly generalisableto the more complex quantum integrable models. The first step towards proving unitarity, ina simpler and systematic way, was achieved in [5] where a quantum inverse scattering basedtechnique for proving the isometry of U was invented. Then, [23] developed a technique, solelybased on the use of natural objects for the quantum inverse scattering, allowing one to proverigorously 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 toestablish 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 [15].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 simplemethod for proving the unitarity of the map realising the separation of variables for the higherspin, non-compact, XXX chains. While focusing on this example, we are deeply convinced of themethod’s generality and hence applicability to many other quantum integrable models possessinginfinite dimensional local Hilbert spaces and which are solvable by the quantum separation ofvariables. In order to illustrate the main features of the method without obscuring them bytechnicalities of the model, as a warm up to our main result, we illustrate how it works in thecase of the Toda chain.The paper is organised as follows. Section 2 outlines, on formal grounds, the key ideas of ourmethod 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 interestto the analysis and establishes their isometry. Finally, Section 4 establishes the isometry oftheir adjoint, and hence completeness of the underlying system of functions giving rise to theirintegral kernels. S.´E. Derkachov, K.K. Kozlowski and A.N. Manashov In this section we illustrate some details of our approach on the example of the open Todachain [16, 37]. which is a one-dimensional system of N particles on the line associated with theHamiltonian H = − N X k =1 ∂ ∂x k + N − X k =1 e x k − x k +1 (2.1)on the Hilbert space L (cid:0) R N , d N x (cid:1) . The model is integrable and can be solved by the Quan-tum Inverse Scattering Method (QISM) [10, 41]. For further discussion it is important thateigenfunctions can be constructed iteratively [19, 20],Ψ λN ( x ) = lim ε → + Z R N − N Y k =1 N − Y j =1 Γ( iλ k − iγ j + ε ) e i (Λ − Γ) x N Ψ γN − ( x ) µ N − ( γ ) N − Y j =1 dγ j , (2.2)where x = ( x , . . . , x N ), λ = ( λ , . . . , λ N ), γ = ( γ , . . . , γ N − ) and Λ = P Nj =1 λ j , Γ = P N − j =1 γ j .The measure µ N ( λ ) – the Sklyanin measure – is given by a product of Γ-functions µ − N ( γ ) = (2 π ) N N ! N Y j
0, Re( β k ) > k . Note, that the integral in the l.h.s. (2.8) is exactly theintegral appearing in (2.6) which, therefore, can be brought to the formlim ε,ε ′ → + (2 π ) δ (Λ − Λ ′ ) Q N +1 k,j =1 Γ( iλ ′ j − iλ k + ε + ε ′ )Γ ( i Λ ′ − i Λ + N ( ε + ε ′ )) = µ − N +1 ( λ ) δ N +1 (cid:0) λ, λ ′ (cid:1) . (2.9)The proof of this identity is already rather straightforward. Some details can be found inAppendix A.It takes a little more work to prove the identity (2.7). Having put α N +1 = L and β N +1 = tL , t >
0, in (2.8) and sending L → + ∞ one arrives at the reduced version of the integral (2.8)which takes the form [9] Z R N t i Λ N Y k,j =1 Γ( α k − iλ j )Γ( iλ j + β k ) µ N ( λ ) d N λ = t A (1 + t ) A + B N Y k,j =1 Γ( α k + β j ) , (2.10)where A ( B ) = P Nk =1 α k ( β k ). At the same time representing the r.h.s. of Eq. (2.7) in thefollowing form (for more details see Lemma 4.3).lim L →∞ lim ε,ε ′ → + N +2 ( L ) Z R N +1 N +1 Y k,j =1 Γ( iγ k − iλ j + ε )Γ( iλ j − iγ ′ k + ε ′ ) e i Λ( x ′ N − x N ) µ N +1 ( λ ) d N +1 λ, where iγ N +1 = − iγ ′ N +1 = L , one can evaluate the integral using Eq. (2.10). Then, after somealgebra, one can show that (2.7) is equivalent to the following identitylim L →∞ lim ǫ → + L i (Γ ′ − Γ) N Y k,j =1 Γ( i ( γ ′ k − γ j ) + ε ) r L π cosh − L (cid:18) x N − x ′ N (cid:19) = µ − N ( γ ) δ ( x N − x ′ N ) δ N (cid:0) γ, γ ′ (cid:1) . (2.11)We recall here that Eqs. (2.9) and (2.11) have to be understood in the sense of distributions andrelegate further details to the Appendix A. S.´E. Derkachov, K.K. Kozlowski and A.N. Manashov In this section we construct the representation of Separated Variables for generic spin chainmodels and recall some elements of the Quantum Inverse Scattering Method (QISM) relevantfor our purposes.The spin chain is a quantum system of interacting spins S ± k , S k , k = 1 , . . . , N , where theindex k enumerates the nodes of the chain. The spin operators are the symmetry generators ofthe SL(2 , R ) group [11] which are determined by a real number (spin) s k > / S − k = − ∂ z k , S k = z k ∂ z k + s k , S + k = z k ∂ z k + 2 s k z k . (3.1)Operators with the index k act in a Hilbert space associated with the k -th site, H k , which isthe Hilbert space of functions holomorphic in the upper complex half-plane, H + . The scalarproduct in the Hilbert space H k = (cid:8) f ∈ L ( H + , d µ s k ) : f is holomorphic on H + (cid:9) is defined asfollows( f, ψ ) = Z ( f ( z k )) † ψ ( z k ) µ s k ( z k ) d z k , (3.2)The measure takes the form µ s ( z k ) = 2 s − π θ (Im z k )(2Im z k ) s − , (3.3)where θ ( x ) is the Heaviside step function. The operators S αk are anti-hermitian with respect tothe scalar product (3.2).Function in H k can be represented by Fourier integrals where the integration runs only overpositive momenta f ( z ) = Z ∞ e ipz F [ f ]( p ) dp. (3.4)Then, in Fourier space, viz . in the momentum representation, the scalar product takes the form( f, ψ ) = Γ(2 s ) Z ∞ (cid:0) F [ f ]( p ) (cid:1) ∗ F [ ψ ]( p ) p − s dp. (3.5)One of the main objects of QISM is the monodromy matrix. For the closed/open spin chainof our interest, the monodromy matrix is given by a product of L -operators [10] which are twoby two matrices, L k ( u ) = u + i (cid:18) S k S − k S + k − S k (cid:19) , (3.6)where u ∈ C is the spectral parameter. The monodromy matrix for the closed chain of length N has the form [10] T N ( u ) = L ( u + ξ ) . . . L N ( u + ξ N ) = (cid:18) A N ( u ) B N ( u ) C N ( u ) D N ( u ) (cid:19) , (3.7)while, for the open spin chain, it is given by the following expression [39] T N ( u ) = T N ( − u ) σ T tN ( u ) σ = (cid:18) A N ( u ) B N ( u ) C N ( u ) D N ( u ) (cid:19) , (3.8)ompleteness of SoV representations 7where σ is the Pauli matrix. The entries of the monodromy matrices form commuting polyno-mial operator families [10, 39] (cid:8) A N ( u ) (cid:9) u ∈ C , (cid:8) B N ( u ) (cid:9) u ∈ C , (cid:8) B N ( u ) (cid:9) u ∈ C :[ A N ( u ) , A N ( v )] = [ B N ( u ) , B N ( v )] = [ B N ( u ) , B N ( v )] = 0 , (3.9)which act on the Hilbert space of the model, H N = N Nk =1 H k . Operators in each of the com-muting families share the same eigenfunctions. These systems of functions have proven to bevery useful for analysing the properties of spin chains. They determine the so-called Sklyaninrepresentation of Separated Variables [40]. For the homogeneous chains, viz . ξ a = 0, the corre-sponding systems for B N , B N and A N operators were constructed in Refs. [1, 6, 7], respectively.Below, we recall these constructions and, on the occasion, extend them to the general case ofinhomogeneous spin chains where the ξ a ’s are generic. Since the technical details are essentiallythe same in all three cases, we consider in some detail the B N -system and only quote the resultsfor the other two.All three families of eigenfunctions can be represented as a convolution of functions of aspecial type. Namely, let us define a function of two complex variables, z, w ∈ H + , and thevariable α ∈ C which is called index, D α ( z, w ) = (cid:18) iz − ¯ w (cid:19) α = 1Γ( α ) Z ∞ e ip ( z − ¯ w ) p α − dp. (3.10)This is a single valued function of z, w which is fixed by the condition arg (cid:0) i/ ( x + iy ) (cid:1) → x →
0. Some properties of the function D α can be found in ref. [9]. B N operator The eigenfunctions can be most conveniently written down in term of the so-called layer op-erators. Let Λ n +1 ( γ, x ) be an operator which maps functions of n complex variables to func-tions of n + 1 variables. It depends on the spectral parameter x ∈ C and the complex vector γ = ( α , . . . , α n , β , . . . , β n ) ∈ C n . Its action takes the form h Λ n +1 ( γ, x ) f i ( z , . . . , z n +1 ) = Z · · · Z n Y k =1 D α k − ix ( z k , w k ) D β k + ix ( z k +1 , w k ) × f ( w , . . . , w n ) n Y a =1 µ αa + βa ( w a ) d w . . . d w n . (3.11)The weight function µ s ( w j ) has been defined in Eq. (3.2). The integral is well defined providedRe( α k − ix ) , Re( β k + ix ) > k = 1 , . . . , n .In the momentum representation obtained by taking the Fourier transform f ( z , . . . , z n ) = Z ∞ · · · Z ∞ F [ f ]( p , . . . , p n ) e i P nk =1 p k z k dp . . . dp n (3.12)the action of the layer operator can be expressed as F h Λ n +1 ( γ, x ) g i ( q , . . . , q n +1 ) = λ n ( γ, x ) Z q dℓ p Z q dℓ p · · · Z q n dℓ n − p n − × F [ g ]( p , p , . . . , p n ) n Y k =1 (cid:18) q k − ℓ k − p k (cid:19) α k − ix − (cid:18) ℓ k p k (cid:19) β k + ix − , (3.13) In many cases it is quite helpful to visualize all further constructions as Feynmann diagrams with the function D α playing the role of a propagator. S.´E. Derkachov, K.K. Kozlowski and A.N. Manashovwhere ℓ k are the “loop” momenta, p k = q k + ℓ k − ℓ k − and ℓ ≡ ℓ n ≡ q n +1 and the factor λ n reads λ n ( γ, x ) = n Y k =1 B − ( α k − ix, β k + ix ) , (3.14)where B is the Euler beta function. Note also that all momenta in (3.13) are positive and P n +1 k =1 q k = P nk =1 p k .The layer operators possess two important properties. Let us define a map t : C n C n − as follows tγ = t ( α , . . . , α n , β , . . . , β n ) = ( α , . . . , α n − , β , . . . , β n ) . (3.15)It can be checked that the operators Λ n satisfy the permutation identity: Λ n +1 ( γ, x ) Λ n ( tγ, x ′ ) = Λ n +1 ( γ, x ′ ) Λ n ( tγ, x ) . (3.16)The derivation is based on integral identities for the functions D α which can be found in [7].Next, for the spin chain of length N , we introduce the following combinations of spins andimpurities s k = s k − iξ k , ¯ s k = s ∗ k = s k + iξ k , k = 1 , . . . , N, (3.17)and define the vector γ N ∈ C N − : γ N = ( s , . . . , s N − , ¯ s , . . . , ¯ s N ) . (3.18)It can be shown, see ref. [7], that the operator Λ N ( γ N , x ) is nullified by B N ( x ), B N ( x ) Λ N ( γ N , x ) = 0 . (3.19)These two properties of the layer operators are crucial for constructing the eigenfunctions of theoperator B N ( u ). Let us define a function of N complex variablesΨ Np,x ( z ) = κ N p S − / h Λ N ( γ N , x ) Λ N − ( tγ N , x ) . . . Λ ( t N − γ N , x N − ) · E p i ( z ) . (3.20)Here x = ( x , . . . , x N − ), z = ( z , . . . , z N ), E p is the exponential function, E p ( w ) = e ipw , p > S ≡ P Nk =1 s k and the normalisation constant κ N reads κ − N = N Y k =1 Γ( s k + ¯ s k ) ! / Y ≤ i
2. Theproof for general N is more involved and presents the main task of this paper.For real x and p the functions Ψ Np,x ( z ) do not belong to the Hilbert space H N . However, theyallow one to define a linear transform from the Hilbert space H BN defined below into H N : H BN = L ( R + ) ⊗ L (cid:16) R N − , dµ BN − ( x ) (cid:17) , dµ BN − ( x ) = µ BN − ( x ) d N − x (3.23)where we agree upon µ BN − ( x ) = 1(2 π ) N − ( N − Q N − k =1 Q Nj =1 Γ( s j − ix k )Γ(¯ s j + ix k ) Q j For any smooth, compactly supported function ϕ on R + × R N − , T BN ϕ ∈ H N and it holds k T BN ϕ k H N = k ϕ k H BN ≡ Z R + Z R N − | ϕ ( p, x ) | µ BN − ( x ) dp d N − x . (3.26) As such, T BN extends to a linear isometry T BN : H BN H N satisfying k T BN ϕ k H N = k ϕ k H BN . (3.27)One may already draw several consequences from this theorem. First of all, (3.26) ensuresthat the system of functions { Ψ Np,x , p ∈ R + , x ∈ R N − } forms an orthogonal system in H N , viz .that (cid:0) Ψ Np ′ ,x ′ , Ψ Np,x (cid:1) H N = δ ( p − p ′ ) δ N − ( x, x ′ ) ( µ BN − ( x )) − , (3.28)where the multi-dimensional Dirac delta-function has been introduced in (2.5) while µ BN ( x ) hasbeen defined in (3.24). This corresponds to the orthogonality relation for the system (cid:8) Ψ Np,x ( z ) (cid:9) .Next, the equality (3.26) implies that k T BN k = 1 and that the image of H BN is a closed subspaceof H N . Now, if one is able to show that T BN is a unitary operator, what amounts to showingT BN H BN = H N , then this will also ensure that the system of functions { ( p, x ) Ψ Np,x ( z ) , z ∈ ( H + ) N } forms an complete system in H N , viz . that given z a = x a + iy a (cid:0) Ψ N ∗ , ∗ ( z ′ ) , Ψ N ∗ , ∗ ( z ) (cid:1) H BN = I ( z, z ′ ) . (3.29)0 S.´E. Derkachov, K.K. Kozlowski and A.N. ManashovThe r.h.s. of this equation, I ( z, z ′ ), is the kernel of the unit operator in H N – the so-calledreproducing kernel, see e.g. [17] – which takes the form I ( z, z ′ ) = Q Nk =1 I k ( z k , z ′ k ), where I k ( z k , z ′ k ) = D s k ( z k , z ′ k ) = (cid:18) iz k − ¯ z ′ k (cid:19) s k . (3.30)It means that for any Ψ( z ) ∈ H N the following identity holdsΨ( z ) = Z I ( z, z ′ )Ψ( z ′ ) N Y k =1 µ s k ( z k ) d z k . (3.31)The unitarity of T BN , viz . that the map has dense range, will be established in Section 4,hence leading to the main result of the paper.In order to prove the theorem, we first establish that (3.26) holds for smooth, compactlysupported functions ϕ on R + × R N − . For that purpose, let us introduce the regularised functionΨ N,ǫp,x ( z ) which is obtained from Ψ Np,x ( z ) by giving small positive imaginary parts to all variables x k , x k → x k + iǫ k , ǫ k > s N → ¯ s ǫN = ¯ s N + P Nk =1 ǫ k in the definition of thevector γ N , Eq. (3.18). Further, let[T B,ǫN ϕ ]( z ) = Z R + Z R N − ϕ ( p, x ) Ψ N,ǫp,x ( z ) µ BN − ( x ) dp d N − x (3.32)One may readily check that [T B,ǫN ϕ ] → [T BN ϕ ]( z ) pointwise as ǫ = ( ǫ , . . . , ǫ N ) → + . We wantto show thatlim ǫ,ǫ ′ → + (cid:0) T B,ǫ ′ N , T B,ǫN (cid:1) H N = k T BN ϕ k H N . (3.33)Let us demonstrate that one can invoke Fubini’s theorem to get k T B,ǫN ϕ k H N = Z ( R + ) dpdp ′ Z R N − d N − xd N − x ′ ϕ ( p, x ) ϕ ∗ ( p ′ , x ′ ) µ BN − ( x ) µ BN − ( x ′ ) (Ψ N,ǫ ′ p ′ ,x ′ , Ψ N,ǫp,x ) H N . The scalar product of the regularised functions Ψ N,ǫp,x may be computed in closed form as [7](Ψ N,ǫ ′ p ′ ,x ′ , Ψ N,ǫp,x ) H N = δ ( p − p ′ ) C ( ǫ,ǫ ′ ) N ( p, x, x ′ ) , (3.34)where C ( ǫ,ǫ ′ ) N ( p, x, x ′ ) = (cid:18) κ N κ ǫN κ ǫ ′ N (cid:19) p P Nk =1 ( ǫ k + ǫ ′ k ) Γ (cid:16) ǫ N + ǫ ′ N + i P N − k =1 ( x k − x ′ k ) (cid:17) Γ (cid:16)P Nk =1 ǫ k + ǫ ′ k (cid:17) × Q N − k,j =1 Γ( i ( x ′ k − x j ) + ǫ kj ) Q N − k =1 Γ(¯ s ǫN + ix k − ǫ k )Γ( s ǫ ′ N − ix ′ k − ǫ k ) Q N − j =1 Γ(¯ s j + ix ′ k + ǫ ′ k )Γ( s j − ix k + ǫ k ) . (3.35)and ǫ kj = ǫ ′ k + ǫ j and κ ǫN = κ N ( s ǫ ).In order to obtain this result it is convenient to perform calculation in the momentum spacerepresentation. Using the momentum representation for the layer operators (3.13) one can obtainompleteness of SoV representations 11an expression for C ( ǫ,ǫ ′ ) N ( p, x, x ′ ) in the form of a multidimensional momentum integral (whichcan be thought of as a Feynman diagram) C ( ǫ,ǫ ′ ) N ( p, x, x ′ ) = Z X f ( ǫ, p, x, x ′ , { ℓ ij } ) Y ij dℓ ij . (3.36)Here the function f is a product of linear combinations of momenta ℓ ij and p raised to somepowers. It is important to note that all these combinations are positive and that the integrationruns over a compact region X . Performing integrations in a special order using the integralidentities for the product of the propagators, see e.g. [6, 7, 9], gives the expression (3.35). Ofcourse one has to justify that the order of integrations does not influence the answer. To thisend we note that the integral of | f | can be written in the form Z | f ( ǫ, p, x, x ′ , { ℓ ij } ) | Y ij dℓ ij = R ( x, x ′ , ǫ ) Z f ( ǫ, p, , , { ℓ ij } ) (cid:12)(cid:12) ξ = ··· = ξ N =0 Y ij dℓ ij , (3.37)where R ( x, x ′ , ǫ ) is some nonsingular factor given by a product of Γ functions. The function f ( ǫ, p, , , { ℓ ij } ) (cid:12)(cid:12) ξ = ··· = ξ N =0 is positive and the integral is a particular case of Eq, (3.36). Thusthis integral can be evaluated, as was discussed before, in a closed form, see Eq. (3.35). Then,by Fubini theorem, the integral (3.36) exists, the integrations can be performed in an arbitraryorder, what thus justifies the result (3.35).Eq. (3.35) therefore leads to (cid:0) T B,ǫ ′ N ϕ, T B,ǫN ϕ (cid:1) H N = Z R + Z R · · · Z R ϕ ( p, x )( ϕ ( p, x ′ )) ⋆ C ( ǫ,ǫ ′ ) N ( p, x, x ′ ) × µ BN − ( x ) µ BN − ( x ′ ) dpd N − x d N − x ′ , (3.38)where we recall that ϕ is a smooth function with a compact support. For ǫ, ǫ ′ → + the integralin the r.h.s. can be easily estimated, see Appendix A for the details, resulting in (cid:0) T B,ǫ ′ N ϕ, T B,ǫN ϕ (cid:1) H N = K + o (1) , (3.39)where K = Z R + Z R N − | ϕ ( p, x ) | µ BN − ( x ) dpdx N − (3.40)and µ BN − is as introduced in (3.24).Since at ǫ → + , one has that [T B,ǫN ϕ ] → [T BN ϕ ]( z ) almost everywhere, it follows from Fatou’stheorem that k T BN ϕ k H N ≤ lim inf ǫ → + k T B,ǫN ϕ k H N = K. Thus, the function T BN ϕ belongs to the Hilbert space H N . Finally, taking into account that(T BN ϕ, T B,ǫN ϕ ) H N = K + o (1) one derives from k T BN ϕ − T B,ǫN ϕ k H N ≥ K ≤ k T BN ϕ k . Thusone gets for the norm of T BN ϕ , k T BN ϕ k H N = K. Finally, the remaining follows from the fact that the set of smooth functions with a compactsupport is dense in the Hilbert space H BN . (cid:3) A N operator In this section we give a brief description of the eigenfunctions of the operator A N . We startwith defining of a layer operator suitable for this case. Let η be a complex vector and η = ( α , . . . , α n +1 , β , . . . , β n ) ∈ C n +1 . (3.41)The layer operator Λ ( σ ) n +1 ( η, x ), which depends on the vector η and two complex parameters, x and σ , Im σ ≥ 0, maps a function of n -complex variables to a function of n + 1 variables asfollows h Λ ( σ ) n +1 ( η, x ) f i ( z , . . . , z n +1 ) = D α n +1 − ix ( z n +1 , σ ) Z · · · Z n Y k =1 D α k − ix ( z k , w k ) D β k + ix ( z k +1 , w k ) × f ( w , . . . , w n ) n Y a =1 µ αa + βa ( w a ) d w . . . d w n . (3.42)The integrals converge provided Re( α k − ix ) , Re( β k + ix ) > k = 1 , . . . , n .Let ̺ be a map: C n +1 C n − , ̺η = ̺ ( α , . . . , α n +1 , β , . . . , β n ) = ( α , . . . , α n , β , . . . , β n ) . (3.43)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 ) . (3.44)Let us put η N ≡ ( s , . . . , s N , ¯ s , . . . , ¯ s N ) (3.45)and define the function Φ ( σ ) x ( z , . . . , z N ) as:Φ Nσ,x ( z ) = κ N (cid:16) Λ ( σ ) N ( η N , x ) Λ ( σ ) N − ( ̺η N , x ) . . . Λ ( σ )1 ( ̺ N − η N , x N ) (cid:17) . (3.46)By virtue of Eq. (3.44) Φ Nσ,x is a symmetric function of x , . . . , x N . It can be shown, see e.g.ref. [7], that the operator A N ( x ) + σB N ( x ) annihilates the layer operator Λ ( σ ) N ( η N , x ), (cid:16) A N ( x ) + σB N ( x ) (cid:17) Λ ( σ ) N ( γ N , x ) = 0 (3.47)and, hence, the function Φ Nσ,x satisfies the equation (cid:0) A N ( u ) + σ B N ( u ) (cid:1) Φ Nσ,x ( z ) = ( u − x ) · · · ( u − x N )Φ Nσ,x ( z ) . (3.48)Thus the function Φ Nx ≡ Φ Nσ =0 ,x diagonalizes the operator A N ( u ). For the separated variables x with small positive imaginary parts the function Φ Nσ,x has a finite norm. Indeed one can findfor the scalar product of two Φ N functions [1] (cid:0) Φ Nσ,y , Φ Nυ,x (cid:1) H N = (cid:18) iσ − ¯ υ (cid:19) i ( ¯ Y − X ) Q Nk,j =1 Γ( i (¯ y k − x j )) Q Nk =1 Q Nj =1 Γ( s j − ix k )Γ(¯ s j + i ¯ y k ) . (3.49)Here X = P Nk =1 x k , Y = P Nk =1 y k .ompleteness of SoV representations 13For real x the functions Φ Nx are orthogonal to each other (see Appendix A for more details) (cid:0) Φ Nx ′ , Φ Nx (cid:1) = lim σ → lim ǫ → + (cid:0) Φ Nσ,x ′ + iǫ , Φ Nx (cid:1) = δ N ( x, x ′ ) ( µ AN ( x )) − , (3.50)where µ AN ( x ) = 1(2 π ) N N ! Q Nk =1 Q Nj =1 (cid:2) Γ( s j − ix k )Γ(¯ s j + ix k ) (cid:3)Q j Let H AN be the Hilbert space of symmetric functions H AN = L sym (cid:16) R N , dµ AN ( x ) (cid:17) , dµ AN ( x ) = µ AN ( x ) d N x. (3.52) The transformation T AN defined for smooth, compactly supported functions χ on R N : Ψ χ ( z ) = [T AN χ ]( z ) = Z R N χ ( x ) Φ Nx ( z ) µ AN ( x ) d N x (3.53) extends into a linear isometry from H AN into H N . In particular it has unit operator norm k T AN k = 1 and satisfies k T AN χ k H N = k χ k H AN = Z R N | χ ( x ) | µ AN ( x ) d N x . (3.54)It will be show in Section 4 that T AN is, in fact, an unitary map between the involved Hilbertspaces. B N operator Let us construct eigenfunctions of the operator B N ( u ), see Eq. (3.8). It can be shown [6] that B N ( u ) = (2 u + i ) b B N ( u ), where b B N ( u ) = b B N ( − u ). In order to write down eigenfunctions of b B N ( u ) we define the corresponding layer operator, e Λ n +1 ( η, x ), where η ∈ C n +1 , see Eq. (3.41)and x ∈ C is a spectral parameter. The layer operator is written in terms of the operators Λ ( σ ) n ,defined in the previous section, Eq. (3.42), as follows:[ e Λ n +1 ( η, x ) f ]( z ) = Z [ Λ ( σ ) n +1 ( η, x ) Λ ( σ ) n ( ̺η, − x ) f ]( z , . . . , z n +1 ) µ αn + αn +12 ( σ ) d σ . (3.55)Above, the product of two layer operators, Λ ( σ ) n +1 ( η, x ) Λ ( σ ) n ( ̺η, − x ), maps a function of n − w , . . . , w n − ) to a function of n + 1 variables, z = ( z , . . . , z n +1 ), as indicated in theabove formula. By virtue of (3.44) the layer operator e Λ n +1 is an even function of x , e Λ n +1 ( η, x ) = e Λ n +1 ( η, − x ) . (3.56)Let ω be a map: C n +1 C n − , defined as ω γ = ω ( α , . . . , α n +1 , β , . . . , β n ) = ( α , . . . , α n , β , . . . , β n , α n +1 ) . (3.57)The layer operators (3.55) satisfy the permutation relation [6] e Λ n +1 ( γ, x ) e Λ n ( ωγ, x ′ ) = e Λ n +1 ( γ, x ′ ) e Λ n ( ωγ, x ) (3.58)4 S.´E. Derkachov, K.K. Kozlowski and A.N. Manashovand is nullified by the operator b B N ( x ) b B N ( x ) e Λ N ( η N , x ) = 0 , (3.59)where the vector η N is given by Eq. (3.45).Given z = ( z , . . . , z N ), define the functionΥ Np,x ( z ) = κ N p S − h e Λ N ( γ N , x ) e Λ N − ( ωγ N , x ) . . . e Λ ( ω N − γ N , x N − ) · E p i ( z ) , (3.60)where S = P Nk =1 s k , x = ( x , . . . , x N − ) and the normalisation factor is κ − N = N Y k =1 Γ( s k + ¯ s k ) ! / Y ≤ i Let H B N be the Hilbert space H B N = L ( R + ) ⊗ L sym (cid:16) ( R + ) N − , dµ B N − ( x ) (cid:17) , dµ B N − ( x ) = µ B N − ( x ) d N − x. (3.65) The transformation T B N defined for smooth, compactly supported functions φ on R + × ( R + ) N − that are symmetric in respect to the last N − variables as: [T B N φ ]( z ) = Z ( R + ) N φ ( p, x ) Υ Np,x ( z ) µ B N − ( x ) dp d N − x (3.66) extends to a linear isometry from H B N into H N . In particular, it has unit operator norm k T B N k = 1 and satisfies k T B N φ k H N = k φ k H B N = Z R N + | φ ( p, x ) | µ B N − ( x ) dp d N − x . (3.67)We will show in Section 4 that T B N is an unitary operator.ompleteness of SoV representations 15 In the previous section we constructed three systems of functions, Ψ x,p , Φ x and Υ x,p . They allowone to define the linear operators, T αN , α = { B, A, B } , which map the Hilbert spaces H αN to theHilbert space H N . For N = 1 the transformations T B and T B are the Fourier transform andT A is the Mellin transform. Thus, these transformations are unitary maps and, in particular,T α H α = H .In order to prove the unitary character of the maps T αN for arbitrary N we use induction on N . Namely, we will show that if T AN is unitary then the maps T BN , T B N and T BN +1 are also unitary.Finally, we show that the unitarity of the map T BN implies the one for T AN . Schematically it isshown on the diagram below B N A N B N +1 A N +1 B N B N +1 (4.1)The backward arrow is dispensable here, but we consider it first because its proof is mosttransparent and all other proofs follow the same scheme.It was shown in the previous section that R (T BN ) is a closed subspace of the Hilbert space H N . If this subspace coincides with the whole H N then the orthogonal completion, R (T BN ) ⊥ , isempty. Since R (T BN ) ⊥ = ker(T BN ) ⋆ it is enough to prove that the kernel of the adjoint operator(T BN ) ⋆ is empty. In order to do it let us consider a linear map from H AN to H BN defined byS BA = (T BN ) ⋆ T AN . (4.2)Since the map T AN is an isometry, by assumption, it maps ker(S BA ) ker(T BN ) ⋆ . Our immediateaim is to show that ker(S BA ) = 0.We prove the following statement: Lemma 4.1. Let S BA be the operator from H AN to H BN defined in Eq. (4.2) . Then, for any χ ∈ H AN it holds k S BA χ k H BN = k χ k H AN . (4.3)First, we calculate the action of the operator S BA on the space of smooth functions witha compact support, χ ( x ). The action of T AN on a function χ ( x ) is given by Eq. (3.53). Infull similarity with the construction in Sect. 3.1.2, we define the regularized function T A,σ,ǫN χ obtained by replacing Φ Nx in (3.53) by Φ Nσ,x + iǫ , where x = ( x , . . . , x N ) and ǫ = ( ǫ , · · · ǫ N ), all ǫ k > A,σ,ǫN χ ( z ) = Z R N χ ( x ) Φ Nσ,x + iǫ ( z ) dµ AN ( x ) . (4.4)As σ, ǫ → k T A,σ,ǫN χ − T AN χ k H N → 0. Since k T BN k = 1, the adjoint to T BN isa bounded operator which acts on a vector Ψ by projecting it on the eigenfunction Ψ Np,y , seeEq. (3.25),(T BN ) ⋆ Ψ = (Ψ Np,y , Ψ) H N ≡ ϕ ( p, y ) . (4.5)Thus we write ϕ ( p, y ) ≡ [S BA χ ]( p, y ) = (cid:2) (T BN ) ⋆ T AN χ (cid:3) ( p, y )= lim σ,ǫ → (cid:2) (T BN ) ⋆ T A,σ,ǫN χ (cid:3) ( p, y ) = lim σ,ǫ → (Ψ Np,y , T A,σ,ǫN χ ) H N (4.6)6 S.´E. Derkachov, K.K. Kozlowski and A.N. ManashovMoreover, one has k ϕ k H BN = lim σ → lim ǫ → + k ϕ σ,ǫ k H BN , ϕ σ,ǫ ( p, y ) = (cid:0) Ψ Np,y , T A,σ,ǫN χ (cid:1) H N . (4.7)The further analysis depends on the remarkable fact that the scalar product of the functionsΨ Np,y and Φ Nσ,x + iǫ can be obtained in a closed form [1]: (cid:0) Ψ Np,y , Φ Nσ,x + iǫ (cid:1) H N = p − / − i Ξ − iX + E e − ip ¯ σ Q Nk =1 Q N − j =1 Γ( i ( y j − x k ) + ǫ k )) Q Nj =1 (cid:16)Q N − k =1 Γ(¯ s j + iy k ) Q Nk =1 Γ( s j − ix k + ǫ k ) (cid:17) , (4.8)where X = P Nk =1 x k and Ξ = P Nk =1 ξ k , and E = P Nk =1 ǫ k . That is ϕ σ,ǫ ( p, y ) = p − / − i Ξ+ E e − ip ¯ σ Q Nj =1 Q N − k =1 Γ(¯ s j + iy k ) Z R N Q Nk =1 Q N − j =1 Γ( i ( y j − x k ) + ǫ k ) Q Nk,j =1 Γ( s j − ix k + ǫ k ) p − iX χ ( x ) dµ AN ( x ) . (4.9)By assumption the function χ is nonzero only in a compact region. Therefore the function ϕ σ,ǫ ( p, y ) grows no faster that some power of y for large y while at large p it decays exponentiallyfast ∼ exp {− Im( σ ) p } ). Taking into account that the measure µ BN − ( y ) decays exponentially fastfor large yµ BN − ( y ) ≃ (4 π ) N ( N − N − ( N − Y ≤ i Let S B A and S B be maps from H AN H BN and H AN ⊗ H A H BN +1 defined asfollows S B A = (T B N ) ⋆ T AN and S B = (T BN +1 ) ⋆ (T AN ⊗ T A ) . (4.16) Provided the map T AN : H AN H N is unitary, k S B A χ k H B N = k χ k H AN and k S B χ ′ k H BN +1 = k χ ′ k H AN ⊗ H A (4.17) hold for any χ ∈ H AN , χ ′ ∈ H AN ⊗ H A . In the proof of these assertions, the main difference from the proof of Lemma 4.1 lies in thetype of the Γ-integrals arising in the process. We briefly discuss these differences below.For the S B A operator the problem is reduced to calculating the norm of the function φ σ,ǫ ( p, y ) = (cid:0) Υ Np,y , T A,σ,ǫN χ (cid:1) H N , (4.18)which is an analogue of the function ϕ σ,ǫ , see Eq. (4.7). The relevant scalar product takes theform (Υ Np,y , Φ Nσ,x + iǫ ) H N = p − / − i Ξ − iX + E e − ip ¯ σ Q ≤ k Nσ,x + iǫ ⊗ Φ σ,x N +1 + iǫ N +1 takes the form (see e.g. [9])(Ψ Np,y , Φ Nσ,x ⊗ Φ σ,x N +1 ) H N = 1 √ p p − iX − i Ξ+ iξ N +1 e − ip ¯ σ N Y k =1 Γ(¯ s k + s N +1 )Γ( s k + ¯ s N +1 ) × s N +1 − ix N +1 ) · N Y k,j =1 Γ( i ( y j − x k ))Γ(¯ s k + iy j )Γ( s j − ix k ) N Y k =1 s N +1 − iy k ) Γ( − i ( y k + x N +1 ))Γ( − i ( x k + x N +1 )) . (4.29)ompleteness of SoV representations 19Calculating the norm of ϕ σ,ǫ we change the order of integration and first take the integral over y .It takes the form of N -fold Gustafson’s integral [15, Theorem 5.1] that we encountered earlier,see Eq. (4.12). After some algebra we obtain k ϕ σ,ǫ k H BN +1 = Z R N +2 (cid:18) (2Im σ ) i ( X − X ′ ) − E Q Nk,j =1 Γ( i ( x ′ k − x j ) + ǫ k + ǫ j ) Q Nk,j =1 Γ(¯ s j + ix ′ k + ǫ k )Γ( s j − ix k + ǫ k ) × Γ( i ( x ′ N +1 − x N +1 ) + 2 ǫ N +1 )Γ(¯ s N +1 + ix ′ N +1 + ǫ N +1 )Γ( s N +1 − ix N +1 + ǫ N +1 ) (cid:19) χ ( x ) ( χ ( x ′ )) ∗ × dµ AN ( x ) dµ AN ( x ′ ) dµ A ( x N +1 ) dµ A ( x ′ N +1 ) . (4.30)The analysis of the above expression in the limit ǫ → + and σ → k ϕ k H BN +1 = k S B χ k H BN +1 = k χ k H AN ⊗ H A = Z R N +1 | χ ( x ) | dµ AN ( x ) dµ A ( x N +1 ) , (4.31)that completes the proof of the Lemma. (cid:3) It follows from Lemma 4.2 ker S B A = 0 and ker S B = 0 and, hence, that the operatorsT B N : H B N H N and T BN +1 : H BN +1 H N +1 are unitary provided that T AN is.The final step required to complete the induction on N is to show that the unitarity of themap T AN follows from that of the map T BN . As it was argued earlier in this section in order toprove this statement it is enough to show that the kernel of the operatorS AB = (T AN ) ⋆ T BN , S AB : H BN H AN (4.32)is empty. Lemma 4.3. Let S AB be the linear operator defined in Eq. (4.32) . If the map T BN : H BN H N is unitary then for any ϕ ∈ H BN the following identity holds: k S AB ϕ k H AN = k ϕ k H BN . Let ϕ ( p, x ) be a smooth function with compact support having the factorised form ϕ ( p, x ) = f ( p ) ˜ ϕ ( x ) . (4.33)The linear span of these functions is dense in H BN . The action of the operator S AB on a function ϕ can be represented as follows χ ( y ) = [S AB ϕ ]( y ) = (cid:0) Φ y , T BN ϕ (cid:1) H N = lim ǫ → + (cid:0) Φ y + iǫ , T BN ϕ (cid:1) H N = lim ǫ → + Z R + Z R N − (cid:0) Φ y + iǫ , Ψ Np,x (cid:1) H N ϕ ( p, x ) dp dµ BN − ( x ) , (4.34)where y + iǫ = ( y + iǫ, . . . , y N + iǫ ) and the scalar product of two eigenfunctions is given byEq. (4.8). We also denote the function given by the integral in the above equation by χ ǫ ( y ), i.e. χ ( y ) = lim ǫ → + χ ǫ ( y ) . (4.35)It can be shown, see [23, Lemma 3.1], that the function χ ( y ) Q i This work was supported by the Russian Science Foundation project No 19-11-00131 and by theDFG grants MO 1801/4-1, KN 365/13-1 (A.M.). The work of K.K.K. is supported by CNRS. A Some representations for multi-dimensional Dirac δ -functions I. Define e C ( ǫ,ǫ ′ ) N ( p, x, x ′ ) = Γ (cid:16) ǫ N + ǫ ′ N + i P N − a =1 ( x a − x ′ a ) (cid:17) Γ (cid:16)P Na =1 ǫ a + ǫ ′ a (cid:17) N − Q a,b Γ( i ( x ′ b − x a ) + ǫ ′ b + ǫ a ) N − Q a = b Γ (cid:0) i ( x ′ a − x ′ b ) (cid:1) Γ (cid:0) i ( x a − x b ) (cid:1) . (A.1)In the following we show that, in the sense of distributions, it holdslim ǫ,ǫ ′ → + n e C ( ǫ ; ǫ ′ ) N (cid:0) p, x, x ′ (cid:1)o = W N − ( x ) · δ N − (cid:0) x, x ′ (cid:1) , (A.2)where W N − ( x ) = (2 π ) N − ( N − N − Y a = b (cid:0) i ( x a − x b ) (cid:1) . (A.3)In other words, given I ( ǫ ; ǫ ′ ) N = Z R + dp Z R N − d N − x Z R N − d N − x ′ e C ( ǫ ; ǫ ′ ) N (cid:0) p, x, x ′ (cid:1) ϕ (cid:0) p, x (cid:1) ϕ ∗ (cid:0) p, x ′ (cid:1) (A.4)it holds thatlim ǫ,ǫ ′ → + I ( ǫ ; ǫ ′ ) N = Z R + dp Z R N − d N − x W ( x ) (cid:12)(cid:12) ϕ (cid:0) p, x (cid:1)(cid:12)(cid:12) . (A.5)In order to establish the result, one starts by reorganising the integral in (A.4) as I ( ǫ ; ǫ ′ ) N = Z R + dp Z R N − d N − x Z R N − d N − x ′ U ( ǫ ; ǫ ′ ) N (cid:0) p, x, x ′ (cid:1) × det N − (cid:20) x ′ k − x j − i ( ǫ j + ǫ ′ k ) (cid:21) · − i P Nk =1 ( ǫ k + ǫ ′ k ) − i ( ǫ N + ǫ ′ N ) + P N − k =1 ( x k − x ′ k ) , (A.6)ompleteness of SoV representations 23where U ( ǫ ; ǫ ′ ) N (cid:0) p, x, x ′ (cid:1) = N − Y a
Let us define S L,εN ( x, x ′ ) = L i P Na =1 ( x ′ a − x a ) Q Na,b =1 Γ( i ( x ′ a − x b ) + ǫ ) Q Na = b Γ (cid:0) i ( x ′ a − x ′ b ) (cid:1) Γ (cid:0) i ( x a − x b ) (cid:1) . (A.19)We will show that in the sense of distributions the following identity holdslim L →∞ lim ǫ → + S L,ǫN ( x, x ′ ) = W N ( x ) · δ N (cid:0) x, x ′ (cid:1) , (A.20)where W N is defined in Eq. (A.3). Namely, given T L,ǫN = Z R N d N x Z R N d N x ′ S L,ǫN (cid:0) x, x ′ (cid:1) ϕ (cid:0) x (cid:1) ϕ ∗ (cid:0) x ′ (cid:1) (A.21)it holds thatlim L →∞ lim ǫ → + T L,ǫN = Z R N d N xW N ( x ) (cid:12)(cid:12) ϕ (cid:0) x (cid:1)(cid:12)(cid:12) . (A.22)Repeating the same arguments as above one can rewrite (A.21) in the formlim L →∞ lim ǫ → + T L,ǫN = N ! lim L →∞ lim ǫ → + Z R N d N x Z R N d N x ′ V N (cid:0) x, x ′ (cid:1) N Y a =1 L i ( x ′ a − x a ) x ′ a − x a − iǫ (A.23)ompleteness of SoV representations 25where V N (cid:0) x, x ′ (cid:1) = ( − i ) N Q Na
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