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Separation of Variables in the Quantum

arXiv:hep-th/9212076v1 11 Dec 1992Separation of Variables in the QuantumIntegrable Models Relatedto the Yangian Y[sl(3)]E. K. Sklyanin ∗Isaac Newton Institute for Mathematical Sciences,Cambridge University, Cambridge, CB3 0EH, U.K.December 8, 1992NI-92013Abstract. There being no precise definition of the quantum integrability, theseparability of variables can serve as its practical substitute.For any quantumintegrable model generated by the Yangian Y[sl(3)] the canonical coordinates andthe conjugated operators are constructed which satisfy the “quantum characteristicequation” (quantum counterpart of the spectral algebraic curve for the L operator).The coordinates constructed provide a local separation of variables.

The conditionsare enlisted which are necessary for the global separation of variables to take place.1IntroductionDespite the huge body of papers on “quantum integrable models” written in lastdecades, ironically enough, there seems still to be no satisfactory definition of quan-tum integrability [1]. In the classical case Liouville’s definition of complete inte-grability allows immediately to integrate the equations of motion in quadratures.In the quantum case, on the contrary, the mere existence of commuting Hamilto-nians provides no help in finding their common spectrum and eigenfunctions whichis a natural analog of integrating equations of motion in the classical mechanics.The discrepancy seems to be due to the difficulty with the concept of independentintegrals of motion in the quantum case [1].There exists, however, another concept in the classical mechanics which is moreor less equivalent to the integrability, namely, the separation of variables in theHamilton-Jacobi equation.

Its advantage is that it has a direct quantum counterpart.Suppose that a quantum mechanical system possesses a finite number D of com-muting Hamiltonians Hj (j = 1, . .

., D). Suppose also that one can introduce D∗On leave from Steklov Mathematical Institute, Fontanka 27, St.Petersburg 191011, Russia.1

pairs of canonically commuting operators xj, pj[xj, xk] = [pj, pk] = 0[pj, xk] = −iδjksuch that:1. The common spectrum of {xj}Dj=1 is simple, that is the whole Hilbert space ofquantum-mechanical states is isomorphic to a space of functions on spec{xj}Dj=1.The momenta pj are then realized as the differentiations pj = −i∂/∂xj.

Thesimplicity of spec{xj}Dj=1 replaces here the classical concept of a Hamiltoniansystem having D degrees of freedom.2. There exist such polynomials Φj of D + 2 variables thatΦj(pj, xj, H1, H2, .

. .

, HD) = 0j = 1, . .

. , D(1)The noncommuting operators in (1) are assumed to be ordered precisely in thesame way they are enlisted, that is pjxjH1H2 .

. .

HD.Now, let Ψ(x1, ..., xD) be a common eigenfunction of the Hamiltonians HjHjΨ = hjΨ(2)Applying the operator identities (1) to Ψ and using (2) and the ordering of H’sin (1) one obtains for Ψ the set of differential equationsΦj(−i ∂∂xj, xj, h1, h2, . .

. , hD)Ψ(x1, .

. ., xD) = 0j = 1, .

. ., D(3)which obviously allows the separation of variables.Ψ(x1, .

. ., xD) =DYj=1ψj(xj)(4)The original multidimensional spectral problem is thus reduced to the set ofone-dimensional multiparameter spectral problemsΦj(−i ∂∂xj, xj, h1, h2, .

. .

, hD)ψj(xj) = 0j = 1, . .

. , D(5)The functions Φj being polynomials, the separated equations (5) are ordinarydifferential equations.

More generally, one can consider Φj as symbols of pseudo-differential operators. If one allows, for instance, Φj to depend on pj exponentially,then (5) become finite-difference equations.The above argument has, however, only a heuristic value since it establishesonly a local separation of variables and for the actual (global) s. o. v. some moreconditions are to be satisfied (see Section 4).It is tempting to adopt the separability of variables as a practical definition ofthe quantum integrability.

To this end, it is necessary to show at least that the2

models generally referred to as “quantum integrable” ones do allow the separationof variables in the above sense. First of all, it concerns the spin chains soluble viaall variants of Bethe ansatz technique which have no apparent resemblance to theseparation of variables.

It is not until recently that an s. o. v. has been constructedfor the models generated by the Yangian Y[sl(2)], see [2, 3] and the referencestherein.The next natural step is to study the Y[sl(3)] case.The relevant quantumintegrable models include SU(3)-invariant spin chains [4, 5, 6] together with theirdegenerated case (Gaudin model) [7, 8, 9], three-wave system (Lee model) [6], SU(2)vector Nonlinear Schr¨odinger equation [10, 11], all of them well studied via coordi-nate Bethe ansatz [4, 7, 10], algebraic BA [6, 8, 9, 11] and analytic BA [5] techniques.The separation of variables for the classical integrable SL(3) magnetic chain wasconstructed in [12], see also [13] for the classical SL(N) case. The present paperis the first in the series devoted to the quantum SL(3) case.

It outlines a generalscheme allowing to construct a separation of variables for the models in question.Intending to make the argument as general as possible we collect here only the resultswhich are common for various representations of Y[sl(2)]. An implementation andadjustment of the presented scheme for the diverse particular models is left forsubsequent publications.Using the experience acquired during the study of the quantum SL(2) case andthe classical SL(3) case we construct the variables xj as the operator zeroes of certainoperator polynomial B(u) having commuting coefficients.

In the classical case theexponents of the conjugated momenta Xj = epj are known to be the eigenvaluesof the L operator T(u) taken at u = xj and, as such, satisfy the correspondingcharacteristic equation. We show that the corresponding quantum variables Xj alsosatisfy a sort of “quantum characteristic equation” which fits the form (1) and thusprovides, in principle, a separation of variables.

The actual separation of variablescan be established, however, only if the representation of the algebra formed byXjxj satisfies several additional conditions which we conjecture to be satisfied forany representation of Y[sl(3)].The conjectured separated equations are 3rd order finite-difference equations.We show that our results agree with those obtained by means of the algebraic Betheansatz.2Description of the algebraConsider the associative algebra T generated by 9(M + 1) generators T αβ,m (α, β ∈{1, 2, 3}, m ∈{0, 1, . .

., M}) and the quadratic relations which are most convenientlydescribed in terms of the polynomials T αβ (u) = PMm=0 umT αβ,m:(u −v)T α1β1 (u)T α2β2 (v) + ηT α2β1 (u)T α1β2 (v)= (u −v)T α2β2 (v)T α1β1 (u) + ηT α2β1 (v)T α1β2 (u)∀u, v(6)3

or, equivalently,3Xβ1,β2=1Rα1α2β1β2 (u −v)T β1γ1 (u)T β2γ2 (v) =3Xβ1,β2=1T α2β2 (v)T α1β1 (u)Rβ1β2γ1γ2 (u −v)whereRα1α2β1β2 (u) = uδα1β1 δα2β2 + ηδα1β2 δα2β1Introducing the 3 × 3 matrixT(u) =T 11 (u)T 12 (u)T 13 (u)T 21 (u)T 22 (u)T 23 (u)T 31 (u)T 32 (u)T 33 (u),the unit operator 1l in C3 and the permutation operator P12 in the tensor productC3 ⊗C31lx = xP12x ⊗y = y ⊗x∀x, y ∈C3,and the notationT1≡T ⊗1lT2≡1l ⊗Tone can rewrite (6) in a compact form:R(u −v)T1(u)T2(v) = T2(v)T1(u)R(u −v)(7)whereR(u) = u1l ⊗1l + ηP12(8)is the well-known SL(3)-invariant solution to the Yang-Baxter equationR12(u)R13(u + v)R23(v) = R23(v)R13(u + v)R12(u)(9)or, at length,3Xβ1,β2,β3=1Rα1α2β1β2 (u)Rβ1α3γ1β3 (u + v)Rβ2β3γ2γ3 (v) =3Xβ1,β2,β3=1Rα2α3β2β3 (v)Rα1β3β1γ3 (u + v)Rβ1β2γ1γ2 (u)It follows from the relations (6) that the leading coefficients T αβ,M of the polyno-mials T αβ (u) belong to the center of the algebra T . We shall suppose that Z ≡TM isa number matrix having distinct nonzero eigenvalues.

Subsequently, some additionalnondegeneracy conditions will be imposed on Z.In what follows we shall consider T αβ,m as linear operators belonging to a repre-sentation of T in a linear space W. The representation theory for the algebra T incase Zaβ = δab is essentially equivalent to that of the Yangian Y[sl(3)], cf. [14, 15, 16].For our purposes it is more convenient to take a generic matrix Z rather then unitone, which is equivalent to taking GL(3) ⊗Y[sl(3)] instead of Y[sl(3)].We conclude this section with a synopsis of properties of quantum minors anddeterminants [5, 14, 17].4

Let P −12 and P −123 be the antisymmetrizers in C3⊗C3 and C3⊗C3⊗C3, respectively,P −12=12(1l ⊗1l −P12)P −123=16(1l ⊗1l ⊗1l + P12P23 + P23P12 −P12 −P13 −P23)and trj stand for the trace over the j-th copy of C3 in C3 ⊗C3 ⊗C3.The quantum determinant d(u) of T(u) is then defined asd(u) = q-detT(u)=tr123P −123T1(u)T2(u + η)T3(u + 2η)and generates, together with TM, the center of the algebra T . We shall assume thatthe coefficients of d(u), like T αβ,M are numbers.The quantum minor T α1α2β1β2 (u) is defined as the quantum determinant of the 2×2submatrix formed by the rows α1 < α2 and the columns β1 < β2 of the matrix T(u)T[α1α2β1β2 ](u)=q-det T α1β1 (u)T α1β2 (u)T α2β1 (u)T α2β2 (u)!=T α2β2 (u)T α1β1 (u + η) −T α1β2 (u)T α2β1 (u + η)=T α1β1 (u)T α2β2 (u + η) −T α2β1 (u)T α1β2 (u + η)=T α1β1 (u + η)T α2β2 (u) −T α1β2 (u + η)T α2β1 (u)=T α2β2 (u + η)T α1β1 (u) −T α2β1 (u + η)T α1β2 (u)(10)Consider the matrix U(u) formed by the quantum minorsU(u) =T[2323](u)−T[2313](u)T[2312](u)−T[1323](u)T[1313](u)−T[1312](u)T[1223](u)−T[1213](u)T[1212](u)(11)An equivalent expression for U(u) is given byU3(u)t = 3tr23P −123T1(u)T2(u + η)where t stands for the matrix transposition.The matrix U(u) allows to invert T(u):d(u)=T(u)tU(u + η) = U(u + η)T(u)t=U(u)tT(u + 2η) = T(u + 2η)U(u)t(12)or, in expanded form,d(u)δαγ=XβT βα (u)Uβγ (u + η) =XβUαβ (u + η)T γβ (u)=XβUβα(u)T βγ (u + 2η) =XβT αβ (u + 2η)Uγβ(u)(13)5

The commutation relations for T(u) and U(u), like (7), can also be written inthe R matrix form ( ˆR(u) = [R(u + η)−1]t2)ˆR(u −v)T1(u)U2(v) = U2(v)T1(u) ˆR(u −v)(14)R(u −v)U1(u)U2(v) = U2(v)U1(u)R(u −v)(15)or, at length,(u −v + η)T α1β1 (u)Uα2β2 (v) −ηδα1α2XγT γβ1(u)Uγβ2(v)= (u −v + η)Uα2β2 (v)T α1β1 (u) −ηXγUα2γ (v)T α1γ (u)δβ1β2(16)(u −v)Uα1β1 (u)Uα2β2 (v) + ηUα2β1 (u)Uα2β1 (v)= (u −v)Uα2β2 (v)Uα1β1 (u) + ηUα2β1 (v)Uα1β2 (u)(17)The quantum minors of U(u) are, in turn, expressed in terms of T(u)U[α1α2β1β2 ](u) = sαsβT α3β3 (u + η)d(u)(18)where the triplets (α1α2α3) and (β1β2β3) are permutations of (123) and sα and sβare the corresponding signatures.The commuting Hamiltonians are given by the matrix traces of T(u) and U(u)t1(u)=trT(u) ≡Pα T αα (u)t2(u)=trU(u) = tr12P −12T1 (u)T2 (u + η)(19)which are operator-valued polynomials of degree M and 2M, respectively. [t1(u), t1(v)] = [t1(u), t2(v)] = [t2(u), t2(v)] = 0Due to the assumption made that the leading coefficient of the polynomial T(u)is a nondegenerate number matrix the total number of commuting Hamiltonians isM + 2M = 3M, cf.

[12].3Construction of canonical variablesThe commuting Hamiltonians being described, the next step is to construct theseparated variables xn. In the classical case [12] they are constructed as the zeroesof the polynomial B(u) = T 23 (u)U31(u) −T 13 (u)U32(u) of degree 3M.

In the quantumcase, let us define the quantum operator-valued polynomial Bc(u) asBc(u) = T 23 (u)U31(u −c) −T 13 (u)U32(u −c)(20)The parameter c is the anticipated quantum correction which will be fixed in thenext section. The results of the present section are valid for any value of c.6

Proposition 1 Bc(u) is a commutative family of operators[Bc(u), Bc(v)] = 0(21)The proof is given by a direct calculation. One takes the product Bc(u)Bc(v) and,after substituting the expressions (20) for Bc and expanding the brackets, tries tobring the u-dependent terms through the v-dependent ones.

To this end one noticesthat the nontrivial commutations occur only inside the pairs (T 23 , T 13 ) and (U31, U32)and applies the relationsT α3 (u)T β3 (v) =u −vu −v −ηT β3 (v)T α3 (u) −ηu −v −ηT α3 (v)T β3 (u)(22)U3α(u)U3β(v) =u −vu −v + ηU3β(v)U3α(u) +ηu −v + ηU3α(v)U3β(u)(23)which follow from (6) and (17). Finally, after multiple cancellations, one arrives atBc(v)Bc(u).Let us assume now that the matrix Z is such that the leading coefficient B ofBc(u) is nonzero.

It means that Z should lie outside of certain algebraic manifoldof codimension 1 which must be true for a generic matrix Z. We can define now thecoordinate operators xn (n = 1, .

. .

, 3M) as the zeroes of the polynomial Bc(u) ofdegree 3M.B(u) = B3MYn=1(u −xn)B(xn) = 0,n = 1, . .

., 3M(24)They commute[xm, xn] = 0(25)by virtue of (21). The detailed description of the construction and the discussion ofthe mathematical subtleties of handling the zeroes of operator-valued polynomialsare given in [3] where the SL(2) case is considered.

Since in the SL(3) case theargument is the same, we do not reproduce it here.We shall assume that the common spectrum of xn is simple and that the rep-resentation space W is realized as some space of functions on spec{xn}3Mn=1. Theprecise description of the functional space depends on the particular representationof T , see [2] for discussion of possibilities in SL(2) case.The next step is the construction of the variables canonically conjugated to xn.Define the rational operator-valued function Ac(u) asAc(u) = −T 23 (u −η)−1U32(u −c) = −U32 (u −c)T 23 (u −η)−1(26)Note that the two expressions for Ac(u) are equivalent since T 23 (u) and U32(v)commute according to (13).

Moreover, since T 23 (u) and U32(u) are themselves com-muting families, Ac(u) is also a commuting family. [Ac(u), Ac(v)] = 0(27)7

Proposition 2 Ac(u) and Bc(v) satisfy the following commutation relation(u −v)Ac(u)Bc(v) −(u −v −η)Bc(v)Ac(u)= ηBc(u)Ac(v)T 23 (u −η)−1T 23 (u)−1T 23 (v −η)T 23 (v)(28)Proof. Notice that the following identity holds(u −v)U32(u −c)Bc(v)T 23 (u −η) −(u −v −η)T 23 (u −η)Bc(v)U32(u −c)= η eBc(u)T 23 (v)U32(v −c),(29)whereeBc(u) ≡T 23 (u −η)U31(u −c) −T 13 (u −η)U32(u −c),(30)which can be verified by a direct calculation in the same manner as in the previousproof.

Using the simple relationT α3 (u −η)T β3 (u) = T β3 (u −η)T α3 (u)(31)which follows from (22) one can verify also another identityeBc(u)T 23 (u) = T 23 (u −η)Bc(u)(32)Now, using (32) and the commutativity of Ac(v) and T 23 (u), we transform theright hand side of (29) intoηT 23 (u −η)Bc(u)T 23 (u)−1T 23 (v)U32(v −c)= −ηT 23 (u −η)Bc(u)Ac(v)T 23 (u)−1T 23 (v)T 23 (v −η)Finally, multiplying the resulting identity by T 23 (u −η)−1 both from the left andfrom the right, we arrive at (28).In the classical case [12] the exponents Xn = epn of the momenta pn canonicallyconjugated to the coordinates xn are given by the formula Xn = A(xn). In orderto transfer the definition to the quantum case it is necessary to describe what isunderstood under the substitution of the operator values u = xn into Ac(u).

Follow-ing [3] we fix the operator ordering putting x’s to the left. So, let for any rationaloperator-valued function F(u) the symbol [F(u)]u = xn be defined as the Cauchyintegral[F(u)]u=xn =12πiZΓ du(u −xn)−1F(u)(33)taken over a closed contour Γ encircling counterclockwise the spectrum of xn andleaving outside the poles of F(u).

Such a contour exists if specxn ∩poles{F(u)} = ∅.For the polynomials F(u) = Pp upFp the definition [F(u)]u = xn = Pp xpnFp givenin [3] is obviously recovered.Define now Xn asXn = [Ac(u)]u=xn(34)8

The condition spec{xn}3Mn=1 ∩poles{Ac(u)} = ∅should be investigated for anyconcrete representation of the algebra T , and here we assume it to be satisfied.The identities (27) and (28) proven above entrain immediately the commutationrelations[Xm, Xn] = 0(35)Xmxn = (xn −ηδmn)Xm(36)The derivation of (35) and (36) is the same as in [3], so we do not reproduce ithere.4Quantum characteristic equationIn the present paragraph we restrict the parameter c to the value c = η so that nowB(u) = T 23 (u)U31(u −η) −T 13 (u)U32(u −η)(37)A(u) = −T 23 (u −η)−1U32(u −η) = −U32(u −η)T 23 (u −η)−1(38)Proposition 3 The following equality (“quantum characteristic equation” for Xn)holdsX3n −X2n[t1(u)]u=xn + Xn[t2(u −η)]u=xn −d(xn −2η) = 0(39)The identity (39) can be thought of as describing the “quantum algebraic spec-tral curve” for the matrix T(u). It is an open question if the proposition 3 canbe generalized for arbitrary value of c and thus if the variables Xn, xn provide aseparation of coordinates for some quantum integrable system.The proof is based on the identityU32(u −η)U32(u −2η)U32(u −3η)+U32(u −η)U32(u −2η)T 23 (u −3η)t1(u −2η)+U32(u −η)T 23 (u −2η)T 23 (u −3η)t2(u −2η)+T 23 (u −η)T 23 (u −2η)T 23 (u −3η)d(u −2η)= ˆB(u)C(u)(40)whereˆB(u) = T 23 (u −3η)U31(u −η) −T 13 (u −3η)U32(u −η)(41)C(u) = T 23 (u −2η)U12(u −2η) −T 21 (u −2η)U32(u −2η)(42)Suppose for a moment that (40) is proven and consider another identityˆB(u)T 23 (u −2η)T 23 (u −η)T 23 (u) = T 23 (u −η)T 23 (u −2η)T 23 (u −3η)B(u)(43)which is derived in the same manner as (32).9

Now let us multiply the equality (40) from the left by T 23 (u −η)−1T 23 (u −2η)−1 ×T 23 (u −3η)−1, then use the identity (43) and the definition (38) of A(u). The resultisA(u)A(u −η)A(u −2η) −A(u)A(u −η)t1(u −2η)+A(u)t2(u −2η) −d(u −2η)= −B(u)T 23 (u −2η)−1T 23 (u −η)−1T 23 (u)−1C(u)(44)To obtain the desired characteristic equation (39) it suffices now to substituteu = xn into (44) from the left and apply the following lemma.Lemma 1 For any operator-valued polynomial F(u)[A(u)F(u −η)]u=xn = Xn[F(u)]u=xn(45)To conclude the proof it remains to prove the identity (40) and the lemma 1.Expanding the right hand side of (40) and reordering commuting factors we getfour termsˆB(u)C(u)=U32(u −η)U32(u −η)T 13 (u −3η)T 21 (u −2η)−U32(u −η)T 13 (u −3η)T 2(u −2η)U12(u −2η)−T 23 (u −3η)U31(u −η)U32(u −2η)T 21 (u −η)+T 23 (u −3η)T 23 (u −2η)U31(u −η)U12(u −2η)(46)Making in the first term substitutionT 13 (u −3η)T 21 (u −2η) = U32(u −3η) + T 23 (u −3η)T 11 (u −2η),see (11) and (10), and replacing T 11 with t1 −T 22 −T 33 we obtain the expressionU32(u −η)U32(u −2η)U32(u −3η) + U32(u −η)U32(u −2η)T 23 (u −3η)t1(u −2η)−U32 (u −η)U32(u −2η)T 23 (u −3η)[T 22 (u −2η) + T 33 (u −2η)]Analogously, in the fourth term of (46) we make the substitutionU31(u −η)U12(u −2η) = U32(u −η)U11(u −2η) + T 23 (u −η)d(u −2η),see (18), and replace U11 with t2 −U22 −U33 obtainingT 23 (u −η)T 23 (u −2η)T 23 (u −3η)d(u −2η) + U32(u −η)T 23 (u −2η)T 23 (u −3η)t2(u −2η)−U32(u −η)T 23 (u −2η)T 23 (u −3η)[U22(u −2η) + U33(u −2η)]Notice that all the four terms of the left hand side of (40) are cancelled by theright hand side terms.

Consider now the second term in (46) and apply the identityT 13 (u −3η)T 23 (u −2η) = T 23 (u −3η)T 13 (u −2η),10

see (31), and then the identityT 13 (u −2η)U12(u −2η) = −T 23 (u −2η)U22(u −2η) −U32(u −2η)T 33 (u −2η),see (16). The result isU32(u−η)T 23 (u−3η)T 23 (u−2η)U22(u−2η)+U32 (u−η)U32(u−2η)T 23 (u−3η)T 33 (u−2η)Analogously, the third term of (46) is transformed, with the use of the identitiesU31(u −η)U32(u −2η) = U32(u −η)U31(u −2η)andU31(u −2η)T 21 (u −2η) = −U32 (u −2η)T 22 (u −2η) −T 23 (u −2η)U33(u −2η),intoU32(u−η)U32(u−2η)T 23 (u−3η)T 22 (u−2η)+U32 (u−η)T 23 (u−2η)T 23 (u−3η)U33(u−2η)Collecting all the terms obtained we observe their total cancellation, the identity(40) being thus proved.To prove the lemma 1 consider the left hand side of (45) and apply the definition(33)[A(u)F(u −η)]u=xn =12πiZΓ du(u −xn)−1A(u)F(u −η) = .

. .Representing F(u −η) as another Cauchy integral over the contour Γ′ encirclingthe point u one rewrites the previous expression as.

. .

=12πiZΓ du 12πiZΓ′ dv(u −xn)−1(v −u)−1A(u)F(v −η) = . .

.Changing the order of integrals and using the definition (34) of Xn and thecommutation relation (36) one obtains finally the right hand side of (45).. . .=12πiZΓ′ dv(v −xn)−1XnF(v −η)=12πiZΓ′ dvXn(v −xn −η)−1F(v −η)=Xn[F(v)]v=xnThe proposition 3 being thus proven, we can discuss its corollaries.

The equation(39) obviously fits the form (1) since the operator ordering in (39) is the same aspostulated for (1). Following the heuristic argument given in the Introduction onecan expect that the quantum characteristic equation (39) yields the separation ofvariables for the 3M commuting Hamiltonians given by the coefficients of the poly-nomials t1(u) and t2(u).

If Xn are realized as shift operators Xn = exp{−η∂/∂xn}11

the resulting separated equations (5) become the third order finite difference equa-tionsψn(xn −3η)−τ1(xn −2η)ψn(xn −2η)+τ2(xn −2η)ψn(xn −η)−d(xn −2η)ψ(xn) = 0where τ1,2(u) are the eigenvalues of the operators t1,2(u).However, one cannot always take Xn as pure shifts.Generally speaking, Xnshould contain a cocycle factor ∆n(x)XnΨ(. .

. , xn, .

. .) = ∆n(x1, .

. ., x3M)Ψ(.

. .

, xn −η, . .

. )There is a liberty of canonical transformations Ψ(x) →ρ(x)Ψ(x) and, respec-tively,∆n(.

. .

, xn, . .

.) →ρ(.

. .

, xn −η, . .

. )ρ(.

. .

, xn, . .

. )∆n(.

. .

, xn, . .

. )(47)where ρ(x) is a nonzero function on spec{xn}3Mn=1.For the finite dimensional representations of T the cocycles ∆n(x) certainlycannot be equivalent to the trivial ones Dn(x) ≡1 since they must have zeroes on theboundary of the finite set which cannot be removed by any canonical transformation(47).For the separation of variables, however, it is enough that the factors ∆n(x) canbe made to depend only on xn which results in the separated equations[Ξ3n −τ1(xn −2η)Ξ2n + τ2(xn −2η)Ξn −d(xn −2η)]ψ(xn) = 0(48)where Ξn stands for the operatorΞnψ(xn) = ∆n(xn)ψ(xn −η)In the SL(2) case the Theorem 3.4 of [3] establishes the property in question.The corresponding problem for the SL(3) is being under study.There is another problem which should be solved before one could establish theseparation of variables in the SL(3) case.

If the common spectrum of the coordinates{xn}3Mn=1 is a bounded (finite) set in C3M and is not a cartesian product of one-dimensional sets specxn then it must have a special geometry for the separationof variables to take place. The simplest example is given by the Laplacian in arectangular triangle with the zero boundary conditions.

The eigenfunctions are notfactorized (4) but, instead, are linear combinations of such products.To sum up, the quantum characteristic equation itself provides only local sep-aration of variables. In order to obtain a global s. o. v. one needs to study moredeeply the spectrum of {xn}3Mn=1 and the representation of the algebra (25), (35),(36).

Their properties depend essentially on the representation of the algebra Ttaken and we leave the problem for a subsequent study. It seems to be a plausibleconjecture that s. o. v. takes place for any representation of T .12

5Comparison with ABAIn the SL(2) case the separated equations are known to be equivalent to Betheequations defining the spectrum of Hamiltonians provided the corresponding rep-resentation of T has a heighest vector and thus allows application of the algebraicBethe ansatz technique [3]. It is natural therefore to investigate an analogous cor-respondence for the SL(3) case.The Bethe ansatz for sl(N) case was developed in the papers [4, 6, 10, 11].

Weshall use the results of Kulish and Reshetikhin [5, 6] who considered the most generalrepresentations of SL(N). As shown in [6] the eigenvalues τ1,2(u) of t1,2(u) togetherwith the quantum determinant d(u) can be written down in the formτ1(u)=Λ1(u) + Λ2(u) + Λ3(u)τ2(u)=Λ1(u)Λ2(u + η) + Λ1(u)Λ3(u + η) + Λ2(u)Λ3(u + η)(49)d(u)=Λ1(u)Λ2(u + η)Λ3(u + 2η) = d1(u)d2(u + η)d3(u + 2η)where the number polynomials d1,2,3(u) are determined by the parameters of therepresentation of T in question.The three “quantum eigenvalues” Λ1,2,3(u) of T(u) can be expressed in terms oftwo polynomials Q1,2(u)Λ1(u) = d1(u)Q1(u + η)Q1(u)Λ2(u) = d2(u)Q1(u −η)Q1(u)Q2(u + η)Q2(u)Λ3(u) = d3(u)Q2(u −η)Q2(u)Eliminating Q2(u) or Q1(u) from (49) one obtains for the polynomials Q1,2(u)the third order finite-difference equations [5].d2(x −2η)d3(x −η)Q1(x −3η) −τ2(x −2η)Q1(x −2η)+τ1(x −η)d1(x −2η)Q1(x −η) −d1(x −η)d1(x −2η)Q1(x)(50)d3(x −2η)d3(x −η)Q2(x −3η) −τ1(x −2η)d3(x −η)Q2(x −2η)+τ2(x −2η)Q2(x −η) −d1(x −2η)d2(x −η)Q2(x)(51)Making the substitutionsQ1(x)=d2(x)d3(x + η)d2(x −η)d3(x)ϕ(x)Q2(x)=d3(x)ψ(x)and using the shift/multiplication operatorsΘ1ϕ(x)=d2(x −2η)d3(x −η)ϕ(x −η)Θ2ψ(x)=d3(x −η)ψ(x −η)13

one can put (50), (51) into the form[Θ31 −τ2(x −2η)Θ21 + τ1(x −η)d(x −2η)Θ1 −d(x −η)d(x −2η)]ϕ(x) = 0(52)[Θ32 −τ1(x −2η)Θ22 + τ2(x −2η)Θ2 −d(x −2η)]ψ(x) = 0(53)One notices immediately that the equation (53) coincides with the hypotheticalseparated equation (48) giving thus a support to the conjecture.In [3] it was conjectured that the separated coordinates should be splitted intotwo subsets producing the two separated equations (52), (53) which seems now tobe an overcomplication. The equation for (52) should be considered rather as theseparation equation for the alternative set of coordinates obtained from the matrixU(u) in the same way as xn are obtained from T(u).Acknowledgments.

I am grateful to V. B. Kuznetsov and V. O. Tarasov for valuableand encourageing discussions. I thank the Isaac Newton Institute for MathematicalSciences for hospitality.References[1] Weigert S.: The problem of quantum integrability.

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