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Mathematical Preprint Series, Report 93-11, University of Amsterdam

arXiv:hep-th/9306155v1 29 Jun 1993hep-th/9306155Mathematical Preprint Series, Report 93-11, University of Amsterdam1This author was supported by the National Dutch Science Organization (NWO) under the Project# 611–306–540.1

Linear r-matrix algebra for classicalseparable systemsJ.C.Eilbecka, V.Z.Enol’skiib, V.B.Kuznetsovc1 and A.V.TsiganovdMarch 12, 2018aDepartment of Mathematics, Heriot-Watt University,Riccarton, Edinburgh EH14 4AS, Scotlande-mail: chris@cara.ma.hw.ac.ukbDepartment of Theoretical Physics, Institute of Metal Physics,Vernadsky str. 36, Kiev-680, 252142, Ukrainee-mail: vbariakhtar@gluk.apc.orgcDepartment of Mathematics and Computer Science, University of Amsterdam,Plantage Muidergraht 24, 1018 TV Amsterdam, The Netherlandse-mail: vadim@fwi.uva.nldDepartment of Earth Physics, Institute for Physics, University of St Petersburg,St Petersburg 198904, Russiae-mail: kuznetso@onti.phys.lgu.spb.suAbstractWe consider a hierarchy of the natural type Hamiltonian systems of n degreesof freedom with polynomial potentials separable in general ellipsoidal and generalparaboloidal coordinates.

We give a Lax representation in terms of 2×2 matricesfor the whole hierarchy and construct the associated linear r-matrix algebra withthe r-matrix dependent on the dynamical variables. A Yang-Baxter equation ofdynamical type is proposed.

Using the method of variable separation we providethe integration of the systems in classical mechanics conctructing the separationequations and, hence, the explicit form of action variables.The quantisationproblem is discussed with the help of the separation variables.Key words: separable systems, linear r-matrix algebras, classical Yang-Baxterequation, Lax representation, Liouville integrable systemsAMS classification: 70H20, 70H33, 58F072

1IntroductionThe method of separation of variables in the Hamilton-Jacobi equation,H(p1, . .

. , pn, x1, .

. .

, xn) = E,pi = ∂W∂xi,i = 1, . .

. n,(1.1)is one of the most powerful methods for the construction of action for the Liouvilleintegrable systems of classical mechanics [3].

We consider below the systems of thenatural form described by the HamiltonianH = 12nXi=1p2i + V (x1, . .

. , xn),pi, xi ∈R .

(1.2)The separation of variables means the solution of the partial differential equation (1.1)for the action function W in the following additive formW =nXi=1Wi(µi; H1, . .

. , Hn),Hn = H,where µi will be called separation variables .

Notice that the partial functions Wi dependonly on their separation variables µi, which define new set of variables instead of theold ones {xk}, and on the set of constants of motion, or integrals of motion , {Hk}. Inthe sequel we shall speak about coordinate separation where the separation variables{µi} are the functions of the coordinates {xk} only.

(The general change of variablesmay include also corresponding momenta {pk}).For a free particle (V = 0), the complete classification of all orthogonal coordinatesystems in which the Hamilton-Jacobi equation (1.1) admits the separation of variablesis known, they are generalized n-dimensional ellipsoidal and paraboloidal coordinates[8, 9] (see also the references therein). It is also known that the Hamiltonian systems(1.2) admitting an orthogonal coordinate separation with V ̸= 0 are separated only inthe samecoordinate systems.The modern approach to finite-dimensional integrable systems uses the language ofthe representations of r-matrix algebras [10, 15, 17, 18].

The classical method of separa-tion of variables can be formulated within this language dealing with the representationsof linear and quadratic r-matrix algebras [11, 12, 17, 19]. For the 2 × 2 L-operatorsthe recipe consists of considerating the zeros of one of the off-diagonal elements as theseparation variables (see also generalization of this approach to the higher dimensionsof L-matrix [19]).

For V = 0 in [11, 12] 2 × 2 L-operators were given satisfying thestandard linear r-matrix algebra [10, 15],{L1(u), L2(v)} = [r(u −v), L1(u) + L2(v)],r(u) = 1u1000001001000001,(1.3)and the link with the separation of variables method was elucidated. Here in (1.3) weuse the familiar notations for the tensor products of L(u) and 2 × 2 unit matrix I,L1(u) = L(u) ⊗I, L2(v) = I ⊗L(v).3

In the present article we construct 2 × 2 L-operators for the systems (1.2) beingseparated in the generalized ellipsoidal and paraboloidal coordinates. In the case whenthe degree N of the potential V is equal to 1 or 2 the associated linear r-matrix algebraappears to be the standard one (1.3).

In the case N > 2 the algebra is of the form{L1(u), L2(v)} = [r(u −v), L1(u) + L2(v)] + [s(u, v), L1(u) −L2(v)],(1.4)with s(u, v) = αN(u, v) σ−⊗σ−, where σ−= σ1 −i σ2 and σi are the Pauli matrices,and αN(u, v) is the function which equals 1 for N = 3 and depends on the dynamicalvariables {xk, pk}nk=1 for N > 3.The study of completely integrable systems admitting a classical r-matrix Poissonstructure with the r-matrix dependent on dynamical variables has attracted some at-tention [5, 6, 13]. It is remarkable that the celebrated Calogero-Moser system, whosecomplete integrability was shown a number of years ago (c.f.

[14]), has been found onlyrecently to possess a classical r-matrix of dynamical type [4].Below we recap briefly how to get the 2 × 2 L-operators for the separable systems(1.2) without potential V [11, 12]. Let us consider the direct sum of the Lie algebraseach of rank 1: A = ⊕nk=1 sok(2, 1).

Generators ⃗sk ∈R3, k = 1, . .

. , n of the A algebrasatisfy the following Poisson brackets:{sik, sjm} = δkm εijl gll slk,g = diag(1, −1, −1).

(1.5)Everywhere below, we will imply the g metric to calculate the norm and scalar productof the vectors ⃗si:⃗s 2i≡(⃗si,⃗si) = (s1i )2 −(s2i )2 −(s3i )2,(⃗si,⃗sk) = s1i s1k −s2i s2k −s3i s3k.Let us fix the values of the Casimir elements of the A algebra: ⃗s 2i = c2i , then variables⃗si lie on the direct product of n hyperboloids in R3. Let ci ∈R and choose the uppersheets of these double-sheeted hyperboloids.

Denote the obtained manifold as K+n . Wewill denote by hyperbolic Gaudin magnet [7] integrable Hamiltonian system on K+ngiven by n integrals of motion Hi which are in involution with respect to the bracket(1.5),Hi = 2nXk=1′(⃗si,⃗sk)ei −ek,{Hi, Hk} = 0,ei ̸= ekifi ̸= k.(1.6)To be more exact one has to call this model an n-site so(2,1)-XXX Gaudin magnet.Notice that all the Hi are quadratic functions on generators of the A algebra and thefollowing equalities are validnXi=1Hi = 0,nXi=1eiHi = ⃗J 2 −nXi=1c2i ,where the new variable ⃗J =Pni=1 ⃗si is introduced which is the total sum of the hy-perbolic momenta ⃗si.

The components of the vector ⃗J obey so(2,1) Lie algebra withrespect to the bracket (1.5) and are in involution with all the Hi. The complete setof involutive integrals of motion is provided by the following choice: Hi, ⃗J 2 and, for4

example, (J3)2. The integrals (1.6) are generated by the 2 × 2 L-operator (as well asthe additional integrals ⃗J )L(u)=nXj=11u −ej is3j−(s1j −s2j)−(s1j + s2j)−is3j!,det L(u) = −nXj=1 Hju −ej+c2j(u −ej)2!,(1.7)satisfying the standard linear r-matrix algebra (1.3).

Let ci = 0, i = 1, . .

. , n, thenthe hyperboloids ⃗s 2i = c2i turn into cones.

The manifold K+n admits in this case thefollowing parameterization (pi, xi ∈R):s1i = p2i + x2i4,s2i = p2i −x2i4,s3i = pi xi2 ,(1.8)where the variables pi and xi are canonically conjugated. Using the isomorphism (1.8)the complete classification of the separable orthogonal coordinate systems was providedin [11, 12] by means of the corresponding L-operators satisfying the standard linear r-matrix algebra (1.3).

The starting point for our investigation are these L-operatorswritten for the cases of free motion on a sphere and in the Euclidean space.The paper is organized as follows. In Section 2 we describe the classical Poissonstructure associated with the hierarchy of natural type Hamiltonians separable in thethree coordinate systems—spherical (for motion on a sphere), and general ellipsoidaland paraboloidal (for n-dimensional Euclidean motion) coordinates.

This structure isgiven in terms of the linear r-matrix formalism, providing new example of the dynamicaldependence of the r-matrices. We also introduce analogue of Yang-Baxter equation forour dynamical r-matrices.

In the Section 3 we derive the Lax representation for allthe hierarchy as a consequence of the r-matrix representation given in the Section 2.Section 3 deals also with the variable separation material. The question of quantisationof the considered systems is briefly discussed.2Classical Poisson structureLet us consider the following ansatz for the 2 × 2 L-operatorLN(u) = A(u)B(u)CN(u)−A(u)(2.1)whereB(u)=ε −nXi=1x2iu −ei,ε = 0, 1, or4(u −xn+1 + B),(2.2)A(u)=˙ε + 12nXi=1xi piu −ei,(2.3)CN(u)=nXi=1p2iu −ei−VN(u),VN(u) =NXk=0Vk uN−k.

(2.4)5

Here the xi, pj are canonically conjugated variables ({pi, xj} = δij), Vk are indetermi-nate functions of the x-variables; B and ei are non-coincident real constants. Note thatdot over ε means differentiation by time, and for natural Hamiltonian (1.2) one has˙xn+1 = pn+1.Theorem 1 Let the curve det(L(u) −λI) = 0 for the L-operator (2.1) have the formλ2 −A(u)2 −B(u)CN(u)=λ2 + ε uN −nXi=1Hiu −ei= 0,forε = 0, 1,and(2.5)λ2 −A(u)2 −B(u)CN(u)=λ2 + 16uN−2(u + B)2 + 8H −nXi=1Hiu −ei= 0,forε = 4(u −xn+1 + B) ,(2.6)with some integrals of motion Hi and Hi, H in the case of (2.6).

Then the followingrecurrence relations for Vk are validVk=nXi=1x2ik−1Xj=0Vk−1−jeji,V0 = 1,forε = 0, 1;(2.7)Vk=(xn+1 + B)Vk−1 + 12nXi=1x2ik−1Xj=1Vk−1−jeji,V0 = 0,forε = 4(u −xn+1 + B). (2.8)The explicit formulae for the integrals Hi have the formHi=−nXj=1′ M2ijei −ej+ ε · p2i + x2iNXk=0VkeN−ki,forε = 0, 1,(2.9)Hi=2x2iN−1Xj=1(−1)j−1eji VN−j + 4pn+1 pi xi −p2i (ei + 4xn+1 −4B)+Xj̸=iM2ijei −ej,forε = 4(u −xn+1 + B),(2.10)where Mij = xipj −xjpi.

The Hamiltonians H are given byH≡nXi=1Hi = ε ·nXi=1p2i + VN+1,forε = 0, 1,(2.11)H=12n+1Xi=1p2i + VN,forε = 4(u −xn+1 + B). (2.12)Proof is straightforward and based on direct computations.We remark, that the above recurrence formulae for the potentials can be written indifferential form.

In particular, for the paraboloidal coordinates we have∂VN∂xi=12∂VN−1∂xn+1xi −Ai∂VN−1∂xi,i = 1, . .

. , n,(2.13)∂VN∂xn+1=VN−1 + 12nXi=1xi∂VN−1∂xi+ (xn+1 −B)∂VN−1∂xn+1.

(2.14)6

Notice that the case of ε = 0 is connected with the ellipsoidal coordinates on a sphereand two other cases ε = 1, 4(u −xn+1 + B) describe the ellipsoidal and paraboloidalcoordinates in the Euclidean space, respectively (see Section 3.2 and [11, 12] for moredetails). Recall that we study now the motion of a particle on these manifolds underthe external field with the potential V that could be any linear combination of thehomogeneous ones Vk.Now we are ready to describe the linear algebra for the L-operator (2.1).Theorem 2 Let the L-matrix be of the form (2.1) and satisfy the conditions of theTheorem 1, then the following algebra is valid for its entries{B(u), B(v)}={A(u), A(v)} = 0,(2.15){CN(u), CN(v)}=−4 αN(u, v) (A(u) −A(v)),(2.16){B(u), A(v)}=2u −v(B(u) −B(v)),(2.17){CN(u), A(v)}=−2u −v(CN(u) −CN(v)) −2 αN(u, v) B(v),(2.18){B(u), CN(v)}=−4u −v(A(u) −A(v)),(2.19)where the function αN(u, v) has the formαN(u, v)=QN(u) −QN(v)u −v=NXk=1Qkuk −vku −v ,(2.20)QN(u)=NXk=0Qk uk,Qk =kXm=0Vm Vk−m.Proof is based on the recurrence relations (2.7), (2.8).We remark that for the paraboloidal coordinates the following formula is validQ(u) = uN−2 −12N−3Xk=0∂VN−k−1∂xn+1,(2.21)therefore in this case we haveQ(u) = 14∂C(u)∂xn+1.

(2.22)The algebra (2.15)-(2.19) can be rewritten in the matrix form as linear r-matrixalgebra{L1(u), L2(v)} = [r(u −v), L1(u) + L2(v)] + [sN(u, v), L1(u) −L2(v)],(2.23)using 4 × 4 notations L1(u) = L(u) ⊗I, L2(v) = I ⊗L(v); the matrices r(u −v) andsN(u, v) are given byr(u −v) =2u −v P,P =1000001001000001,(2.24)sN(u, v) = 2 αN(u, v) σ−⊗σ−,σ−= 0010!.7

The algebra (2.15)-(2.19) or (2.23)-(2.24) contains all the information about the systemunder consideration. From it there follows the involutivity of the integrals of motion.Indeed, the determinant d(u) ≡det L(u) is the generating function for the integrals ofmotion and it is simply to show that{d(u), d(v)} = 0.

(2.25)In particular, the integrals Hi (2.9), (2.10) are the residues of the function d(u):Hi = res|u=ei d(u),i = 1, . .

. , n.The Hamiltonians H (2.11),(2.12) appear to be a residue at infinity.

Let us rewrite therelation (2.23) in the form{L1(u), L2(v)} = [d12(u, v), L1(u)] −[d21(u, v)L2(v)],(2.26)with dij = rij + sij, dji = sij −rij at i < j.Theorem 3 The following equations (dynamical Yang-Baxter equations) are valid forthe algebra (2.26)[d12(u, v), d13(u, w)] + [d12(u, v), d23(v, w)] + [d32(w, v), d13(u, w)] ++{L2(v), d13(u, w)} −{L3(w), d12(u, v)} ++[c(u, v, w), L2(v) −L3(w)] = 0,(2.27)where c(u, v, w) is some matrix dependent on dynamical variables. Other two equationsare obtained from (2.27) by cyclic permutations.Proof Let us write the Jacobi identity as{L1(u), {L2(v), L3(w)}} + {L3(w), {L1(u), L2(v)}}+{L2(v), {L3(w), L1(u)}} = 0(2.28)with L1(u) = L(u) ⊗I ⊗I, L2(v) = I ⊗L(v) ⊗I, L3(w) = I ⊗I ⊗L(w).

The extendedform of the (2.28) reads [13],[L1(u), [d12(u, v), d13(u, w)] + [d12(u, v), d23(v, w)] ++[d32(w, v), d13(u, w)]] +(2.29)+[L1(u), {L2(v), d13(u, w)} −{L3(w), d12(u, v)}] ++cyclic permutations = 0.Further we restrict ourselves proving the (2.27) only in the paraboloidal case (othercases can be handled in a similar way). To complete the derivation of (2.27) we shallprove the following equality for all the members of the hierarchy{L2(v), s13(u, w)}−{L3(w), s12(u, v)} = 2βN(u, v, w)[P23, s13 + s12]−∂βN(u, v, w)∂xn+1[s, L2(v) −L3(w)](2.30)8

(with cyclic permutations). In (2.30) the matrix s = σ−⊗σ−⊗σ−andβN(u, v, w) = QN(u)(v −w) + QN(v)(w −u) + QN(w)(u −v)(u −v)(v −w)(w −u).

(2.31)In the extended form (2.30) can be rewritten as{Q(u), Q(v)}={B(u), Q(v)} = 0,(2.32){A(u), Q(v)}=4 αN(u, v) −12∂αN(u, v)∂xn+1B(u),(2.33){A(w), αN(u, v)}=4u −v(αN(w, u) −αN(w, v))−12B(w)u −v ∂αN(w, v)∂xn+1−∂αN(w, u)∂xn+1!,(2.34){Q(u), C(v)}+{C(u), Q(v)} = ∂αN(u, v)∂xn+1(A(u) −A(v)). (2.35)The equality (2.32) is trivial and equation (2.33) is derived by differentiating (2.18).Equation (2.34) follows from the definition of Q(u) and (2.33).

To prove (2.35) we writeit using the explicit form of CN(u) and A(u) as followsnXi=1pi(v −ei)(u −ei) (u −ei)∂αN(w, u)∂xi−(v −ei)∂αN(w, v)∂xi!= 12nXi=1pixi(u −ei)(v −ei) ∂αN(w, u)∂xn+1−∂αN(w, v)∂xn+1!. (2.36)Using the identityuwk −ukw −u −vwk −vkw −v= wk+1 −uk+1w −u−wk+1 −vk+1w −v,and the recurrence relation (2.13), we find that the equality (2.36) is valid.

Thereforethe equations (2.27) follow with the matrix c(u, v, w) = ∂β(u, v, w)/∂xn+1 σ−⊗σ−⊗σ−.The proof is completed.We remark that validity of the equations (2.27) with an arbitrary matrix c(u, v, w)is sufficient for the validity of (2.28) and, therefore, (2.27) can be interpreted as somedynamical classical Yang-Baxter equation, i.e. the associativity condition for the linearr-matrix algebra.

These equations have an extra term [c, Li −Lj] in comparison withthe extended Yang-Baxter equations in [13].We would like to emphasize that all statements of the Section can be generalized tothe following form of the potential term VN(u) in (2.4)VMN(u) =NXk=−Mfk Vk uN−k,fk ∈Cthat corresponds to the linear combinations of homogeneous terms Vk as potential Vand also includes the negative degrees to separable potential. See the end of Section3.1 for more details.9

3Consequences of the r-matrix representation3.1Lax representationFollowing the article [5] we can consider the Poisson structure (2.26) for the powers ofthe L-operator{(L1(u))k, (L2(v))l} = [d(k,l)12 (u, v), L1(u)] −[d(k,l)21 (u, v)L2(v)],(3.1)withd(k,l)ij(u, v) =(3.2)k−1Xp=0l−1Xq=0(L1(u))k−p−1(L2(v))k−q−1dij(u, v)(L1(u))p(L2(v))q.As an immediate consequence of (3.1)-(3.2) we obtain that the conserved quantities H,Hi are in involution. Indeed, we have{Tr(L1(u))2, Tr(L2(v))2} = Tr{(L1(u))2, (L2(v))2},(3.3)and after applying the equality (3.1) at k = l = 2 to this equation and taking thetrace, we obtain the desired involutivity.

Further, let us define differentiation by timeas follows:˙L(u) = ddtL(u) = Tr2{L1(u), (L2(u))2},(3.4)where the trace is taken over the second space. Applying the equation (3.1) at k =1, l = 2 to (3.4), we obtain the Lax representation in the form ˙L(u) = [M(u), L(u)]with the matrix M(u) given asM(u) = 2 limv→u Tr2L1(v)(r(u −v) −s(u, v)).

(3.5)After the calculation in which we take into account the asymptotic behaviour of theL-operator (2.1), we obtain the following explicit Lax representation:˙L(u)=[M(u), L(u)],L(u)= A(u)B(u)CN(u)−A(u),M(u) =01QN(u)0,(3.6)where QN(u) was defined by the equations (2.20). Lax representations for the higherflows can be obtained in a similar way.It follows from the equation (3.6) thatA(u) = −12˙B(u),CN(u) = −12¨B(u) −B(u)QN(u),so our L-matrix can be given in the formL(u) = −12 ˙B(u)B(u)−12 ¨B(u) −B(u)QN(u)12 ˙B(u)!.

(3.7)10

The equations of motion, which follow from (3.6) with the L-matrix from (3.7), havethe formB1[QN] · B(u) = 0,(3.8)whereB1[QN] ≡14∂3 + 12{∂, QN},∂≡ddt,with curly brackets standing for the anticommutator. Operator B1 is the Hamiltonianoperator of the first Hamiltonian structure for the coupled KdV equation [1, 2].

Equa-tion (3.8), considered as one for the unknown function B(u), was solved in the threecases (2.2),B(u) = ε −nXi=1x2iu −ei,ε = 0, 1, 4(u −xn+1 + B),in [1] and [2]. General solution of this equation as one for the Q(u) has the formQMN(u) =NXk=−Mfk Qk uk,fk ∈C,(3.9)where the coefficients Qk are defined from the generating function ˜Q(u)˜Q(u) ≡B−2(u) =+∞Xk=−∞Qkuk.

(3.10)Recall that we can write the element CN(u) of the L-matrix (2.1) in two different forms(using Q or V functions)CN(u) = −12¨B(u) −B(u)QN(u) = −nXi=1p2iu −ei−V (u)where function V (u) =PNk=0 Vk uN−k was defined in (2.4). The general form of thefunction V (u) is as follows:VMN(u) =NXk=−Mfk Vk uk,fk ∈C,(3.11)where coefficients Vk are defined by the generating function ˜V (u)˜V (u) ≡B−1(u) =+∞Xk=−∞Vk uk.

(3.12)Potentials Vk are connected with coefficients Qk. Indeed, using generating functions(3.10) and (3.12), we have˜Q = B−2(u) = ˜V (u) · ˜V (u) =+∞Xk=−∞Vk uk+∞Xk=−∞Vk uk=+∞Xk=−∞ukkXj=0Vk Vk−j,and, therefore, Qk = Pkj=0 Vk Vk−j.

Thus we have recovered the formula (2.20) for thes-matrix.11

3.2Separation of VariablesLet K denote the number of degrees of freedom: K = n −1 for ellipsoidal coordinateson a sphere, K = n for ellipsoidal coordinates in the Eucludean space, and K =n + 1 for paraboloidal coordinates in the Euclidean space. The separation of variables(c.f.

[12, 18]) is understood in the context of the given hierarchy of Hamiltonian systemsas the construction of K pairs of canonical variables πi, µi, i = 1, . .

. , K,{µi, µk} = {πi, πk} = 0,{πi, µk} = δik,(3.13)and K functions Φj such thatΦjµj, πj, H(1)N , .

. .

, H(K)N= 0,j = 1, 2, . .

. , K,(3.14)where H(i)Nare the integrals of motion in involution.The equations (3.14) are theseparation equations .

The integrable systems considered admit the Lax representationin the form of 2 × 2 matrices (3.6) and we will introduce the separation variables πi, µiasB(µi) = 0,πi = A(µi),i = 1, . .

. , K.(3.15)Below we write explicitly these formulae for our systems.

The set of zeros µj, j = 1, . .

. Kof the function B(u) defines the spherical (ε = 0), general ellipsoidal (ε = 1) and generalparaboloidal (ε = 4(u −xn+1 + B)) coordinates given by the formulae [8, 9, 12]x2m=cQn−1j=1(µj −em)Qk̸=m(em −ek),m = 1, .

. .

, n,wherec =nXk=1x2k,forε = 0;(3.16)x2m=−4Qnj=1(µj −em)Qk̸=m(em −ek),m = 1, . .

. , n,forε = 1;(3.17)xn+1=−nXi=1ei + B +n+1Xi=1µi,x2m=−4Qn+1j=1(µj −em)Qk̸=m(em −ek),m = 1, .

. .

, n,forε = 4(u −xn+1 + B). (3.18)Theorem 4 The coordinates µi, πi given by (3.15) are canonically conjugated.Proof Let us list the commutation relations between B(v) and A(u),{B(u), B(v)}={A(u), A(v)} = 0,(3.19){A(u), B(v)}=2v −u(B(u) −B(v)).

(3.20)12

The equalities {µi, µj} = 0 follow from (3.19). To derive the equality {µi, πj} = −δijwe substitute u = µj in (3.20), obtaining thus{πj, B(v)} = −2v −µjB(v),which together with the equation0 = {πj, B(µi)} = {πj, B(v)} |v=µi +B′(µi){πj, µi}gives{πj, µi} = −1B′(µi){πj, B(v)} |v=µi= δij.Equalities {πi, πj} = 0 can be verified in the similar way:{πi, πj}={A(µi), A(µj)}={A(µj), A(v)} |v=µi +A′(µi){µi, A(µj)}=A′(µj){A(µi), µj)} + A′(µi){µi, A(µj)} = 0.The separation equations have the formπ2i = d(µi),(3.21)where the function d(u) is the determinant of the L-operator (2.25).3.3QuantizationThe separation of variables has a direct quantum counterpart [11, 20].To pass toquantum mechanics we change the variables πi, µi to operators and the Poisson brackets(3.13) to the commutators[µj, µk] = [πj, πk] = 0,[πj, µk] = −iδjk.

(3.22)Suppose that the common spectrum of µi is simple and the momenta πi are realizedas the derivatives πj = −i ∂∂µj . The separation equations (3.21) become the operatorequations, where the noncommuting operators are assumed to be ordered precisely inthe order as those listed in (3.14), that is πi, µi, H(1)N , .

. .

, H(K)N . Let Ψ(µ1, .

. .

, µK) bea common eigenfunction of the quantum integrals of motion:H(i)N Ψ = λiΨ,i = 1, . .

. , K.(3.23)Then the operator separation equations lead to the set of differential equationsΦj(−i ∂∂µj, µj, H(1)N , .

. .

, H(K)N )Ψ(µ1, . .

. , µK) = 0,j = 1, .

. .

, K,(3.24)which allows the separation of variablesΨ(µ1, . .

. , µK) =KYj=1ψj(µj).

(3.25)13

The original multidimensional spectral problem is therefore reduced to the set of one-dimensional multiparametric spectral problems which have the following form in thecontext of the problems under consideration d2du2 + εuN +nXi=1λiu −ei!ψj(u; λ1, . .

. , λn) = 0,forε = 0, 1,(3.26) d2du2 + 16uN−2(u + B)2 + 8λn+1 +nXi=1λiu −ei!ψj(u; λ1, .

. .

, λn+1) = 0,forε = 4(u −xn+1 + B),(3.27)with the spectral parameters λ1, . .

. , λn+1.

The problems (3.26),(3.27) must be solvedon the different intervals—“permitted zones”—for the variable u.4ConclusionWe remark that all systems considered yield the algebra which has general propertiesbeing independent on the type of the system.Therefore it would be interesting toconsider its Lie-algebraic origin within the general approach to the classical r-matrices[16].There exists an interesting link of the algebra studied here with the restricted flowformalism for the stationary flows of the coupled KdV (cKdV) equations [1].TheLax pairs which have been derived in the paper from the algebraic point of view wererecently found in [2] by consideration of the bi-Hamiltonian structure of cKdV.It seems to be interesting to study the same questions for the generalized hierarchyof differential operators of Gelfand-Dickey for which the corresponding L-operators haveto be the n × n matrices.AcknowledgementsThe authors are grateful to E K Sklyanin and A P Fordy for valuable discussions. Wealso would like to acknowledge the EC for funding under the Science programme SCI-0229-C89-100079/JU1.

One of us (JCE) is grateful to the NATO Special ProgrammePanel on Chaos, Order and Patterns for support for a collaborative programme, and tothe SERC for research funding under the Nonlinear System Initiative. VBK acknowl-edges support from the National Dutch Science Organization (NWO) under the Project#611-306-540.References[1] M Antonowicz and S Rauh-Wojciechowski.

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