Quantum integrable systems generalizing Calogero-Sutherland systems were
introduced by Olshanetsky and Perelomov (1977). Recently, it was proved that
for systems with trigonometric potential, the series in the product of two wave
functions is a deformation of the Clebsch-Gordan series. This yields recursion
relations for the wave functions of those systems. In this note, this approach
is used to compute the explicit expressions for the three-body
Calogero-Sutherland wave functions, which are the Jack polynomials. We
conjecture that similar results are also valid for the more general
two-parameters deformation introduced by Macdonald.Comment: 10 page

A M Perelomov

Calogero F

E Ragoucy

Harish-Chandra

Helgason S

Herz C

Jack H

James A

Koornwinder T

Lassalle M

Macdonald I G

Olshanetsky M A

Patera J

Perelomov A M

Ph Zaugg

Prati M-C

Sekiguchi J

Stanley R P

Sutherland B

Vilenkin N Ja

Vretare L

Weyl H

English

arXiv

Quantum integrable systems generalizing Calogero-Sutherland systems were introduced by Olshanetsky and Perelomov (1977). Recently, it was proved that for systems with trigonometric potential, the series in the product of two wave functions is a deformation of the Clebsch-Gordan series. This yields recursion relations for the wave functions of those systems. In this note, this approach is used to compute the explicit expressions for the three-body Calogero-Sutherland wave functions, which are the Jack polynomials. We conjecture that similar results are also valid for the more general two-parameters deformation introduced by Macdonald.Quantum integrable systems generalizing Calogero-Sutherland systems were introduced by Olshanetsky and Perelomov (1977). Recently, it was proved that for systems with trigonometric potential, the series in the product of two wave functions is a deformation of the Clebsch-Gordan series. This yields recursion relations for the wave functions of those systems. In this note, this approach is used to compute the explicit expressions for the three-body Calogero-Sutherland wave functions, which are the Jack polynomials. We conjecture that similar results are also valid for the more general two-parameters deformation introduced by Macdonald

Perelomov, A.M.

Ragoucy, E.

Zaugg, P.

CERN Document Server

arXiv:hep-th/9805149v1  22 May 1998February 12, 2014CERN-TH/98-141LAPTH-681/98Explicit solution of the quantumthree-body Calogero-Sutherland modelA.M. Perelomova,1, E. Ragoucyb,2, and Ph. ZauggaaLaboratoire de Physique The´orique LAPTH, Chemin de Bellevue, BP 110, 74941 Annecy-le-Vieux Cedex, FrancebTheory Division, CERN, 1211 Geneva 23, SwitzerlandAbstractThe class of quantum integrable systems associated with root systems was introduced in [OP 1977]as a generalization of the Calogero–Sutherland systems [Ca 1971, Su 1972]. In a recent note[Pe 1998], it was proved that for such systems with a potential v(q) = κ(κ − 1) sin−2 q, theseries in the product of two wave functions is the κ-deformation of the Clebsch–Gordan series.This yields recursion relations for the wave functions of those systems and, related to them, forgeneralized zonal spherical functions on symmetric spaces.In the present note, this approach is used to compute the explicit expressions for the three-bodyCalogero–Sutherland wave functions, which are the Jack polynomials. We conjecture that sim-ilar results are also valid for the more general two-parameters deformation ((q, t)-deformation)introduced by Macdonald [Ma 1988].CERN-TH/98-141May 981On leave of absence from Institute for Theoretical and Experimental Physics, 117259Moscow, Russia.Current e-mail address: perelomo@posta.unizar.es.2On leave of absence from Laboratoire de Physique The´orique LAPTH, 74941 Annecy-le-Vieux,France. E-mail address: ragoucy@lapp.in2p3.fr11 IntroductionThe class of quantum integrable systems associated with root systems was introducedin [OP 1977] (see also [OP 1978]) as generalization of the Calogero–Sutherland systems[Ca 1971, Su 1972]. Such systems depend on one real parameter κ (for the type An,Dn and E6, E7, E8), on two parameters (for the type Bn, Cn, F4 and G2) and on threeparameters for the type BCn. These parameters are related to the coupling constants ofthe quantum system.For special values of this parameter κ, the wave functions correspond to the charactersof the compact simple Lie groups (κ = 1) [We 1925/26], or to zonal spherical functionson symmetric spaces (κ = 1/2, 2, 4) [Ha 1958, He 1978]. At arbitrary values of κ, theyprovide an interpolation between these objects.This class has many remarkable properties. Let us only mention that the wave func-tions of such systems are a natural generalization of special functions (hypergeometricfunctions) to the case of several variables. The history of the problem and some resultscan be found in [OP 1983]. It was recently shown in [Pe 1998], that the product of twowave functions is a finite linear combination of analogous functions, namely of functionsthat appear in the corresponding Clebsch–Gordan series. In other words, this deforma-tion (κ-deformation) does not change the Clebsch–Gordan series. For rank 1, one obtainsthe well-known cases of the Legendre, Gegenbauer and Jacobi polynomials and the lim-iting cases of the Laguerre and Hermite polynomials (see for example [Vi 1968]). Someother cases were also considered in [He 1955, Ja 1970, Ko 1974, Ja 1975, Se 1977, Vr 1976,Vr 1984, Ma 1988, La 1989] and [St 1989]. In this note we use this property in order toobtain the explicit expressions for the Jack polynomials3 of type A2 which give the solu-tion of the three-body Calogero–Sutherland model. For special values of κ = 1/2, 2, 4 weobtain the explicit expressions for zonal polynomials of type A2.We conjecture that these results remain valid for the Macdonald polynomials of typeA2 [Ma 1988] and this will be the subject of a separate communication [PRZ 1998].2 General descriptionThe systems under consideration are described by the Hamiltonian (for more details see[OP 1983]):H =12p2 + U(q), p2 = (p, p) =l∑j=1p2j , (2.1)3We use the name of Jack polynomials, although, strictly speaking, they are slightly different fromthose introduced by Jack [Ja 1970]. Another possible denomination is generalized Gegenbauer polyno-mials [Ja 1975].2where p = (p1, ..., pl), pj = −i ∂∂qj , is a momentum vector operator, and q = (q1, ..., ql) is acoordinate vector in the l-dimensional vector space V ∼ Rl with standard scalar product(α, q). They are a generalization of the Calogero–Sutherland systems [Ca 1971, Su 1972]for which {α} = {ei − ej}, {ej} being a standard basis in V . The potential U(q) isconstructed by means of a certain system of vectors R+ = {α} in V (the so-called rootsystem):U =∑α∈R+g2α v(qα), qα = (α, q), g2α = κα(κα − 1). (2.2)The constants satisfy the condition gα = gβ, if (α, α) = (β, β). Such systems are com-pletely integrable for five types of potential [OP 1983]. In this note we consider only theA2 case with potential v(q) = sin−2 q.3 The Clebsch–Gordan seriesIn this section, we recall the main results of [Pe 1998] and specialize them to the A2 case.The Schro¨dinger equation for this quantum system with v(q) = sin−2 q has the formH Ψκ = E(κ)Ψκ; H = −∆2 + U(q1, q2, q3), ∆2 =3∑j=1∂2∂q2j(3.1)withU(q1, q2, q3) = κ(κ− 1)(sin−2(q1 − q2) + sin−2(q2 − q3) + sin−2(q3 − q1)). (3.2)The ground state wave function and its energy areΨκ0(q) =(3∏j<ksin(qj − qk))κ, E0(κ) = 8κ2. (3.3)Substituting Ψκλ = ΦκλΨκ0 in (3.1) we obtain−∆κ Φκλ = ελ(κ) Φκλ, ∆κ = ∆2 +∆κ1 , ελ(κ) = Eλ(κ)− E0(κ). (3.4)Here the operator ∆κ1 takes the form∆κ1 = κ3∑j<kcot(qj − qk)(∂∂qj− ∂∂qk). (3.5)It is easy to see that ∆κ leaves invariant the set of symmetric polynomials in variablesexp(2iqj). Such polynomials mλ are labelled by an SU(3) highest weight λ = mλ1 + nλ2,with m,n non-negative integers and λ1,2 the two fundamental weights. In general,Φκλ =∑P+∋µ≤λCµλ (κ)mµ, mµ =∑λ′∈W ·µe2i(q,λ′), (3.6)3where P+ denotes the cone of dominant weights, W the Weyl group, and Cµλ (κ) are someconstants, taking care of the wave function normalization.The most remarkable result of [Pe 1998] is that the product of two wave functions isa finite sum of wave functions (a sort of the κ-deformed Clebsch–Gordan series)Φκµ Φκλ =∑ν∈Dµ(λ)Cνµλ(κ) Φκν . (3.7)In this equation, Dµ(λ) = (Dµ + λ) ∩ P+, where Dµ is the weight diagram of the repre-sentation with highest weight µ.Since Φκµ are symmetric functions of exp(2iqj), it is convenient to work with a new setof variablesz1 = e2iq1 + e2iq2 + e2iq3 , z2 = e2i(q1+q2) + e2i(q2+q3) + e2i(q3+q1),z3 = e2i(q1+q2+q3).(3.8)As we are in the centre-of-mass frame (∑i pi = 0), the wave functions depend on twovariables only, which we choose to be z1 and z2 (it is consistent to set z3 = 1). In thesevariables, and up to a normalisation factor, ∆κ reads (∂i = ∂/∂zi):∆κ = (z21 − 3z2) ∂21 + (z22 − 3z1) ∂22 + (z1z2 − 9) ∂1∂2 + (3κ+ 1) (z1∂1 + z2∂2). (3.9)Its eigenvalues areεm,n(κ) = m2 + n2 +mn + 3κ(m+ n). (3.10)For the rest of this note, we will use a different normalization for the polynomials Φκλand denote them by P κm,n. In [Pe 1998] their general structure was outlinedP κm,n(z1, z2) =∑p,qCp,qm,n(κ) zp1 zq2 = zm1 zn2 + lower terms, (3.11)with p+ q ≤ m+ n and p− q ≡ m− n(mod 3). The first polynomials are easy to find:P κ0,0 = 1, Pκ1,0 = z1, Pκ0,1 = z2. (3.12)In [Pe 1998] simple instances of (3.7) for P κλ = Pκ1,0 or Pκ0,1 were givenz1 Pκm,n = Pκm+1,n + am,n(κ)Pκm,n−1 + cm(κ)Pκm−1,n+1, (3.13)z2 Pκm,n = Pκm,n+1 + a˜m,n(κ)Pκm−1,n + cn(κ)Pκm+1,n−1, (3.14)wheream,n(κ) = a˜n,m(κ) = cn(κ) cm+n+κ(κ), (3.15)cm(κ) =e(m)e(κ +m), e(m) =mm− 1 + κ. (3.16)In the next section, we will build the polynomials with the help of these recursion relations.44 ResultsAs a first step towards the complete solution, it is instructive to compute the simpler P κm,0polynomials, which were considered first by Jack [Ja 1970] (see also [Pr 1984]). Combiningthe recursion relations (3.13) and (3.14), we getP κm+1,0 = z1 Pκm,0 − cm z2 P κm−1,0 + dm P κm−2,0, (4.1)where dm = cm cm−1 cm−1+κ (for brevity we drop the κ dependence in cm). From thegeneral structure (3.11) of P κm,n, it is natural to decompose Pκm,0 asP κm,0 =[m3]∑l=0zm−3l1 Qκ,ml (y), y =z2z21, (4.2)Qκ,ml (y) being a polynomial in y. Then the recursion relation (4.1) implies that theseQκ,ml satisfyQκ,m+10 (y) = Qκ,m0 (y)− cm y Qκ,m−10 (y), (4.3)Qκ,m+1l (y) = Qκ,ml (y)− cm y Qκ,m−1l (y) + dmQκ,m−1l−1 (y). (4.4)The first relation involves only Qκ,ml with l = 0 and can be readily solved with the helpof the Gegenbauer polynomials Cκm(t) asQκ,m0 (y) =[m2]∑i=0(−1)ii!m!(m− 2i)!Γ(κ+m− i)Γ(κ +m)yi=ym/2e(m+ 1)−mCκm(12√y), (4.5)where 1/e(x + 1)−i denotes the product4 e(x) e(x − 1) · · · e(x − i + 1). For higher l, wetry the following ansatzQκ,ml (y) = αml Qκ+l,m−3l0 (y), (4.6)which solves (4.4), provided that the constants αml areαml =m!l! (m− 3l)!Γ(κ+m− 2l)Γ(κ+m). (4.7)Therefore, we conclude that the polynomials P κm,0(z1, z2) are just some particular linearcombinations of the one-variable Gegenbauer polynomials. One obtains the other set ofpolynomials P κ0,n using the relation Pκ0,n(z1, z2) = Pκn,0(z2, z1).4Similarly, for positive i, e(x)i = e(x) e(x + 1) · · · e(x + i − 1). This is a functional generalization ofthe Pochhammer symbol (x)i = Γ(x+ i)/Γ(x), i ∈ Z used later in the text.5The recursion relation (4.1) is also very useful to derive a generating function for theP κm,0 polynomials. Indeed, plugging in (4.1) the following functionGκ0(u) =∞∑m=0e(m+ 1)−m um P κm,0, (4.8)we obtain the first-order differential equation, easily solved byGκ0(u) = (1− z1u+ z2u2 − u3)−κ. (4.9)This generating function is perfectly suited to prove some basic properties of thesepolynomials, such as∂1Pκm,0 = mPκ+1m−1,0, ∂2Pκm,0 = −m(m− 1)κ+m− 1 Pκ+1m−2,0. (4.10)We will build the general polynomials with the help of the P κm,0, using the propertyP κm,0 Pκ0,n =min(m,n)∑i=0γim,n Pκm−i,n−i. (4.11)This is a consequence of Eq. (3.7), with the notable difference that the sum on the right-hand side is over a restricted domain (actually, it parallels exactly the SU(3) Clebsch–Gordan decomposition).For the proof, we proceed by iteration, assuming that (4.11) is valid up to (m,n).Then, with repeated use of (3.13) and (3.14), we getP κm,0 Pκ0,n+1 =min(m,n+1)∑i=0γim,n+1 Pκm−i,n+1−i + cn δim,n+1 Pκm+1−i,n−1−i, (4.12)where we definedγim,n+1 = γim,n + a˜m−i+1,n−i+1 γi−1m,n − cn cm−i+1 γi−1m,n−1, (4.13)δim,n+1 = c−1n cn−i γim,n − γim,n−1 − am−i+1,n−i γi−1m,n−1+ cn−1 cκ+n−1 γi−1m,n−2. (4.14)From the polynomials normalisation, we already know that γ0m,n = 1, and after a straigth-forward computation, the solution to (4.13) is found to beγim,n =e(2κ+m+ n+ 1− i)−ie(1)i e(m+ 1)−i e(n + 1)−i, (4.15)which implies that δim,n+1 = 0 in (4.14).6The constructive aspect of this formula lies in its inverted form.Theorem 1. The Jack polynomials P κm,n of type A2 are given by the formulaP κm,n =min(m,n)∑i=0βim,n Pκm−i,0 Pκ0,n−i, (4.16)where the constants areβim,n =(−1)ii! (κ + 1)−i3κ+m+ n− 2i3κ+m+ n− i(κ+m)−i (κ+ n)−i (2κ+m+ n)−i(m+ 1)−i (n + 1)−i (3κ+m+ n)−i. (4.17)Note that the βim,n are obtained using the relationβim,n = −i−1∑j=0βjm,n γi−jm−j,n−j. (4.18)From this theorem, we see that the construction of a general polynomial P κm,n is similar tothe construction of SU(3) representations from tensor products of the two fundamentalrepresentations.Likewise, one can explicitly study other types of decompositions, such asP κm,0 Pκn,0 =min(m,n)∑i=0γ˜im,n Pκm+n−2i,i, (4.19)with the coefficientsγ˜im,n =e(κ+m+ n + 1− i)−ie(1)i e(m+ 1)−i e(n+ 1)−i. (4.20)The proof is essentially the same as for (4.11). Here again, the summation on the right-hand side is on a restricted domain, compared to (3.7).Theorem 2. There is another formula for polynomials P κm,n at m ≥ n:γ˜nm+n,n Pκm,n =n∑i=0β˜imn Pκm+n+i,0 Pκn−i,0, (4.21)whereβ˜im,n =(−1)ii! (κ+ 1)−im+ 2im(κ+m+ n)i(m+ n + 1)i(m)i(κ+m+ 1)i(κ+ n)−i(n+ 1)−i. (4.22)This theorem is a simple consequence of (4.19), and the coefficients β˜im,n are foundusingβ˜im,n = −(γ˜n−im+n+i,n−i)−1 i−1∑j=0β˜jm,n γ˜n−im+n+j,n−j. (4.23)7As a by-product of (4.16), specializing it to the case κ = 1, where P κm,n are nothingbut the SU(3) characters, we getP 1m,n = P1m,0 P10,n − P 1m−1,0 P 10,n−1. (4.24)From this we easily deduce the generating function for SU(3) characters (see e.g. [PS 1978])G1(u, v) =∞∑m,n=0um vn P 1m,n =1− uv(1− z1 u+ z2 u2 − u3) (1− z2 v + z1 v2 − v3) . (4.25)5 ConclusionIn this letter we have solved the quantum three-body Calogero–Sutherland model exactly.The wave functions are known to be Jack polynomials, and our construction gives explicitexpansion of them. They appear to be constructed with Gegenbauer polynomials.Since the wave functions correspond, for special values of κ, to zonal spherical poly-nomials, we have obtained, as a by-product, explicit expression for zonal spherical func-tions of the symmetric spaces SU(3)/SO(3) (κ = 1/2), SU(3) × SU(3)/SU(3) (κ = 1),SU(6)/Sp(3) (κ = 2), and E6(−78)/F4 (κ = 4).Due to the algebraic framework, many aspects of this work can be applied to theN -body model, for instance Eq. (4.11) is easy to generalize to the SU(N) case.Let us also remark that preliminary investigations indicate that relations similar to(3.13) hold in the case of the Macdonald polynomials.Appendix: Explicit expressions for P κmn with m+ n ≤ 4In addition to those already given in the main text, we list here the first few polynomialsP κm,n with m+ n ≤ 4:P κ2,0 = z21 −2κ + 1z2P κ1,1 = z1z2 −32κ+ 1P κ3,0 = z31 −6κ + 2z1z2 +6(κ+ 1)(κ+ 2)P κ2,1 = z21z2 −2κ+ 1z22 −3κ+ 1(κ+ 1)2z1P κ4,0 = z41 −12κ + 3z21z2 +12(κ+ 2)(κ+ 3)z22 +24(κ+ 2)(κ+ 3)z1P κ3,1 = z31z2 −6κ+ 2z1z22 −3(3κ+ 2)(κ+ 2)(2κ+ 3)z21 +30(κ+ 2)(2κ+ 3)z28P κ2,2 = z21z22 −2κ+ 1(z31 + z32)−12(κ− 1)(κ+ 1)(2κ+ 3)z1z2 +9(κ− 1)(κ+ 1)2(2κ+ 3)AcknowledgementsOne of the authors (A.P.) would like to thank Prof. P. Sorba and the Laboratoire dePhysique The´orique LAPTH for hospitality.References[Ca 1971] Calogero F., J. Math. Phys. 12, 419–436[Ha 1958] Harish–Chandra; Amer. J. Math. 80, 241–310 and 553–613[He 1955] Herz C., Ann. of Math. 61, 474–523[He 1978] Helgason S., Differential Geometry, Lie Groups and Symmetric Spaces, Acad.Press: N.Y.[Ja 1970] Jack H., Proc. Roy. Soc. Edinb. A69, 1–18[Ja 1975] James A., pp 497–520 in: Theory and Application of Special Functions, Acad.Press: N.Y.[Ko 1974] Koornwinder T., Proc. Ned. Acad. Wet. A77, 48–66, 357–381[La 1989] Lassalle M., C.R. Acad. Sci., ser. I, 309, 941–944[Ma 1988] Macdonald I.G., Orthogonal polynomials associated with root systems,preprint[OP 1977] Olshanetsky M.A. and Perelomov A.M., Lett. Math. Phys. 2, 7–13[OP 1978] Olshanetsky M.A. and Perelomov A.M., Funct. Anal. Appl. 12, 121–128[OP 1983] Olshanetsky M.A. and Perelomov A.M., Phys. Rep. 94, 313–404[PRZ 1998] Perelomov A.M., Ragoucy E. and Zaugg Ph., in preparation[PS 1978] Patera J. and Sharp R.T., Proceedings of the VII International Colloquiumon Group Theoretical Methods in Physics, Austin, Texas.[Pe 1998] Perelomov A.M., J. Phys. A31, L31–L379[Pr 1984] Prati M.–C., Lett. Nuovo Cim. 41, 275[Se 1977] Sekiguchi J., Publ. RIMS Kyoto Univ. Suppl. 12, 455[St 1989] Stanley R.P., Adv. Math. 77, 76–115[Su 1972] Sutherland B., Phys. Rev. A4, 2019–2021[Vi 1968] Vilenkin N.Ja., Special Functions and the Theory of Group Representations,Amer. Math. Soc. Transl. Monographs 22, Providence, RI[Vr 1976] Vretare L., Math. Scand. 39, 343–358[Vr 1984] Vretare L., SIAM J. Math. Analysis 15, 805–833[We 1925/26] Weyl H., Math. Zs. 23, 271–309 (1925); 24, 328–376 and 377–395 (1926)10

Explicit solution of the quantum three-body Calogero-Sutherland model

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Explicit solution of the quantum three-body Calogero - Sutherland model

Explicit solution of the quantum three-body Calogero-Sutherland model

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