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The Lie algebraic structure of extended Sutherland models Kazuyuki Oshima Faculty of Education, Gifu Shotoku Gakuen University 6 0 2078 Takakuwa, Yanaizu-cho, Hashima-gun 0 Gifu 501-6194, Japan 2 e-mail: [email protected] n a J 7 Abstract ] We disclose the Lie algebraic structure of two extended Sutherland I models. Their Hamiltonians are BCN, and AN Sutherland Hamiltonians S with some additional terms. We show that both Hamiltonians can be . n written inthequadraticformsofgenerators oftheLiealgebragl(N+1). i l n Keywords : Quasi-exactly-solvability; Sutherland models; Lie algebra. [ PACS numbers : 02.20.Sv, 03.65.Fd. 4 v 2 4 1 Introduction 0 1 The quasi-exactly-solvable models have been developed considerably over last 1 decade. Tenyearsagothecrucialdefinitionofquasi-exact-solvabilitywasmade.1 4 0 The author defined the quasi-exact-solvability as follows. Let Pn be the space / of polynomials of degree ≤n. n i l Definition 1 Let us name a linear differential operator of the k-th order, T n k quasi-exactly-solvable, if it preserves the space P . Correspondingly, the oper- : n v ator E , which preserves the infinite flag P ⊂ P ⊂ P ⊂ ··· ⊂ P ⊂ ··· of k 0 1 2 n i X spaces of all polynomials, is named exactly-solvable. r a In particular, if operatorscan be representedby the elements of the enveloping algebra of a certain Lie algebras realized in terms of differential operators pre- serving the space P , they are quasi-exactly-solvable. Following this concept, n variousquasi-exactly-solvablemodelshavebeenproposed. Inreference,2theau- thorsobtainthesl -baseddeformationoftheCalogeromodels. A , BC , B , 2 N N N C , and D Calogero and Sutherland Hamiltonian, as well as thier supersym- N N metricgeneralizationsareshowntohaveexpressioninthequadraticpolynomials in the generators of the Lie algebra or the Lie superalgebra for the supersym- metriccase.3Moreoverthesl(N+1)deformationoftheCalogeromodels,whose Hamiltonians havethe quadraticandsextic self-interactionterms,areshownto be quasi-exact-solvable.4 Recentlythe quasi-exact-solvablemodelsareclassified 1 2 K. Oshima by using the new family of A -type Dunkl operators which includes the usual N Dunkl operator.5 The purpose of this paper is to obtain the explicit Lie algebraic form of the extended Sutherland Hamiltonians, following the method explained in the reference.3 We consider here two kinds of Hamiltonians. One of them is an extended Hamiltonian of the BC -Sutherland model, which has an additional N term: N g cos22x +2g cos2x . (1) 1 i 2 i i=1 X(cid:8) (cid:9) The other is an extended Hamiltonian of the A -Sutherland model, which has N an additional term: N {g cos4x +g cos2x +g sin4x +g sin2x }. (2) 1 i 2 i 3 i 4 i i=1 X These Hamiltonians are indicated in the reference.5 This paper is arranged as follows. In section 2, we consider an extended BC -Sutherland Hamiltonian. First we rotate it with the ground state eigen- N function as a gauge fuctor. Then we rewrite the gauge rotated Hamiltonian in terms of the elementary symmetric polynomials. Finally we arrive at the explicit form of the Hamiltonian in quadratic form of generators which generate the Lie algebra gl(N +1). In section 3, we show an extended A -Sutherland N Hamiltonian has Lie algebraic form as the same technique as in section 2. 2 Extended BC -Sutherland model N In this section we begin with the case of the extended BC -Sutherland Hamil- N tonian. 2.1 Hamiltonian The Hamiltonian for the Sutherland model of type BC is3 N N ∂2 1 1 H = − +g + ∂x2 sin2(x −x ) sin2(x +x ) i=1 i i6=j(cid:26) i j i j (cid:27) X X N N 1 1 +4g +g (3) 3 sin22x 4 sin2x i=1 i i=1 i X X where g, g and g are coupling constants. From the Hamiltonian (3) we can 3 4 deduce the Hamiltonian of the Sutherland models of type B , C , D as fol- N N N lows: g =0 : B , 3 N g =0 : C , 4 N g =g =0 : D . 3 4 N We will consider a Hamiltonian with the external potential5 N g cos22x +2g cos2x , (4) 1 i 2 i i=1 X(cid:8) (cid:9) The Lie algebraic structure of extended Sutherland models 3 that is, N ∂2 1 1 H = − +g + (5) ∂x2 sin2(x −x ) sin2(x +x ) i=1 i i6=j(cid:26) i j i j (cid:27) X X N 1 1 + g cos22x +2g cos2x +4g +g 1 i 2 i 3sin22x 4sin2x i=1(cid:26) i i(cid:27) X where g = ν(ν−1), g = −ν2, 1 1 g = ν {1+2n+2(N −1)ν+ν +ν } (n∈Z), 2 1 3 4 g = −ν (ν −1), 3 3 3 g = ν (ν −1)−ν (ν −1). 4 3 3 4 4 Let us call the Hamiltonian (5) the extended BC -Sutherland Hamiltonian. N 2.2 Gauge rotation First we make a gauge rotation of the Hamiltonian (5) with the ground state eigenfunction ν µ = sin(x −x )sin(x +x ) (6) i j i j   Yi<j  N ν3 N ν4 N × cosx sinx  exp −ν1 cos2x i i i 2 ( ) ( ) iY=1 iY=1 Yi=1 (cid:16) (cid:17) as gauge factor, H =µ−1Hµ. Using the relation 2 Ne cos(x −x ) cos(x +x ) i j i j + (7)  sin(x −x ) sin(x +x )  Xi=1 jX(6=i)(cid:18) i j i j (cid:19)  cos2(x −x ) cos2(x +x )  i j i j =2 + +constant, sin2(x −x ) sin2(x +x ) i<j(cid:18) i j i j (cid:19) X it is not difficult to obtain that N ∂2 N cos(x −x ) cos(x +x ) i j i j H = − −2 ν + (8) ∂x2 sin(x −x ) sin(x +x ) Xi=1 i Xi=1( jX(6=i)(cid:18) i j i j (cid:19) e cos2x 1 ∂ i +(ν +ν ) −(ν −ν ) +ν sin2x 3 4 sin2x 3 4 sin2x 1 i ∂x i i ) i N +4nν cos2x , 1 i i=1 X hereafter we will omit the constant terms. 4 K. Oshima 2.3 Lie algebraic form InordertodisclosetheLiealgebraicstructureofthegaugerotatedHamiltonian (8), it is expedient to rewrite the Hamiltonian H in terms of the elementary symmetric polynomials. Let e be the i-th elementary symmetric polynomials i of cos2x, e N e = cos2x , 1 i i=1 X e = cos2x cos2x , 2 i j i<j X e = cos2x cos2x cos2x , 3 i j k i<j<k X . . . e = cos2x cos2x ···cos2x . N 1 2 N Lemma 1 Inthesevariables {e ,e ,...,e }, thegaugerotatedHamiltonian H 1 2 N becomes e N H = −4 Ne e − (k−i)e e +(l−1+i)e e k−1 l−1 k−i l+i k−1−i l−1+i ( k,l=1 i≥0(cid:20) X X e ∂ ∂ −(k−2−i)e e −(l+1+i)e e k−2−i l+i k−1−i l+1+i ∂e ∂e (cid:21)) k l N +4 ν(N −k+2)(N −k+1)e k−2 ( k=1 X + νk(2N −k−1)+2ν k+k e 3 k (cid:20) (cid:21) ∂ +ν −e e +(N −k+1)e +(k+1)e 1 1 k k−1 k+1 ∂e (cid:20) (cid:21)) k +4nν e . (9) 1 1 Proof. After the change of variables x →e , the Hamiltonian (8) is i i N N ∂e ∂e ∂ ∂ N N ∂2e ∂ k l k H = − − (10) ∂x ∂x ∂e ∂e ∂x2 ∂e k,l=1i=1 i i k l k=1i=1 i k X X XX e N cos(x −x ) ∂e ∂e i j k k −ν − sin(x −x ) ∂x ∂x k=1i6=j ( i j (cid:18) i j(cid:19) XX cos(x +x ) ∂e ∂e ∂ i j k k + + sin(x +x ) ∂x ∂x ∂e i j (cid:18) i j(cid:19)) k N N cos2x ∂e ∂ i k −2(ν +ν ) 3 4 sin2x ∂x ∂e i i k k=1i=1 XX The Lie algebraic structure of extended Sutherland models 5 N N 1 ∂e ∂ k +2(ν −ν ) 3 4 sin2x ∂x ∂e i i k k=1i=1 XX N N N ∂e ∂ k −2ν sin2x +4nν cos2x . 1 i∂x ∂e 1 i i k k=1i=1 i=1 XX X As an example, let us demonstrate how to express N ∂e k X = sin2x (11) k i ∂x i i=1 X in terms of the elementary symmetric polynomials. Let us consider the generating function of X : k N X(t)= X tk. k k=0 X Note that the generating function of elementary symmetric polynomials e is k N ∞ (−1)n+1 E(t)= e tk =exp p (cos2x)tn , (12) k n n " # k=0 n=1 X X where p are the power sums: n N p (cos2x)= cosn2x . n i i=1 X Calculating the generating function X(t), we have N N ∂e X(t) = sin2x k tk (13) i ∂x " i# k=0 i=1 X X N N ∂e = sin2x ktk i ∂x i i=1 k=0 X X N ∂E(t) = sin2x i ∂x i i=1 X ∞ ∞ = 2 (−1)np (cos2x)tn− (−1)np (cos2x)tn E(t). n−1 n+1 ( ) n=1 n=1 X X It is not difficult to show that ∂ ∂ X(t)=2 p (cos2x)−Nt+t2 − E(t). (14) 1 ∂t ∂t (cid:18) (cid:19) Substituting E(t)= N e tk in (14), we find that k=0 k X =P2{e e −(N −k+1)e −(k+1)e }. (15) k 1 k k−1 k+1 One can show the other parts similarly. 2 6 K. Oshima NextwewillrealizetheLiealgebragl(N+1)intermsofelementarysymmet- ric polynomials. One of the simplest realization of gl(N +1) is the differential realization. The generators can be represented in the following form: ∂ J− = (i=1,2,...,N), (16) i ∂e i ∂ J0 = e (i,j =1,2,...,N), (17) ij i∂e j N ∂ J0 = n− e , (18) i ∂e i i=1 X J+ = e J0 (i=1,2,...,N), (19) i i where the parameter n is an integer. Ifnis anon-negativeinteger,the generatorsactontherepresentationspace of polynomials in N variables of the following type N P =span en1en2···enN 0≤ n ≤n . (20) n 1 2 N i ( (cid:12) ) (cid:12) Xi=1 (cid:12) (cid:12) With simple algebraic transformations(cid:12)from (9), we obtain the following result. Proposition 1 We obtain the Lie algebraic form of the gauge rotated extended BC -Sutherland Hamiltonians (9) as follows: N H = N −e4 NJk0−1,lJl0−1,k− (k−i) Jk0−i,lJl0+i,k−δi0Jk0,k ( k,l=1 i≥0(cid:20) X X (cid:0) (cid:1) +(l−1+i) J0 J0 −δ J0 k−1−i,l l−1+i,k i1 k−2,k (cid:0) (cid:1) −(k−2−i) J0 J0 −δ J0 −(l+1+i)J0 J0 k−2−i,l l+i,k i0 k−2,k k−1−i,l l+1+i,k (cid:21)) (cid:0) (cid:1) N +4 ν(N −k+2)(N −k+1)J0 +ν (N −k+1)J0 k−2,k 1 k−1,k ( k=1 X + νk(2N −k−1)+2ν k+k J0 +ν (k+1)J0 3 k,k 1 k+1,k ) (cid:20) (cid:21) +4ν J+. (21) 1 1 Here we define J0 =J− (i=1,2,...,N). 0i i 3 Extended A -Sutherland Hamiltonian N In this section we devote to a consideration of the extended A -Sutherland N Hamiltonian with using the same approachas in the previous section. The Lie algebraic structure of extended Sutherland models 7 3.1 Hamiltonian A second Hamiltonian we deal with is an extension of A -Sutherland model: N N ∂2 1 H =− +g . (22) ∂x2 sin2(x −x ) i=1 i i6=j i j X X Considering the external potential5 N 2 {g cos4x +2g cos2x +2g sin4x +2g sin2x }, (23) 1 i 2 i 3 i 4 i i=1 X we define a Hamiltonian N ∂2 1 H = − +g ∂x2 sin2(x −x ) i=1 i i6=j i j X X N +2 {g cos4x +2g cos2x +2g sin4x +2g sin2x } (24) 1 i 2 i 3 i 4 i i=1 X where coupling constants g,g ,g ,g ,g satisfy 1 2 3 4 g = ν(ν−1), g = (β−γ)(β+γ), 1 g = (γ+αβ+nγ+νγ(N −1)), 2 g = βγ, 3 g = (αγ−β−nβ−νβ(N −1)). 4 We call the Hamiltonian (24) the extended A -Sutherland Hamiltonian. N 3.2 Gauge rotation Similarlytowhatwasdoneinsection2fortheextendedBC -SutherlandHamil- N tonian, we first make a gauge rotation of (24) with the ground state eigenfunction N µ(x)= sinν(x −x ) cosnx exp[αx +βsin2x −γcos2x ] (25) i j i i i i i<j i=1 Y Y as a gauge fuctor H =µ−1Hµ. It is not difficult to obtain N ∂2 N cos(x −x ) sinx H = e− −2 ν i j −n i ∂x2 sin(x −x ) cosx i=1 i i=1( i6=j i j i X X X e ∂ +(α+2βcos2x +2γsin2x ) (26) i i ∂x ) i N −n(n−1) tan2x +2νn tanx tanx i i j i=1 i<j X X N +2n(α−2β) tanx . i i=1 X 8 K. Oshima 3.3 Lie algebraic form Next we rewritethe gaugerotatedHamiltonian (26) interms of the elementary symmetric polynomials of tanx. N e = tanx , 1 i i=1 X e = tanx tanx , 2 i j i<j X e = tanx tanx tanx , 3 i j k i<j<k X . . . e = tanx tanx ···tanx . N 1 2 N Numerous but straightforward calculations similar to what we made in the proof of lemma 1 leads to the following representation. N H = − Ne e −e e e −e e e +(e2−2e )e e k−1 l−1 1 k+1 l 1 k l+1 1 2 k l ( k,l=1 X e + (k−2−i)e e −(l−1+i)e e k−2−i l+i k−1−i l−1+i i≥0(cid:20) X +2(k−i)e e −2(l+1+i)e e k−i l+i k−1−i l+1+i ∂ ∂ +(k+2−i)e e −(l+3+i)e e k+2−i l+i k−1−i l+3+i ∂e ∂e (cid:21)) k l N +ν (N −k+2)(N −k+1)e k−2 ( k=1 X ∂ −2 k(N −k)+e e +(k+1)ke 2 k k+2 ∂e (cid:20) (cid:21) ) k N ∂ − 2 (k+2)e −e e +(k+e2−2e )e k+2 1 k+1 1 2 k ∂e k=1( (cid:20) (cid:21)) k X N ∂ +2n (k+2)e −e e +(k+e2−2e )e k+2 1 k+1 1 2 k ∂e ( ) k k=1 X N ∂ −2α N −(k−1) e +e e −(k+1)e k−1 1 k k+1 ∂e k=1((cid:20) (cid:21) ) k X N ∂ −4β N −(k−1) e −e e +(k+1)e k−1 1 k k+1 ∂e k=1((cid:20) (cid:21) ) k X N ∂ −4γ 2ke k ∂e k k=1 X −n(n−1)(e2−2e )+2νne +2n(α−2β)e . (27) 1 2 2 1 The Lie algebraic structure of extended Sutherland models 9 Now we use the same realization (16)-(19) of Lie algebra gl(N +1). With simple algebraic transformations from (27), we derive the following result. Proposition 2 We obtain the Lie algebraic form of the gauge rotated extended A -Sutherland Hamiltonian (27) as follows: N H = N −e NJk0−1,lJl0−1,k+ (k−2−i)(Jk0−2−i,lJl0+i,k−δi0Jk0−2,k) ( k,l=1 i≥0(cid:20) X X −(l−1+i)(J0 J0 −δ J0 ) k−1−i,l l−1+i,k i1 k−2,k +2(k−i)(J0 J0 −δ J0 )−2(l+1+i)J0 J0 k−i,l l+i,k i0 k,k k−1−i,l l+1+i,k +(k+2−i)(J0 J0 −δ J0 )−(l+3+i)J0 J0 k+2−i,l l+i,k i0 k+2,k k−1−i,l l+3+i,k (cid:21)) N + ν(N −k+1)(N −k+2)J0 − 2kν(N −k)+8γk J0 k−2,k k,k k=1( (cid:20) (cid:21) X −2(α+2β)(N −k+1)J0 k−1,k + νk(k+1)+2(n−1)(k+2) J0 (28) k+2,k (cid:20) (cid:21) ) N N −2 J0 J++2(n+ν+1)J++2(α−2β)J+−2 J0 J+−J+J+. k+1,k 1 2 1 k,k 2 1 1 k=1 k=1 X X Here we define J0 =J− (i=1,2,...,N). 0i i Acknowledgements I am thankful to Professor Hidetoshi Awata for his continuous encouragement and helpful comments. Note added. After submission of this paper, an article6 has been brought to the my attention. In that article, the complete classification of the quasi- exactly solvable models from the sl(M+1) generators are given. I am thankful to Professor Toshiaki Tanaka for informing his articles. References 1. A. Turbiner, Contemp.Math. Vol.160 263-310(1994). 2. A.Minzoni,M.RosenbaumandA.Turbiner,Mod.Phys.Lett.A111977-1984 (1996). 3. L. Brink, A. Turbiner and N. Wyllard, J.Math.Phys. 39 1285-1315(1998). 4. Xinrui Hou and M. Shifman, Int.J.Mod.Phys. A14 2993-3004(1999). 5. F. Finkel, D. Gómez-Ullate, A. González-Lopéz, M. A. Rodr´ıguez and R. Zhdanov, Comm.Math.Phys. 221 472-496 (2001). 10 K. Oshima 6. Toshiaki Tanaka, Annals Phys. 309 239-280 (2004).