Exchange operator formalism for an infinite family of solvable and integrable quantum systems on a plane

EX CHANGE OPERA TOR F ORMALISM F OR AN INFINITE F AMIL Y OF SOL V ABLE AND INTEGRABLE QUANTUM SYSTEMS ON A PLANE C. QUESNE Ph ysique Nucl ´ eaire T h´ e orique et Physique Ma th ´ ematique, Univ ersit´ e Libre de Bruxelles, Campus de la Plaine CP229, Boulev ard du T riomphe, B-1050 Brussels, Belgium cquesne@ulb.ac.b e Abstract The exc hange op erato r formalism in p olar co o rdinates, previously consid- ered f o r the Calogero-Ma rc hioro-W olfes problem, is generalized to a recen tly in tro duced, infinite family of exactl y solv able and in tegrable Hamiltonians H k , k = 1, 2, 3, . . . , on a plane. The elemen ts of the dihedr al group D 2 k are realized as op erators on this p lane and u sed to defin e some differenti al- difference op erators D r and D ϕ . The latter serv e to construct D 2 k -extended and in v ariant Ha miltonians H k , from whic h th e s ta r ting Ha miltonians H k can b e retrieve d by p ro jection in the D 2 k iden tity represent ation space. Running head: Exc hange Operato r F ormalism Keyw ords: Quantum Hamiltonians; in tegrability ; exc hange op erators P A CS Nos.: 03.65.F d 1 1 In tro d uction In a recen t w ork, a n infinite family of exactly solv able and in tegrable quantum Hamiltonians H k , k = 1, 2, 3, . . . , on a plane has b een intro duce d [1]. Such a family includes all previously kno wn Hamiltonians with the ab o v e prop erties, con t a ining rational p oten tials and allow ing separation of v ariables in p olar co ordinates. These corresp ond to the Smoro dinsky-Win ternitz (SW) system ( k = 1 ) [2, 3], the rational B C 2 mo del ( k = 2) [4, 5], and the Calogero-Marc hioro -W olfes (CMW ) mo del ( k = 3 ) [6, 7] (reducing in a sp ecial case to the three-particle Calog ero one [8]). F urthermore, it has b een conjectured (and prov ed for the first few cases) that all mem b ers of the family are also sup erin tegrable. In agreemen t with suc h a conjecture, all b ounded classical tra jectories hav e b een sho wn to b e closed and the classical motion to b e p eriodic [9]. Since the pioneering w ork of Olshanetsky and Perelomo v [4, 5] on the in tegrabil- it y of Calogero-Sutherland type N -bo dy mo dels, i.e. , the existenc e of N w ell- de fined, comm uting in tegrals of motion including the Hamiltonian, there ha v e b een sev eral studies of suc h a problem using v arious approache s (see, e.g. , Refs. [10, 11] for some recen t ones). One of the most in teresting metho ds is ba s ed on the use of some differen tial-difference op erators o r co v arian t deriv ativ es, kno wn in the mathematical literature as Dunkl op erators [1 2]. These op erators w ere indep enden tly redisco v- ered b y P olyc hrona k os [13] and Brink et al. [14] in the context of the N -b o dy Calogero mo del. Later o n, they w ere generalized to the CMW mo del [15] and, in suc h a context, an in teresting exc hange op erator formalism in p olar co ordinates w as in tro duced [16]. Since the CMW Hamiltonian is one of the mem b ers of the infinite family of Hamiltonians H k , k = 1, 2, 3, . . . , considered in Ref. [1], it is w o rth while to extend the latter f ormalism to the whole family and to study some of its consequences. This is the purp ose o f this letter. T o solv e the problem, w e shall hav e to distin- guish b et w een o dd and eve n k v alues and to prov e se v eral non trivial trigonometric iden tities. 2 2 Odd k Hamiltonians Let us consider the subfamily of Hamiltonians H k = − ∂ 2 r − 1 r ∂ r − 1 r 2 ∂ 2 ϕ + ω 2 r 2 + k 2 r 2 [ a ( a − 1) sec 2 k ϕ + b ( b − 1) csc 2 k ϕ ] , (2.1) corresp onding to k = 1, 3, 5, . . . . Here ω , a , b are three parameters suc h t ha t ω > 0, a ( a − 1) > − 1 / (4 k 2 ), b ( b − 1) > − 1 / (4 k 2 ), and the configuration space is g iv en b y the sector 0 ≤ r < ∞ , 0 ≤ ϕ ≤ π / (2 k ). In cartesian co ordinates x = r cos ϕ , y = r sin ϕ , the Hamiltonian (2.1 ) can b e rewritten as H 1 = − ∂ 2 x − ∂ 2 y + ω 2 ( x 2 + y 2 ) + a ( a − 1) x 2 + b ( b − 1 y 2 , (2.2) H 3 = − ∂ 2 x − ∂ 2 y + ω 2 ( x 2 + y 2 ) + 9( x 2 + y 2 ) 2  a ( a − 1 x 2 (3 y 2 − x 2 ) 2 + b ( b − 1) y 2 (3 x 2 − y 2 ) 2  , (2.3) and more and more complicated expressions as k is increasing. H 1 is kno wn as t he SW Hamiltonian [2, 3], while H 3 is the relativ e motio n Hamiltonian in the CMW problem, as sho wn below. 2.1 The k = 3 case The three-par t icle Hamiltonian of t he CMW problem is giv en b y [7] H CMW = 3 X i =1 ( − ∂ 2 i + ω 2 x 2 i ) + 2 a ( a − 1)  1 x 2 12 + 1 x 2 23 + 1 x 2 31  + 6 b ( b − 1)  1 y 2 12 + 1 y 2 23 + 1 y 2 31  , (2.4) where x i , i = 1, 2, 3, denote the particle co ordinates, x ij = x i − x j , i 6 = j , and y ij = x i + x j − 2 x k , i 6 = j 6 = k 6 = i . The range of the particle co ordinates is appropriately restricted as explained in Ref. [15]. I n terms of the v ariables x = x 12 / √ 2, y = y 12 / √ 6, and X = ( x 1 + x 2 + x 3 ) / √ 3, the Hamiltonian can b e separated in to a cen tre-of-mass Hamiltonian H cm = − ∂ 2 X + ω 2 X 2 and a relative one H rel , coinciding with H 3 giv en in (2.3). 3 The CMW Hamiltonian (2.4) is know n [4, 5] to b e related to the G 2 Lie algebra, whose W eyl group is the dihedral gro up D 6 . The 12 operato rs of the la t ter can be realized either in terms o f the particle p erm uta t io n op erators K ij and the inv ersion op erator I r in relativ e co ordinate space [15] or in t erms of t he r o tation op erator R = exp  1 3 π ∂ ϕ  through angle π / 3 in the pla ne ( r , ϕ ) and the op erator I = exp(i π ϕ∂ ϕ ) c hanging ϕ into − ϕ [16]. These exch ange op erators can then b e used to extend the partial deriv a tiv es ∂ i or ∂ r , ∂ ϕ in to differential-difference op erators D i or D r , D ϕ , resp ec tiv ely . In terms o f the former D i = ∂ i − a X j 6 = i 1 x ij K ij − b X j 6 = i 1 y ij K ij − X j,k i 6 = j 6 = k 6 = i 1 y j k K j k ! I r , (2.5) the latter can be defined as D r = − r 2 3 X i sin  ϕ − i 2 π 3  D i , D ϕ = − r 2 3 r X i cos  ϕ − i 2 π 3  D i . (2.6) On using the corresp o nde nces K ij ↔ R 2 k + 3 I and K ij I r ↔ R 2 k I with ( ij k ) = (123), Eq. (2 .6) can b e r ewritten as D r = ∂ r − 1 r ( a R + b )(1 + R 2 + R 4 ) I , (2.7) D ϕ = ∂ ϕ + a  tan ϕ R 3 + tan  ϕ + π 3  R 5 + tan  ϕ + 2 π 3  R  I − b  cot ϕ + cot  ϕ + π 3  R 2 + cot  ϕ + 2 π 3  R 4  I . (2.8) Note that the small discrepancies existing b et w een Eqs. (2.6) – (2.8) and the corre- sp onding expressions (13) and (15) of Ref. [16] come from the fact that here w e use the conv en tional definition of p olar co ordinates, while our former work was based on W olf es’ definition [7]. W e can no w build on this exc hange op erator f ormalism in p olar co ordinates to go further and extend the Hamiltonian (2.4 ) itself. As a first step, w e not e that from t he c hara cte ristic relations of D 6 , R 6 = I 2 = 1 , I R = R 5 I , R † = R 5 , I † = I , (2.9) 4 it is easy to pro v e that D r and D ϕ satisfy the equations D † r = − D r − 1 r [1 + 2( a R + b )(1 + R 2 + R 4 ) I ] , R D r = D r R , I D r = D r I , (2.10) D † ϕ = − D ϕ , R D ϕ = D ϕ R , I D ϕ = − D ϕ I . (2.11) It ma y b e observ ed that the first r elat io n in (2.1 1 ) is similar to that fulfilled b y the partial deriv ativ e op erator ∂ ϕ . Ho w ev er, the first relation in (2.10) differs from the corresp onding result fo r ∂ r , namely ∂ † r = − ∂ r − 1 r . F urthermore, in con trast with ∂ r and ∂ ϕ whic h commu te with one a no ther, D r and D ϕ satisfy the commutation relation [ D r , D ϕ ] = − 2 r ( a R + b )(1 + R 2 + R 4 ) I D ϕ . (2.12) The next stage consists in expressing D 2 ϕ in terms of ϕ , ∂ ϕ , R , and I . This can be done using Eqs. (2.9) and (2.11 ), as w ell as some well-kno wn trigonometric iden tities. The res ult reads D 2 ϕ = ∂ 2 ϕ − h sec 2 ϕ a ( a − R 3 I ) + sec 2  ϕ + π 3  a ( a − R 5 I ) + sec 2  ϕ + 2 π 3  a ( a − RI ) i − h csc 2 ϕ b ( b − I ) + csc 2  ϕ + π 3  b ( b − R 2 I ) + csc 2  ϕ + 2 π 3  b ( b − R 4 I ) i + 3( a 2 + b 2 + 2 ab R )(1 + R 2 + R 4 ) . (2.13) Finally , w e may intro duc e some generalized CMW Hamiltonian, defined by H CMW = H cm + H rel , (2.14) where H rel = H 3 = − ∂ 2 r − 1 r ∂ r − 1 r 2 [ D 2 ϕ − 3( a 2 + b 2 + 2 ab R )(1 + R 2 + R 4 )] + ω 2 r 2 = − D 2 r − 1 r [1 + 2( a R + b )(1 + R 2 + R 4 ) I ] D r − 1 r 2 D 2 ϕ + ω 2 r 2 . (2.15) Suc h a D 6 -extended Hamiltonian is endow ed with t w o in teresting prop erties: (i) it is left in v arian t under D 6 and (ii) its pro jection in the represen tation space of the D 6 iden tity represen tatio n, obtained b y replacing b oth R and I b y 1, g ives bac k the starting CMW Hamiltonian. 5 2.2 Generalization to other o dd k v alues W e no w plan to sho w that the formalism dev elop ed for k = 3 in Sec. 2.1 can b e extended to an y other o dd k v alue (including k = 1 ). F or suc h a purp ose, let us in tro duce t he tw o o p erators R = exp  1 k π ∂ ϕ  and I = exp(i π ϕ∂ ϕ ), satisfying the defining relations R 2 k = I 2 = 1 , I R = R 2 k − 1 I , R † = R 2 k − 1 , I † = I (2.16) of the dihedral group D 2 k , whose elemen ts may b e realized as R i and R i I , i = 0, 1, . . . , 2 k − 1. It is then straightforw ard to show t ha t the differen tial-difference op erators D r = ∂ r − 1 r ( a R + b ) k − 1 X i =0 R 2 i ! I , (2.17) D ϕ = ∂ ϕ + a k − 1 X i =0 tan  ϕ + i π k  R k +2 i I − b k − 1 X i =0 cot  ϕ + i π k  R 2 i I (2.18) still fulfil Eq. (2.11), while Eqs . (2.10) and (2.1 2) a r e g en eralized in to D † r = − D r − 1 r " 1 + 2( a R + b ) k − 1 X i =0 R 2 i ! I # , R D r = D r R , I D r = D r I (2.19) and [ D r , D ϕ ] = − 2 r ( a R + b ) k − 1 X i =0 R 2 i ! I D ϕ , (2.20) resp ec tiv ely . The extension of Eq. ( 2 .13), how ev er, turns out to b e more tric ky , b ecause in the calculation there app ear some inv olv ed sums o f trigonometric functions. T he simplest ones, k − 1 X i =0 sec 2  ϕ + i π k  = k 2 sec 2 k ϕ, k − 1 X i =0 csc 2  ϕ + i π k  = k 2 csc 2 k ϕ, (2.21) ha ve b een prov ed (under a sligh tly differen t form) in Ref. [17] b y using some ele- gan t metho d. Inspired b y this type o f approach, w e hav e demonstrated the three 6 additional iden tities k − 1 X i =0 tan h ϕ + ( i + j ) π k i tan h ϕ + ( i + 2 j ) π k i = − k , j = 1 , 2 , . . . , k − 1 , (2.22) k − 1 X i =0 cot h ϕ + ( i + j ) π k i cot h ϕ + ( i + 2 j ) π k i = − k , j = 1 , 2 , . . . , k − 1 , (2.23) k − 1 X i =0 n tan h ϕ + ( i + j ) π k i cot h ϕ + ( i + 2 j ) π k i + cot h ϕ + ( i + j ) π k i tan h ϕ + ( i + 2 j ) π k io = 2 k , j = 1 , 2 , . . . , k − 1 . (2.24) It is worth stressing tha t in Eqs. (2.21) – (2 .2 4 ), k is restricted to o dd v alues. As it has b een sho wn in R ef. [17 ], the counte rpart of the first relation in Eq. (2.21), for instance, lo oks en tir ely differen t for ev en k . On ta king adv antage of suc h results, we ar riv e at the equation D 2 ϕ = ∂ 2 ϕ − k − 1 X i =0 sec 2  ϕ + i π k  a ( a − R k +2 i I ) − k − 1 X i =0 csc 2  ϕ + i π k  b ( b − R 2 i I ) + k ( a 2 + b 2 + 2 ab R ) k − 1 X i =0 R 2 i , (2.25) from which we can build a D 2 k -extended Hamiltonian H k = − ∂ 2 r − 1 r ∂ r − 1 r 2 " D 2 ϕ − k ( a 2 + b 2 + 2 ab R ) k − 1 X i =0 R 2 i # + ω 2 r 2 = − D 2 r − 1 r " 1 + 2( a R + b ) k − 1 X i =0 R 2 i ! I # D r − 1 r 2 D 2 ϕ + ω 2 r 2 , (2.26) left in v ariant under D 2 k and giving back H k b y pro jection in the iden tit y represen- tation space. 3 Ev en k Hamiltonians Considering next the subfamily of Hamilto nia ns (2.1) for k = 2, 4, 6, . . . , w e note that the only mem b er kno wn in the literature b efore the work of Ref. [1] was the 7 B C 2 Hamiltonian H 2 , whose expression in cartesian co ordinates reads [5] H 2 = − ∂ 2 x − ∂ 2 y + ω 2 ( x 2 + y 2 ) + 2 a ( a − 1)  1 ( x − y ) 2 + 1 ( x + y ) 2  + b ( b − 1)  1 x 2 + 1 y 2  . (3.1) W e shall therefore start our study of the ev en k case b y reviewing this example. 3.1 The k = 2 case F or the B C 2 mo del, the W eyl gro up is the dihedral group D 4 , whose eight op erators can b e realized either in terms of the op erator K in terc hang ing x with y and the reflection op erators I x : x → − x , I y : y → − y , or in terms of the rotation o perator R = exp  π 2 ∂ ϕ  through angle π / 2 and t he op erator I = exp(i π ϕ∂ ϕ ) changing ϕ in to − ϕ . F rom the former, we can construct differen tial-difference o p erators in cartesian co ordinates D x = ∂ x − a  1 x − y + 1 x + y I x I y  K − b x I x , (3.2) D y = ∂ y + a  1 x − y − 1 x + y I x I y  K − b y I y . (3.3) By pro ceeding as in the CMW mo del [16], w e can then intro duc e corresp onding op erators in p olar co ordinates D r = cos ϕD x + sin ϕD y , D ϕ = r ( − sin ϕD x + cos ϕD y ) . (3.4) On ta king a dv antage of the corr esp ondences K ↔ R 3 I , I x ↔ R 2 I , I y ↔ I , I x I y K ↔ RI , suc h new op erators can b e rewritten as D r = ∂ r − 1 r ( a R + b )(1 + R 2 ) I , (3.5) D ϕ = ∂ ϕ + a [(tan 2 ϕ + sec 2 ϕ ) R 2 + tan 2 ϕ − sec 2 ϕ ] RI + b (tan ϕ R 2 − cot ϕ ) I , (3.6) in terms of r , ϕ , ∂ r , ∂ ϕ , R , and I . 8 Here R and I satisfy Eq. (2 .16) with k = 2, wh ile D r and D ϕ fulfil Eqs. (2.11), (2.19), a nd (2.20) with k = 2 in the last t w o ones. F urt hermore, it can b e easily pro ved that D 2 ϕ = ∂ 2 ϕ − 2  1 (cos ϕ − sin ϕ ) 2 a ( a − R 3 I ) + 1 (cos ϕ + sin ϕ ) 2 a ( a − RI )  −  1 cos 2 ϕ b ( b − R 2 I ) + 1 sin 2 ϕ b ( b − I )  + 2( a 2 + b 2 + 2 ab R )(1 + R 2 ) . (3.7) Hence the B C 2 Hamiltonian (3.1) can b e extended in to a D 4 -in v ariant generalized Hamiltonian H 2 = − ∂ 2 r − 1 r ∂ r − 1 r 2 [ D 2 ϕ − 2( a 2 + b 2 + 2 ab R )(1 + R 2 )] + ω 2 r 2 = − D 2 r − 1 r [1 + 2( a R + b )(1 + R 2 ) I ] D r − 1 r 2 D 2 ϕ + ω 2 r 2 (3.8) and realizes t he latter in the D 4 iden tity represen tat io n. Although this result is the exact counterpart o f tho s e obtained for o dd k v alues, it has b een obtained at t he price of mo difying the expression of D ϕ in terms o f ϕ , R , and I . F r o m Eq. (3.6), it is indeed o b vious that the term propo rtional to a cannot b e written in a form similar to tha t of the corresp onding one in (2.18), alb eit w e can re-express the term prop ortional to b as − b  cot ϕ + cot  ϕ + π 2  R 2  I , in agreemen t with (2.18). Suc h a discrepancy is related to the ab o v e- me n tioned dep endence of some tr igonometric iden t it ies on the par ity of k . 3.2 Generalization to other ev en k v alues The exte nsion of the formalism dev eloped for the B C 2 mo del to higher k v alues is not so straightforw ard as that carried out in the previous section for o dd k ones. As b efore w e start from a D 2 k group with elemen ts R and I satisfying Eq. (2.16) and w e define D r with prop erties (2.19) a s in Eq. (2.17). T o find a generalization of D ϕ in (3.6), w e are led by the condition that Eq. ( 2 .11) remain true. F or suc h a purp ose, it is conv enien t to in tro duce a linear com bination of p o w ers of R , S = ( k − 2) / 2 X i =0 R 4 i , (3.9) 9 whic h is suc h that RS = S R , R 4 S = S , S 2 = 1 2 k S , I S = S I , S † = S . (3.10) In terms o f it, w e can indeed write D ϕ = ∂ ϕ + a [(tan k ϕ + sec k ϕ ) R 2 k − 1 + (tan k ϕ − sec k ϕ ) R ] S I − b k − 1 X i =0 cot  ϕ + i π k  R 2 i I , (3.11) whic h reduces to (3.6 ) for k = 2 and satisfies Eqs. (2.1 1) and (2.20 ). On using the trigo no me tric iden tities 1  cos k 2 ϕ − sin k 2 ϕ  2 + 1  cos k 2 ϕ + sin k 2 ϕ  2 = 2 sec 2 k ϕ, (3.12) k − 1 X i =0 csc 2  ϕ + i π k  = k 2 csc 2 k ϕ, (3.13) k − 1 X i =0 cot  ϕ + i π k  = k cot k ϕ, (3.14) for ev en k , w e obtain the relat io n D 2 ϕ = ∂ 2 ϕ − k  1  cos k 2 ϕ − sin k 2 ϕ  2 S a ( a − R 2 k − 1 I ) + 1  cos k 2 ϕ + sin k 2 ϕ  2 S a ( a − RI )  − k − 1 X i =0 csc 2  ϕ + i π k  b ( b − R 2 i I ) + k ( a 2 + b 2 + 2 ab R ) k − 1 X i =0 R 2 i , (3.15) from which it follo ws that the ab o v e-men tioned connection betw een H k and the D 2 k -extended Hamiltonian H k , defined in (2.26), is also v alid for an y even k v alue. 4 Conclus ion Here w e ha ve generalized the exc hange op erator formalism in p olar co ordinates, pre- viously in tr oduced for the CMW mo del [16], to any mem b er H k , k = 1, 2, 3, . . . , of the infinite family of exactly solv able and integrable quan tum Hamiltonians on 10 a pla ne considered in Ref. [1] b y r ealizing the elemen ts of the dihedral group D 2 k as op erators on this plane. W e ha ve then constructed some differen tial-difference op erators D r and D ϕ , serving as building blo c ks for defining an infinite family of D 2 k -extended and inv ariant Hamiltonians H k , k = 1 , 2, 3, . . . . The start ing Hamil- tonians H k can b e recov ered by pro j ecting the corresp onding H k in the D 2 k iden tity represen tation. As a final p oin t , it is w orth observing that the in tegrabilit y o f H k , i.e. , the ex- istence of an in tegral of motion X k (see Eq. ( 2 3) of Ref. [1]), has an immediate coun terpart for H k since the commuting opera t or − D 2 ϕ giv es back X k − k 2 ( a + b ) 2 b y pro jection in the D 2 k iden tity represen ta tion. Whether the new formalism in tro- duced here may pro vide a framew ork for provin g t he superin t egr a bilit y conjecture of Ref. [1] remains an inte resting op en question for future in v estigation. 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