Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 4 (2008), 008, 21 pages Mo dels for Quadratic Algebras Asso ciated with Second Order Sup e rin tegrable Systems in 2D ⋆ Ernest G. KALNINS † , Wil lar d MILLE R Jr. ‡ and Sar ah PO ST ‡ † Dep artment of Mathematics, Unive rsity of Waikato, Hamilton, New Ze al and E-mail: math02 36@math.wa ikato.ac.nz URL: http://w ww.math. waikato. ac.nz ‡ Scho ol of Mathematics, University of Minnesota, Minne ap olis, Minnesota, 55455, USA E-mail: mil ler@ima.umn.e du , p ostx052@math.umn.e du URL: http://w ww.ima.u mn.edu/ ~ miller/ Received Octob er 25, 2007 , in f inal form Ja nuary 15, 200 8; Published online Ja nuary 18, 200 8 Original article is a v aila ble at h ttp:/ /www. emis. de/journals/SIGMA/2008/008/ Abstract. There are 13 equiv alence classe s of 2D second order quantum and c la ssical sup e rintegrable systems with nontrivial p otential, each asso c iated with a quadr atic a lgebra of hidden symmetries. W e s tudy the f inite and inf inite irreducible r e presentations o f the quantum quadr atic alg ebras thoug h the construction of mo dels in which the symmetries act on spaces of functions o f a single complex v ariable via either dif feren tial op er ators or dif ference operator s. In a nother paper w e ha v e already carried out parts of this analysis for the generic nondegenerate super int egrable system on the complex 2-sphere. Here w e carry it out for a degenerate sup er int egrable system on the 2-spher e. W e p oint out the connection betw een o ur re sults and a p osition dep endent mas s Hamiltonia n studied by Quesne. W e also show ho w to deriv e simple mo dels of the c la ssical quadr atic algebra s for sup erintegrable systems and then obtain the quantum models fr o m the classical mo dels, even though the classical and quantu m quadratic algebra s are distinct. Key wor ds: sup erintegrability; quadratic alge br as; Wilson po lynomials 2000 Mathematics Subje ct Classific ation: 20C99; 20C3 5; 22E 70 1 In tro duction A classica l (or quantum) m th o rder sup erinte grable system is a n in tegrable n -dimensional Hamil- tonian system w ith p otent ial that ad m its 2 n − 1 fun ctionally indep end en t co nstan ts of the motion, the maxim um p ossible, and suc h that the constant s of the motion are p olynomial of at most order m in the momenta . Suc h systems are of sp ecial signif icance in mathematical physic s b ecause the tra jectories of the classical motions can b e determined b y algebraic means alone, w hereas the quantum eigen v alues for the energy and the other symm etry op erators can also b e determined b y algebraic metho d s. In con trast to merely integrable systems, they can b e solv ed in multiple wa ys. The b est kno wn (and historically most imp ortan t) examples are the classical Kepler system an d the quan tum Coulomb (h ydrogen atom) system, as w ell as the isotropic oscillator. F or th ese examples m = 2 and the most complete classif icati on and struc- ture results are kno wn for the second order ca se. There is an exte nsiv e literature on the sub ject [1, 2, 3, 4 , 5, 6, 7 , 8, 9, 10 , 11, 12, 13, 14, 1 5, 16, 17, 18, 19, 20, 21, 22], with a r ecen t new burst of activit y [23 , 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. All suc h s ystems ha v e b een classif ied for real and complex Riemannian spaces with n = 2 and ⋆ This pap er is a contribution to the Pro ceedings of th e Seven th I nternational Conference “Symmetry in Nonlinear Mathematical Physics ” (June 24–30, 2007, Kyiv, U kraine). The full collection is av ailable at http://w ww.emis.de/j ournals/SIGMA/symmetry2007.h tml 2 E.G. Kalnins, W. Miller Jr. and S. Post their associated quadr atic algebras of symmetries computed [15, 29, 30, 24, 31]. F or non constant p oten tials there are 13 equiv alence classes of suc h stems (un der the St¨ ac k el transform b etw een manifolds), 7 with nond egenerate (3-parameter) p otentia ls and 6 wit h degenerate (1-parameter) p oten tials [44, 30]. The constan ts of the motion for eac h system generate a quadr atic algebra that closes at order 6 in the nond egenerate case and at order 4 in the degenerate case. The represen tatio n theory of suc h algebras is of great inte rest b ecause it is this quadratic al- gebra “hidden symmetry” that accoun ts for the degeneracies of the energy lev els of the quan tum systems and the abilit y to compute all asso ciated sp ectra of su c h sys tems by algebraic means alone. In prin ciple, all of these quad r atic alge bras c an b e obtained from the quad r atic alge bra o f a sin gle generic 3-parameter p otent ial on the complex tw o- sphere b y prescrib ed limit op erations and thr ough St¨ ac k el transforms. Ho wev er , these limiting op erations are not ye t suf f icien tly understo o d. Eac h equiv alence class has sp ecial pr op erties, and eac h of the 13 cases is worth y of s tu dy in its o wn right . A p o w erful tec hnique for carrying out this stu d y is the use of “one v ariable mo dels”. In the quantum case these are realiza tions of the quadratic algebra (on an energy e igenspace) in terms of dif feren tial or dif ference op erators a cting on a space o f functions of a single complex v ariable, and for w hic h the energy eigen v alue is constan t. Eac h mo del is adapted to the sp ectral decomp osition of one of the s ymmetry op erators, in p articular, one that is asso ciated w ith v ariable separation in the original quantum system. The p ossib le irre- ducible represen tatio ns can b e constr u cted on these spaces with function space inner or bilinear pro du cts (as appropriate) and intert wining op erators to map the represent ation space to the solution sp ace of the asso ciated quan tum system. (There ha v e b een several elegan t treatmen ts of th e r epresen tation theory of some quadratic algebras, e.g. [16, 17, 18, 5, 6, 19, 37]. Ho w ev er these ha v e almost alw a ys b een restricted to f inite dimensional and unitary representati ons and the qu estion of determining all one v ariable m o dels has not b een addr essed .) In th e classical case these are r ealizations of th e quadratic algebra (restricted to a constan t energy surface) b y functions of a single pair of canonical conjugate v ariables. In [39] we ha v e already carried out parts of this analysis for the generic nondegenerate sup er- in tegrable s y s tem on the complex 2-sphere. There the p oten tial was V = a 1 /s 2 1 + a 2 /s 2 2 + a 3 /s 2 3 where s 2 1 + s 2 2 + s 2 3 = 1, and the one v ariable qu antum mod el w as expressed in term s of dif ference op erators. It ga v e exactly the algebra that describ es the Wilson and Racah p olynomials in their full generalit y . In this pap er w e treat a sup erint egrable case with a degenerate p oten tial. Our example is again on the complex 2-sphere, but no w the p otent ial is V = α/s 2 3 . Though this p oten tial is a restriction of the generic p oten tial, the degenerate case admits a Killing v ector so the q u adratic algebra structur e c hanges d ramatically . The asso ciated quadratic algebra closes at lev el 4 and has a r ic her r epresen tation theory than the nond egenerate case. No w we f ind one v ariable mo dels for an irreducible representati on expr essed as either dif ference or dif ferent ial op erators, or sometimes b oth. W e sho w that this system can occur in u nob vious wa ys, such as in a p osition d ep endent mass Hamiltonian recen tly introd u ced b y Quesne [37]. The second p art of the pap er concerns mo d els of classical quadr atic algebras. Here we inau- gurate this study , in particular its relationship to quan tum m o dels of sup erint egrable systems. W e f irst describ e ho w these classical mo dels arise out of standard Hamilton–Jacobi th eory . In [44, 27] w e ha v e shown that for second o rder sup erintegrable systems in t wo dimensions there is a 1-1 relationship b et w een classical quadr atic algebras and qu an tum quadratic algebras, ev en though th ese algebras are not isomorphic. In this sense the quant um quadratic algebra, the sp ectral theory for its irreducible repr esen tatio ns and its p ossible one v ariable mo dels are already uniqu ely determined by the classical system. W e mak e this concrete by s h o wing ex- plicitly ho w the p ossible classica l mo d els of the classical sup erin tegrable system with p oten tial V = α/s 2 3 lead d irectly to the p ossible one v ariable dif ferent ial or dif ference op erator mo dels for the q u an tum qu adratic algebra. Then we rep eat this analysis for the nondegenerate p otent ial V = a 1 /s 2 1 + a 2 /s 2 2 + a 3 /s 2 3 where the quantum mo d el is essen tiall y the Racah algebra QR(3) Mo dels for Q u adratic Algebras 3 and its inf inite dimensional extension to d escrib e the Wilson p olynomials. Ou r results sh ow that th e Wilson p olynomial structur e is already im b edded in th e classical system with p otent ial V = a 1 /s 2 1 + a 2 /s 2 2 + a 3 /s 2 3 , even though this p oten tial admits no Lie symmetries. Th us the prop erties of the Wilson p olynomials in their full generalit y could ha v e b een deriv ed directly from classical m ec hanics! This w ork is part of a long term pr o ject to s tudy th e structure and repr esen tation theory for quadratic alge bras asso ciated with sup er integrable systems in n dimensions [23, 24, 25, 26, 27, 35 , 36]. Th e analysis for n = 3 d imensions will b e m uc h more challe nging, bu t also a goo d indication of b eha vior for general n . 2 The structure equations for S 3 Up to a St¨ ac k el transform, ev ery 2D second order s u p erintegrable system with nonconstant p o- ten tial is equiv alent to one of 13 systems [44]. There is a representa tiv e from eac h equiv alence class on either the complex 2-sphere or complex Eu clidean sp ace. In sev eral pap ers , in particu- lar [15], we h a v e classif ied all of the constant cu rv atur e su p erintegrable systems, and this p ap er fo cuses on t w o sys tems con tained in that list: S9 and S3. The quadr atic algebra of the generic nondegenerate system S9 was already treated in [39] and we w ill return to it again in this pap er. First w e study the quadratic algebra represen tation theory for the degenerate p oten tial S3. This one-parameter p oten tial 2-sph ere system corresp onds to the p oten tial V = α s 2 3 , where s 2 1 + s 2 2 + s 2 3 = 1 is the im b edd ing of the sphere in Euclidean space. The quan tum degenerate sup erinte grable s ystem is H = J 2 1 + J 2 2 + J 2 3 + V ( x, y ) = H 0 + V , where J 3 = s 1 ∂ s 2 − s 2 ∂ s 1 and J 2 , J 3 are obtained by cyclic p erm utations of th e in d ices 1 , 2 , 3. The basis symmetries are L 1 = J 2 1 + αs 2 2 s 2 3 , L 2 = 1 2 ( J 1 J 2 + J 2 J 1 ) − αs 1 s 2 s 2 3 , X = J 3 , H = J 2 1 + J 2 2 + J 2 3 + V , where J 3 = s 2 ∂ s 1 − s 1 ∂ s 2 plus cyclic p ermutations. Th ey generate a quadratic algebra that closes at order 4. Th e quadratic algebra relations are [ H, X ] = [ H , L j ] = 0 and [ L 1 , X ] = 2 L 2 , [ L 2 , X ] = − X 2 − 2 L 1 + H − α, (1) [ L 1 , L 2 ] = − ( L 1 X + X L 1 ) −  1 2 + 2 α  X. The Casimir relation is C ≡ 1 3  X 2 L 1 + X L 1 X + L 1 X 2  + L 2 1 + L 2 2 − H L 1 +  α + 11 12  X 2 − 1 6 H +  α − 2 3  L 1 − 5 α 6 = 0 . (2) W e kno w that the quan tum S c hr¨ odinger equation separates in spherical co ord inates, and that corresp ondin g to a f ixed ener gy eigen v alue H , the eige n v alues of X tak e the linear form λ n = A n + B , 4 E.G. Kalnins, W. Miller Jr. and S. Post where n is an in tege r, so we w ill lo ok for irreducible rep resen tations of the quadratic alg ebra suc h that the represen tation space h as a b asis of eigenv ectors f n with corresp onding eigen v alues λ n . (Indeed, from the analysis of [16] or of [6] the structure equations imply that th e s p ectrum of X m ust b e of th is form.) W e will use the abstract s tr ucture equations to list th e corresp onding represent ations and compute the action of L 1 and L 2 on an X basis. Thus, we assume that there is a b asis { f n } , for the r epresen tation space such that X f n = λ n f n , L 1 f n = X j C ( j, n ) f j , L 2 f n = X j D ( j, n ) f j . Here, A , B are n ot y et f ixed. W e do not imp ose any in ner pro duct sp ace structure. F rom these assumptions we can compute the action of L 1 and L 2 on the basis. Ind eed, [ L 1 , L 2 ] f n = X j,k ( C ( j, k ) D ( k , n ) − D ( j, k ) C ( k, n )) f j , (3) [ L 1 , X ] f n = X j ( λ n − λ j ) C ( j, n ) f j , [ L 2 , X ] f n = X j ( λ n − λ j ) D ( j, n ) f j . (4) On the other h and, f rom the equations (1) we h av e [ L 1 , X ] f n = 2 X j D ( j, n ) f j , (5) [ L 2 , X ] f n = − 2 X j C ( j, n ) f j + ( − λ 2 n + H − α ) f n , (6) [ L 1 , L 2 ] f n = − X j ( λ n + λ j ) C ( j, n ) f j −  1 2 + 2 α  λ n f n . (7) No w we equate equations (4) with (5) or (6). F or j = n , equating co ef f icien ts of f n in the resulting identitie s yields the conditions D ( n, n ) = 0 , C ( n, n ) = − λ 2 n + H − α 2 . Similarly , equating co ef f icien ts of f j in the case j 6 = n yields A ( n − j ) D ( j, n ) = − 2 C ( j, n ) , A ( n − j ) C ( j, n ) = 2 D ( j, n ) , or  A 2 ( n − j ) 2 + 4  C ( j, n ) = 0 , j 6 = n. Th us, either C ( j, n and D ( j, n ) v anish or A 2 ( n − j ) 2 = − 4. W e can scale A suc h that the smallest nonzero jum p is for j = n ± 1, in whic h case A = ± 2 i . By rep lacing n by − n if necessary , we can assume A = 2 i . (W e also set B = iµ .) Th us the only p ossible nonzero v alues of C ( j, n ), D ( j, n ) are for j = n, n ± 1 and there are the r elations D ( n + 1 , n ) = − iC ( n + 1 , n ) , D ( n − 1 , n ) = iC ( n − 1 , n ) . Comparing (3 ) and (7) and equating co ef f icien ts of f n ± 2 , f n ± 1 , resp ectiv ely , on b oth sides of the resulting iden tities, w e do n ot obtain new conditions. Ho w ev er, equating co ef f icien ts of f n results in the condition F n +1 − F n = 1 2 (2 n + µ )  4 n 2 + 4 µn + µ 2 + H + α + 1 2  , Mo dels for Q u adratic Algebras 5 where F n = C ( n, n − 1) C ( n − 1 , n ). The general s olution of this d if ference equ ation is F n = n 4 + (2 µ − 2 ) n 3 +  3 2 µ 2 − 3 µ + H 2 + α 2 + 5 3  n 2 +  µ 3 2 − 3 µ 2 2 +  H 2 + α 2 + 5 4  µ − H 2 − α 2 − 1 4  n + κ, where κ is an arb itrary constan t. T o determine κ w e su bstitute th ese results in to the Casimir equation (2) and set equal to zero the co ef f icien ts of f j in the expr ession C f n = 0. F or j 6 = n w e get nothing new. How ever, j = n we f ind κ = 1 16 µ 4 − 1 4 µ 3 + H + α + 5 / 2 8 µ 2 − 1 / 2 + H + α 4 µ + 1 8 H + α 8 + H 2 16 − H α 8 + α 2 16 . (8) Th us, F n = C ( n, n − 1) C ( n − 1 , n ) is an explicit 4th order p olynomial in n . By factoring this p olynomial in v arious wa ys, and r e-normalizing th e b asis v ectors f n appropriately via f n → c ( n ) f n , we can ac hiev e a realizat ion of the action of L 1 and L 2 suc h that L 1 f n = C ( n + 1 , n ) f n +1 + C ( n, n ) f n + C ( n − 1 , n ) f n − 1 , L 2 f n = D ( n + 1 , n ) f n +1 + D ( n, n ) f n + D ( n − 1 , n ) f n − 1 (9) and all of th e co ef f icien ts are p olynomials in n . Th e 4 ro ots of F n are 1 − µ 2 ± 1 4 q 2 − 4( H + α ) ± 2 p 1 − 4( H + α ) + 16 H α , so a con v enien t factorization is C ( n, n − 1) =  n + µ − 1 2  2 − 1 8  1 − 2( H + α ) + p 1 − 4( H + α ) + 16 H α  , C ( n − 1 , n ) =  n + µ − 1 2  2 − 1 8  1 − 2( H + α ) − p 1 − 4( H + α ) + 16 H α  . F rom these expressions and fr om D ( n, n ) = 0 , C ( n, n ) = (2 n + µ ) 2 + H − α 2 , D ( n ± 1 , n ) = ∓ iC ( n ± 1 , n ) w e see that we ca n f ind C , D co ef f icien ts in which the the dep end ence on n is alw a ys as a p olynomial. There are raising and lo w ering op erators A † = L 1 + iL 2 + 1 2 ( X 2 − H + α ) , A = L 1 − iL 2 + 1 2 ( X 2 − H + α ) . Indeed, A † f n = 2 C ( n + 1 , n ) f n +1 , Af n = 2 C ( n − 1 , n ) f n − 1 , and [ A, A † ] f n = 2( F n +1 − F n ) f n , so [ A, A † ] is a third order p olynomial in X . 6 E.G. Kalnins, W. Miller Jr. and S. Post T o get a one-v ariable mo del of the quadr atic algebra in terms of second order dif feren tial op erators, w e can simply make the c hoices f n ( t ) = t n , X = i (2 t d dt + µ ) and def ine L 1 from expressions (9) v ia the p r escription L 1 f n ( t ) =  tC  t d dt + 1 , t d dt  + C  t d dt , t d dt  + t − 1 C  t d dt − 1 , t d dt  f n ( t ) , (10) with a similar pro cedure for L 2 . In general the irreducible represent ations that we ha v e def ined are inf inite dimensional and the basis v ec tors f n o ccur for all p ositiv e and negativ e integ ers n . W e can obtain represen tations b ound ed b elo w, and with lo w est weig h t µ for − iX and corresp ondin g lo w est w eigh t v ector f 0 , simply b y requiring F 0 = 0, which amounts to setting κ = 0. F or con v enience we set α = 1 / 4 − a 2 . Then w e ha v e F n = C ( n, n − 1) C ( n − 1 , n ) = n ( n + µ − 1)( n + µ − 1 + a )( n − a ) . (11) Since κ = 0, (8), H must b e a solution of this qu ad r atic equation: H = − ( µ − 1 + a ) 2 + 1 4 . (12) A con v enien t c hoice is C ( n − 1 , n ) = n ( n + µ − 1 + a ) , C ( n + 1 , n ) = ( n + µ )( n + 1 − a ) , C ( n, n ) = 2 n 2 + 2 nµ − µ a + a + µ − 1 2 . If µ is not a negativ e inte ger then this b ou n ded b elo w represen tation is in f inite dimensional. Ho w ev er, if there is a highest weig h t vect or f m then w e m ust h a v e F m +1 = 0, or µ = − m , m = 0 , 1 , . . . . Th us the f in ite dimensional represent ations are indexed b y the nonnegativ e in teger m and the eigen v alues of − iX are m, m − 2 , . . . , − m . The d imension of th e r epresen tation sp ace is m + 1. A t this p oin t it is wo rth p ointi ng out that al l of the f inite dimensional, inf inite dimensional b ound ed b elow, and general inf inite dimensional irred ucible represen tations and mo dels of the quadratic algebra asso ciated with the su p erintegrable system S3 are of direct interest and ap- plicabilit y . A s im ilar argument w as made in [39] where we ga v e examples of v ario us analytic function expansions and distinct u n itary str uctures associated with one sup erin tegrable s y s - tem. The original S 3 qu an tum system is give n in term s of complex v ariables w ith no sp ecif ic inner pro duct str u cture imp osed. One could u se the r epresen tation theory results to describ e eigenfunction exp ansions simply in terms of analytic fu nctions. If one wan ts an inner p ro du ct structure or bilinear pro duct structure, it is merely necessary to imp ose the s tr ucture on a single eigenspace of H , and there are a v ariety of wa y s to do this. F or examp le, one could restrict the complex system to the real sph ere and imp ose the standard inner p ro duct for that case. Alternativ ely , one could restrict to the real hyp erb oloid of one sheet, or the real hyper b oloid of t w o sh eets. In all cases the mo d els of the irr ed ucible repr esen tatio ns are r elev an t, though not necessarily to one sp ecial case, suc h as th e r eal sphere with the stand ard inner pro du ct. While w e h av e no d irect p r o of that all mo dels of irreducible repr esen tations of quadratic algebras ob- tained in this w a y lead to representat ions f or some v ersion of the original sup erinte grable system, w e ha v e n o counterexamples. F or a deep er analysis we n eed to construct intert wining op erators that relate b asis f u nctions for the m o del with eigenfunctions of th e quantum Hamiltonian. Mo dels for Q u adratic Algebras 7 3 Dif feren tia l op erator mo dels A con v enien t realiz ation of the f inite dimensional represen tations b y dif ferent ial op erators in one complex v ariable is L 1 =  t 3 + 2 t 2 + t  d 2 dt 2 +  (2 − a − m ) t 2 + 2(1 − m ) t + a − m  d dt + m ( a − 1) t + a ( m + 1) − m − 1 2 , X = i (2 t d dt − m ) , L 2 = i  − t 3 + t  d 2 dt 2 + i  ( a + m − 2) t 2 + a − m  d dt − im ( a − 1) t. (13) This mo del is also corr ect for inf inite dimensional b oun ded-b elow representa tions, except that no w th e low est w eigh t is µ = − m where m 6 = 0 , 1 , 2 , . . . is a complex num b er. The r aising and lo w ering op erators for the mo del are A † = 2 t 3 d 2 dt 2 + 2(2 − a − m ) t 2 d dt + 2 m ( a − 1 ) t, A = 2 t d 2 dt 2 + 2( a − m ) d dt . In the f inite dimens ional case, for example, the eigen v alues of L 1 are χ n = a 2 − 1 4 −  n − a + 1 2  2 , n = 0 , 1 , . . . , m, (14) and the corresp on d ing unnorm alized eigenfunctions are (1 + t ) n 2 F 1  n − a n − m a − m ; − t  . No w, motiv ated by the quant um mec hanical system on the real 2-sphere, w e imp ose a Hilbert space structure on the irredu cible represent ations s u c h that L 1 and L 2 are self-adjoin t and X is sk ew adjoint : h L j f n , f n ′ i = h f n , L j f n ′ i , j = 1 , 2; h X f n , f n ′ i = −h f n , X f n ′ i . W riting φ n = k n f n where φ n has norm 1, we hav e the recursion relation k 2 n = ( n − 1 + µ )( n − a ) n ( n − 1 + µ + a ) k 2 n − 1 . F or inf inite dimensional b ounded below represen tati ons k 2 n m ust b e p ositiv e for all in te gers n ≥ 0, and w e normalize k 0 = 1. T h us k 2 n = ( µ ) n (1 − a ) n n !( a + µ ) n . F or f inite dimens ional repr esen tations we hav e µ = − m . Normalizing k 0 = 1, (p ossible for a < 1 or for a > m ), we f ind that an orthonormal basis in the one v ariable mo d el is given by φ n ( t ) = k n f n ( t ) = k n t n , n = 0 , 1 , . . . , m where k n = s ( − m ) n (1 − a ) n n !( − m + a ) n = s m !(1 − a ) n (1 − a ) m − n (1 − a ) m n !( m − n )! . Note the ref lection symm etry || f n || = || f m − n || . 8 E.G. Kalnins, W. Miller Jr. and S. Post T o deriv e a realization of the Hilb ert space for the dif ferent ial op erator m o dels of th e f inite di- mensional and inf inite dimensional b ound ed b elo w unitary repr esen tations in terms of a f unction space inner p ro duct h p, q i = K Z Z p ( t ) q ( t ) ρ ( tt ) dt dt , where p , q are p olynomials and K is a normalization constan t, w e us e the formal self- and sk ew-adjoin t requirements and obtain a dif ferent ial equation for the w eigh t fun ction: ( − ζ 2 + ζ ) d 2 ρ ( ζ ) dζ 2 + ( − µ − a + 1 + ( − 1 + µ − a ) ζ ) dρ ( ζ ) dζ + ( − 2 + µ − 2 a + aµ ) ρ ( ζ ) = 0 , where ζ = t t . The solution th at v anishes at ζ = 1 for a < 1 / 2 and is integrable at ζ = 0 for a + µ > − 1 is ρ 1 ( ζ ) = (1 − ζ ) 1 − 2 a 2 F 1  − µ +3 a − Q 2 + 1 , − µ +3 a + Q 2 + 1 2 − 2 a ; 1 − ζ  , where Q = p a 2 + (2 µ − 8 ) a + µ 2 + 4 µ − 8. (Note that the in tegral is an ev en fu nction of Q .) A t ζ = 0 this fu nction has a b ranc h p oin t with b ehavior ζ a + µ . W e write t = r e iθ , t = r e − iθ , ζ = r 2 and choose our conto urs of in tegrat ion for the inner pro duct as the un it circle | e iθ | = 1, i.e., 0 ≤ θ ≤ 2 π and, in the complex ζ -plane, a conto ur that starts at ζ = 1 and trav els jus t ab o v e the real ζ -axis to circle ζ = 0 once in the countercloc kwise direction and retur ns to ζ = 1 just b elo w the real ζ -axis. W e require that h 1 , 1 i = 1. By c hoosing a regime where a + µ > − 1 w e can sh r ink th e ζ -cont our ab out ζ = 0 so that the n orm tak es the form h 1 , 1 i = − 4 π K 1 e iπ ( a + µ ) sin[ π ( a + µ )] Z 1 0 ρ 1 ( ζ ) dζ = − π K 1 a − 1 e iπ ( a + µ ) sin[ π ( a + µ )] 2 F 1  − µ − 3 a + Q 2 + 1 − µ − 3 a − Q 2 + 1 3 − 2 a ; 1  = 4 π K 1 e iπ ( a + µ ) sin[ π ( a + µ )] Γ(2 − 2 a )Γ( a + µ + 1) Γ(2 − a − µ + Q 2 )Γ(2 − a − µ − Q 2 ) , where w e ha v e integrate d term-b y-term a nd then made use of Gauss’ Theorem for the sum m ation of 2 F 1 (1). Th is giv es us the v alue for K 1 suc h that h 1 , 1 i = 1. No w, the result extends for the original con tour by analytic con tin uation. T his def ines a pre-Hilb ert sp ace inner pr o duct that then can b e extended to obtain a true Hilb ert space. The cont our integ ral for the inner pro d uct obtained in the pr evious paragraph requir es Re a < 1 for con v ergence, and this do esn’t hold for s ome of the unitary irred u cible represent a- tions def ined ab ov e. Accordingly , w e consider a second solution of the w eigh t fun ction equatio n. The solution that v anishes at ζ = 0 for a + µ > 0 and is integrable at ζ = 1 for a < 1 is ρ 2 ( ζ ) = ζ µ + a 2 F 1  − µ +3 a − Q 2 , − µ +3 a + Q 2 µ + a + 1 ; ζ  . A t ζ = 1 this f unction has a branc h p oin t with b eha vior (1 − ζ ) 1 − 2 a . W e write t = r e iθ , t = r e − iθ , ζ = r 2 and c hoose our con tours of in tegrati on for th e in n er p ro duct as the unit circle | e iθ | = 1 and, in the complex ζ -plane, a con tour that starts at ζ = 0 and trav els ju st b elo w the real ζ -axis to circle ζ = 1 once in the count erclo c kwise dir ection and return s to ζ = 0 just ab ov e the real ζ -axis. This integ ral con v erges for Re ( a + µ ) > − 1. W e requir e that h 1 , 1 i = 1. By choosing a r egime wher e a < 1 we can sh rink the ζ -con tour ab out ζ = 1 so that the norm tak es the form h 1 , 1 i = 4 π K 2 e iπ (2 a − 1) sin[ π (2 a − 1)] Z 1 0 ρ 2 ( ζ ) dζ Mo dels for Q u adratic Algebras 9 = π K 2 µ + a + 1 e iπ (2 a − 1) sin[ π (2 a − 1)] 2 F 1  µ +3 a + Q 2 µ +3 a − Q 2 a + µ + 2 ; 1  = − 4 π K 2 e iπ (2 a − 1) sin[ π (2 a − 1)] Γ(2 − 2 a )Γ( a + µ + 1) Γ(2 − a − µ + Q 2 )Γ(2 − a − µ − Q 2 ) . This giv es us the v alue for K such that h 1 , 1 i = 1, and the result extends by analytic con tin u ation to all v alues of a , µ for whic h th e original con tour int egral conv erges. Th us we ha v e an explicit pr e-Hilb ert fu nction space in ner pro du ct for eac h of our dif ferential op erator mo d els. In the f inite dimensional case we ha v e the r ep ro ducing k ernel fu n ction δ ( t, s ) = m X n =0 φ n ( t ) φ n ( s ) = 2 F 1  − m 1 − a − m + a ; t s  . In the inf inite dim en sional b ound ed b elo w case w e ha v e the repr o ducing k ernel fun ction δ ( t, s ) = ∞ X n =0 φ n ( t ) φ n ( s ) = 2 F 1  µ 1 − a µ + a ; t s  whic h con v erges as an analytic function and in the Hilb ert space norm for | s | < 1. Here, || δ ( s, s ) || = 2 F 1  µ 1 − a µ + a ; | s | 2  . In eac h case h f ( t ) , δ ( t, s ) i = f ( s ) for f in the Hilb ert s pace. 4 Dif ference op erator mo dels There are also dif ference op erator mo dels for the rep r esen tations of the S 3 qu adratic algebra. W e f irst give the d etails for the f inite dimensional represent ations indexed by the nonn egativ e in teger m . Here the op erator L 1 is diagonalized: L 1 = − λ ( t ) + a − 1 2 , λ ( t ) = t ( t − 2 a + 1) , − iX = ( t − 2 a + 1)( t − m ) 2 t − 2 a + 1 T 1 − t ( t + m − 2 a + 1) 2 t − 2 a + 1 T − 1 , L 2 = ( t − a + 1)( t − 2 a + 1)( t − m ) 2 t − 2 a + 1 T 1 + t ( t − a )( t + m − 2 a + 1) 2 t − 2 a + 1 T − 1 , where T k is the dif ference op erator T k f ( t ) = f ( t + k ). The b asis fun ctions are f n ( t ) = ( − 1) n p n ( λ ) where p n ( λ ( t )) = 3 F 2  − n − t t − 2 a + 1 − m 1 − a ; 1  . Here f n is a p olynomial of order n in the v ariable λ ( t ), a sp ecial case of the family of d ual Hahn p olynomials [45, p. 346]. These p olynomials are o rthogonal with r esp ect to a measur e with supp ort at th e v alues t = 0 , 1 , . . . , m , in agreemen t with equation (14) for the eigen v alues of L 1 . Indeed, we hav e (for a < 1) m X t =0 (1 − 2 a ) t (3 / 2 − a ) t ( − m − 1) t ( − 1) t (1 / 2 − a ) t (2 + m − 2 a ) t t ! p n ( λ ( t )) p n ′ ( λ ( t )) = (2 − 2 a ) m ( a − m ) n n ! (1 − a ) m (1 − a ) n ( − m ) n δ nn ′ . 10 E.G. Kalnins, W. Miller Jr. and S. Post F or th e inf inite dim en sional, b ounded b elo w, case w e h a v e L 1 = t 2 + a 2 − 1 4 , − iX = (1 / 2 − a − it )( µ + a − 1 / 2 − it ) 2 t T i − (1 / 2 − a + it )( µ + a − 1 / 2 + it )) 2 t T − i , L 2 = − i (1 − 2 it )(1 / 2 − a − it )( µ + a − 1 / 2 − it ) 4 t T i − i (1 + 2 it )(1 / 2 − a + it )( µ + a − 1 / 2 + it )) 4 t T − i . (15) The basis fun ctions are f n ( t ) = ( − 1) n s n ( t 2 ) where s n ( t 2 ) = 3 F 2  − n 1 2 − a + it 1 2 − a − it µ 1 − a ; 1  . Here f n is a p olynomial of order n in the v ariable t 2 , a sp ecial case of the family of con tin uous dual Hahn p olynomials [45, p. 331]. T he orthogonalit y and n ormalization are giv en by 1 2 π Z ∞ 0     Γ(1 / 2 − a + it )Γ( µ + a − 1 / 2 + it )Γ(1 / 2 + it ) Γ(2 it )     2 s n ( t 2 ) s n ′ ( t 2 ) dt = Γ( n + µ )Γ( n + 1 − a )Γ( n + µ + a ) n ! ( µ ) 2 n | (1 − a ) n | 2 δ nn ′ , where µ > 1 / 2 − a > 0. In su mmary , w e h a v e foun d the follo wing p ossibilities for b ound ed b elo w ir r educible repre- sen tations such that L 1 , L 2 are self-adjoin t and X is ske w adjoint, together with asso ciated one v ariable mo dels. (Here, n 0 is a p ositiv e in teger.) represent ation parameter range mo del f inite dimensional µ = − m, m = 0 , 1 , 2 , . . . dif feren tial op erators either a < 1 or a + µ > 0 dif ference op erators inf . dim . b dd . b elo w µ > 0 dif feren tial op erators a < 1 and a + µ > 0 dif ference op erators inf , dim . b dd . b elow 0 > µ = − n 0 + t , t ∈ (0 , 1) dif f eren tial op erators a = n 0 + s , s ∈ (0 , 1) inf . dim . b dd . b elo w 0 > µ = − n 0 + t , t ∈ (0 , 1) dif f eren tial op erators − t < a < 1 − t 5 Quesne’s p osition d ep endent mass (PDM) system in a t w o-dimensio nal semi-inf inite la y er In [37] Quesne considered a sup erintegrable exactly solv able p osition dep endent mass (PDM) system in a t w o-dimen s ional semi-inf inite la y er. Her system is equiv al en t via a gauge transfor- mation to a standard qu antum mec hanical problem on the real 2-sph er e with p oten tial of the form S3. In d eed, in Qu esne’s pap er we are given th e Hamiltonian − H Q = cosh 2 q x ( ∂ 2 x + ∂ 2 y ) + 2 q cosh q x sinh q x∂ x + q 2 cosh 2 q x − q 2 k ( k − 1) sinh 2 q x . Mo dels for Q u adratic Algebras 11 W e adopt co ord inates on the u nit sphere as s 1 = sin q y cosh q x , s 2 = cos q y cosh q x , s 3 = tanh q x, where s 2 1 + s 2 2 + s 2 3 = 1 and the metric is ds 2 = q 2 ( dx 2 + dy 2 ) / cosh 2 q x . The Laplacian b ecomes ∆ S = cosh 2 q x q 2 ( ∂ 2 x + ∂ 2 y ) . In these co ord inates, the degenerate su p erinteg rable system S3 b ecomes H S = cosh 2 q x q 2 ( ∂ 2 x + ∂ 2 y ) + 1 4 − α 2 tanh 2 q x . By a gauge transf orm H O = (cosh q x ) − 1 H S cosh q x , w e get H O = 1 q 2 ( cosh 2 q x ( ∂ 2 x + ∂ 2 y ) + 2 q cos q x sinh q x∂ x + q 2 cosh 2 q x + q 2 1 4 − α 2 sinh 2 q x ) + 1 4 − α 2 . Th us w e ha v e H Q = − q 2 H 0 + q 2 (1 / 4 − α 2 ) = − q 2 (cosh q x ) − 1 H S cosh q x + q 2 (1 / 4 − α 2 ), with 1 / 4 − α 2 = − k ( k − 1) which has solutions k = a + 1 / 2 or k = − a + 1 / 2. Sin ce k is assumed p ositiv e and a is r equired to b e less than 1, w e take a < 0. Supp ose w e f ind an eigen v ector f or H S with eigen v alue λ S , call it v λ S . Then λ Q will b e th e eigen v alue of v λ Q for H Q . W e ha v e the transform ations v λ Q = v λS / cosh q x and λ Q = − q 2 λ S + q 2 (1 / 4 − α 2 ) = − q 2 λ S − q 2 k ( k − 1). Checking th e tw o eigen v alues, we ha v e λ S = − ( µ − 1+ a ) 2 + 1 / 4 and λ Q = q 2 ( N + 2) ( N + 2 k + 1). W e note that these t w o v al ues coincide wh en − µ = m = N + 1 with m an intege r. Using the ab o v e calculations and the eigenfunctions given in the pap er, we can obtain eigen- functions for the S3 case as v λ S = N ( k ) n,ℓ (tanh q x ) − a + 1 2 (cosh q x ) − ℓ − 1 P − a,ℓ +1 n ( − tanh 2 q x ) χ ℓ ( y ) , or in co ord in ates on the sphere v λ S = N ( k ) n,ℓ ( s 3 ) − a + 1 2 ( s 2 1 + s 2 2 ) ℓ +1 2 P − a,ℓ +1 n ( − s 2 3 ) χ ℓ ( y ) , where m = 2 n + ℓ + 1, and χ ℓ ( y ) = sin[( ℓ + 1) q y ] or cos [( ℓ + 1) qy ]. W e can rewrite these by noting 1 / cosh 2 q x = 1 − tanh 2 q x so that we can w rite sin q y = s 1 / p 1 − s 2 3 and cos q y = s 2 / p 1 − s 2 3 , then w e obtain χ ℓ ( y ) = a n T ℓ +1 s 2 p s 2 1 + s 2 2 ! + b n s 1 p s 2 1 + s 2 2 U ℓ s 2 p s 2 1 + s 2 2 ! , where T ℓ and U ℓ are th e Cheb yshev p olynomials of the f irst and second kind, resp ectiv ely . Quesne found the S3 quadr atic algebra (which closes at order 4) but did n ot use it f or sp ectral analysis purp oses b ecause her p roblem inv ol v ed a b ound ary condition th at brok e the full quadratic algebra symmetry . Instead she consid er ed her sys tem as a s p ecial case of S9 and u sed the more complicat ed S9 sym m etry algebra that closes at ord er 6 to f ind the f inite dimensional represent ations. (Note that although the 1-paramete r S3 p otent ial is a limit of the 3- parameter S9 p oten tial as t w o of the parameters go to 0, a discontin u it y o ccur s in the stru ctur e of the qu adratic a lgebra. A f irst order symmetry app ears and the num b er o f seco nd order symmetries j umps from 3 to 4.) Quesn e’s p oin t of view has merit, bu t it complicates the sp ectral 12 E.G. Kalnins, W. Miller Jr. and S. Post analysis of the p roblem, since the only one-v ariable mo del is in terms of d if feren ce op erators and Racah p olynomials. F rom our v an tage p oin t of one one v ariable dif ferential op erator analysis for the mo d el, Qu esne’s b oun dary conditions amoun t to decomp osing an irr educible subsp ace corresp ondin g to an m -dimensional representati on in to a d irect su m o f eve n and o d d parit y subspaces V + , V − . (Indeed her b ou n dary conditions r equire choic e of χ ℓ ( y ) in the cosine form for ℓ ev en and in the sine f orm for ℓ odd .) Let P b e the op er ator P f ( t ) = t m f (1 /t ) . Since P 2 = I and || f n || = || f m − n || , it is clear that P is u nitary . W e def ine unit vect ors Φ + ℓ = 2 − 1 / 2 ( φ ℓ + ( − 1) m φ m − ℓ ) and Φ − ℓ = 2 − 1 / 2 ( φ ℓ − ( − 1) m φ m − ℓ ) for ℓ = 0 , 1 , . . . , [ m/ 2]. T hen for m = 2 k the ve ctors Φ + ℓ , ℓ = 0 , . . . , k form an orthonorm al basis for V + m and the vect ors Φ − ℓ , ℓ = 0 , . . . , k − 1 form an on basis for V − m . F or m = 2 k − 1, the v ectors Φ + ℓ , ℓ = 0 , . . . , k − 1 form an ON basis for V + m and the v ecto rs Φ − ℓ , ℓ = 0 , . . . , k − 1 form an orthonormal basis for V − m . These basis v ectors are very easily exp ressible in term s of the one v ariable d if feren tial op erator mo del, where they are sums of tw o monomials. T he b asis u sed b y Quesne corresp onds to the V − subspaces.Thus our mo dels can b e used to carry out the sp ectral an alysis f or this PDM system, and they yield a simplif icatio n. 5.1 Classical mo dels for S3 No w we describ e ho w the metho ds of classical mec hanics lead dir ectly to the quant um mo d els. The classical system S3 on the 2-sph ere is determined by the Hamiltonian H = J 2 1 + J 2 2 + J 2 3 + α ( s 2 1 + s 2 2 + s 2 3 ) s 2 3 , where J 1 = s 2 p 3 − s 3 p 2 and J 2 , J 3 are cyclic p erm utations of this exp r ession. F or computational con v enience we hav e im b edded the 2-sphere in Euclidean 3-space. Th us w e use the Poi sson brac k et {F , G } = 3 X i =1 ( − ∂ s i F ∂ p i G + ∂ p i F ∂ s i G ) for our compu tations, bu t at the end we restrict to the sphere s 2 1 + s 2 2 + s 2 3 = 1. The classical basis for the constants of the motion is L 1 = J 2 1 + α s 2 2 s 2 3 , L 2 = J 1 J 2 − α s 1 s 2 s 2 3 , X = J 3 . The structure r elations are {X , L 1 } = − 2 L 2 , {X , L 2 } = 2 L 1 − H + X 2 + α, {L 1 , L 2 } = − 2( L 1 + α ) X , (16) and the Casimir relatio n is L 2 1 + L 2 2 − L 1 H + L 1 X 2 + α X 2 + α L 1 = 0 . (17) F rom the results of [46] we kno w that additive separation of v ariables in the Hamilton–Jacobi equation H = E is p ossible in sub group t yp e co ordin ates in which X , L 1 or S = 2( L 1 − i L 2 ) − H + X 2 , resp ectiv ely , are constan ts of separation. This corresp onds to t w o c hoices of spherical co ordinates and one of horosph er ical co ord inates, r esp ectiv ely . W e seek t w o v ariable mo d els for the P oisson b rac k et relations (16), (17). There is also separation in ellipsoidal co ordinates (i.e., non-subgroup type co ordin ates) bu t we will n ot m ake use of this here. The justif ication for th ese mo dels comes f r om Hamilton–Jacobi theory . The phase sp ace f or our p roblem is 4-dimensional. Thus it is p ossib le to f ind canonical v ariables H , I , Q , P suc h Mo dels for Q u adratic Algebras 13 that {I , H} = {P , Q} = 1 and all other P oisson br ac k ets v anish. In terms of H and the other canonical v ariables the P oisson br ac k et can b e expr essed as {F , G } = − ∂ H F ∂ I G + ∂ I F ∂ H G − ∂ Q F ∂ P G + ∂ P F ∂ Q G . (18) (As follo w s from standard theory [47] one can constru ct a set of suc h canonical v ariables f rom a complete in tegral of the Ha milton–Jacobi equatio n. Ou r 2D second order sup erintegrable systems are alwa ys multiseparable, and eac h s eparable solution of the Hamilton–Jaco bi equation pro vides a c omplete in tegral. Th us w e can f ind these canonica l v ariables in sev eral distinct w a ys.) No w w e restrict our atten tio n to the algebra of constan ts of the motion. This alge bra is generated b y H , L 1 , L 1 , X , sub ject to the relation (17). T h us, consider ed as fun ctions of the canonical v ariables, the constan ts of the motion are indep endent of H . If w e f urther restrict the system to the constan t energy space H = E then w e can consider H as non v arying and ev ery constant of the motion F can b e expressed in the form F ( E , Q , P ). This means th at the P oisson br ac k et of t w o constan ts of the motion, F , G can b e computed as {F , G } = − ∂ Q F ∂ P G + ∂ P F ∂ Q G . Th us all fu nctions d ep end on only t w o canonically conjugate v ariables Q , P and the parameter E . This sho ws the existence and the form of t w o v ariable mo d els of conjugate v aria bles. How ever the pro of is not constructiv e and, furth ermore, it is not un ique. Two obtain constru ctiv e r esults w e will use the strategy of setting Q equal to one of the constan ts of th e motion that corresp onds to separation of v ariables in some co ordinate system, and then u s e (18) for the Po isson br ac k et and r equire that relations (16), (17) h old. In order to m ake clear that w e are computing on the constan t energy hyp ersurface expressed in canonical v ariables w e will use a dif ferent notation. W e will set Q E = c , P E = β so, F ( H , Q , P ) = f ( c, β ), G ( H , Q , P ) = g ( c, β ), and {F , G } E = { f , g } = − ∂ c f ∂ β g + ∂ β f ∂ c g . F or our f irst mo del w e r equire X ≡ X E = c . Sub stituting this requirement and H = E into the structur e equ ations w e obtain the r esult I : L 1 = 1 2 ( E − c 2 − α ) + 1 2 p c 4 − 2 c 2 ( E + α ) + ( E − α ) 2 sin 2 β , (19) X = c, L 2 = 1 2 p c 4 − 2 c 2 ( E + α ) + ( E − α ) 2 cos 2 β . In this mo del, and in all other classical mo dels, β is not un iquely determin ed : we can replace it b y β ′ = β + k ( c ) for any function k ( c ) and the v ariables c and β ′ remain canonically conjugate. F or a second mo d el w e require L 1 ≡ ( L 1 ) E = c and pro ceed in a similar fashion. T h e result is I I : L 1 = c, L 2 = p c ( E − c − α ) sin(2 √ c + αβ ) , (20) X = r c ( E − c − α ) c + α cos(2 √ c + αβ ) . F or the third and last mo del we need to diagonalize the symmetry S = 2( L 1 − i L 2 ) − H + X 2 corresp ondin g to separation in horospherical co ordinates. F or this it is con v enien t to rewrite the structure equations (16) , (17) in terms of the new basis S , L 1 + i L 2 , X : {S , X , } = 2 i ( S + α ) , {S , L 1 + i L 2 } = − 2 i X ( S − 2 X 2 + 2 H + 3 α ) , {L 1 + i L 2 , X } = − i  X 2 + 2( L 1 + i L 2 ) − H + α  . The Casimir relation is − 2 S ( L 1 + i L 2 ) − S X 2 + X 4 + H S − 2 X 2 H + H 2 − α  2( L 1 + i L 2 ) + S + 3 X 2 + H  = 0 . 14 E.G. Kalnins, W. Miller Jr. and S. Post F or m o del I I I we s et S = c and obtain I I I : S = c, X = − 2 i ( c + α ) β , L 1 + iL 2 = 8( c + α ) 3 β 4 + 2( c + α )(3 α + c + 2 E ) β 2 − ( c + E )( α − E ) 2( c + α ) . 5.2 Classical mo del → quan tum mo del What ha v e we ac hiev ed with these classical mo dels? F or one thing they sho w u s h o w to paramete- rize the constan ts of the motion an d exhib it their fun ctional dep enden ce. More imp ortan t for our purp oses, they giv e us a rational means to deriv e the p ossible one-v ariable quan tum mo- dels. This ma y seem surp rising. Ho w can classical mec hanics determine quantum m echanics uniquely? Ho w can structures suc h as the Wilson family of orthogonal p olynomials, con taining the Hahn p olynomials, b e deriv ed d irectly from classical mechanics? The p oin t is that the struc- tures w e are studyin g are second order sup erin tegrable systems in 2D. In pap ers [30, 44, 27] it has b een sho wn that there is a 1-1 relationship b et w een the quan tum and classica l v ersions f or suc h systems, for all 2D Riemannian spaces. (Similarly there is a 1-1 relationship in 3D for nondegenerate p otentia ls on conformally f lat spaces.) T he stru ctures are not iden tical, since as w e can see fr om the examples in this pap er, the s tructure r elations in the classical and quant um cases are not identical ; there are quantum m o dif ications of the classical equations. Although w e kn ow of no direct prescription for their determination, nonetheless the quantum structure equations are uniquely determined b y the classical structure equ ations. Gi v en a s econd system of s econd order constan ts of th e motion w e w rite do wn the corresp ond ing quantum system via the u sual corresp ond ence, wh ere pro du cts of classical functions are replaced by symmetrized quan tum op erators, and generate th e quadratic algebra by taking rep eated comm utators. Eve n order classical symmetries corresp ond to formally self-adjoin t qu an tum sym m etries, and o dd order classical symmetries corresp ond to formally sk ew-adjoin t qu an tum symmetries. (This re- lationship no longer h olds for third order sup erinte grable systems [21, 40].) W e will demonstrate here ho w to get qu an tum mo dels from th e classical ones th at w e hav e deriv ed. The basic pr escription for the transition from the classical case to the op erator case is to replace a pair of canonically conju gate v ariables c , β b y c → t , β → ∂ t . (Th ere is n o obstruction to quantiza tion for second order sup erinte grable systems.) Once an app ropriate c hoice of β is made in a classical mo del, w e can use this pr escription to go to a dif ferentia l op erator mo del of the quan tum structur e equations. In particular mo del I I I ab ov e suggests a op erator mo d el su c h that S is m ultiplication by c , X is a f irst order dif feren tial op erator in c and L 1 + iL 2 is a fourth order d if feren tial op erator. T h e result, whose existence is implied by the 1-1 classical/quan tum relationship for second ord er sup erin tegrable systems, is I I I : S = t, X = − 2 i ( t + α ) ∂ t + 2 i, (21) L 1 + iL 2 = 8( t + α ) 3 ∂ 4 t + 2( t + α ) (3 α + t + 2 E + 9) ∂ 2 t − 2( t + 5 α + 4 E + 18 ) ∂ t +  2 + E 2 − α 2  + E 2 + 2 E (9 − α ) + ( α + 12)( α + 6) 2( t + α ) . The leading order dif feren tial op erators terms agree with the classical case but there are lo wer order correction terms needed to correct for the noncommutivit y of t and ∂ t . W e can realize v arious irr educible represen tations of the quadratic algebra b y c hoosing subsp aces of fun ctions of t on whic h the op erators act. Th is mod el agrees with (10), (11), (12) in the case where C ( n − 1 , n ) = 1 and C ( n + 1 , n ) is four th order. Ho we v er, there w e had a sp ace spanned b y a coun table num b er of eigen v ecto rs of th e skew-adjoin t symmetry X whereas here we w an t the sp ectral decomp osition of the self-adjoint symmetry S to go v ern the mo d el. T his forces L 2 to b e sk ew-adjoin t and X to b e self-adjoin t. Thus, though the d if feren tial op erators are formally Mo dels for Q u adratic Algebras 15 the same, the Hilb ert spaces and th e sp ectral analysis are dif ferent. All the representa tions are inf inite-dimensional. O ne class can b e realized by closing the dense sub space of C ∞ functions with compact supp ort on 0 < t < ∞ wh ere the measure is dt/t . The the sp ectrum of S is con tin uous and ru n s ov er the p ositive real axis. Here X also h as contin u ous real sp ectra co v erin g the fu ll real axis. In p articular the generalized eigenfunction of X with real eigen v alue λ is prop ortional to t − iλ , and µ is pur e imaginary . Thus the sp ectral analysis of X is give n by the Mellin transf orm . There is a similar irredu cible represen tation def ined on − ∞ < t < 0. By a canonical tr an s formation we can also get mo dels of these representa tions in whic h b oth C ( n − 1 , n ) = 1 and C ( n + 1 , n ) are second order. (W e shall illustrate this explicitly for mo del I.) Then the sp ectral decomp osition of S is giv en b y the Hank el transform. Since these particular eigenspaces of H admit no d iscrete sp ectrum for any of the symm etries of in terest, we shall n ot analyze them fur ther. No w w e consider mo d el I, (19 ). Due to the presence of trigonometric terms in β w e cannot realize this as a f inite order dif ferentia l op erator mo d el. Ho w ev er , w e can p erf orm a ho dograph transformation, i.e. use the prescription β → t , c → − ∂ t to realize th e mo del. This would seem to make no sense due to the app earance of functions of c u nder the squ are r o ot s ign. Ho wev er, b efore using the p r escription we ca n make use of the freedom to make a replacemen t β ′ = β + g ( c ) whic h preserves canonical v ariables. W e c ho ose e − 2 iβ → e − 2 iβ / p c 4 − 2 c 2 ( E + α ) + ( E − α ) 2 but lea v e c un c hanged. Then we f ind L 1 = 1 2 ( E − c 2 − α ) − i 4 h  c 4 − 2 c 2 ( E + α ) + ( E − α ) 2  e 2 iβ − 1 i , L 2 = − i 4 h  c 4 − 2 c 2 ( E + α ) + ( E − α ) 2  e 2 iβ + 1 i , (22) with X as b efore. No w w e app ly th e qu an tizatio n prescription β → t , c → − ∂ t and obtain a mo del in which b oth L 1 and L 2 are fourth ord er and X is a f irst ord er dif feren tial op erator. This is, in fact, id entical to with in a co ord inate change to mo del (2 1). One migh t also try to obtain a dif ference op erator mo d el from (22) with the r eplacemen t c → t , β → ∂ t , so that e 2 iβ w ould b ecome a dif ference op erator. Ho w ev er, this dif ference op erator quan tum mod el is equiv ale n t to what would get fr om the β → t , c → − ∂ t mo del by taking a F ourier tr ansform. Th us w e don’t r egard it as new. There is an alternate wa y to obtain a quantum realizatio n from mo del I. W e us e th e fact that c 4 − 2 c 2 ( E + α ) + ( E − α ) 2 = ( c 2 − ( E + α )) 2 − 4 α and set φ = arctan  √ − 4 α c 2 − ( E + α ) 2  . No w w e let 2 β → 2 β + φ to obtain L 1 = 1 2 ( E − c 2 − α ) + 1 2  ( c 2 − ( E + α ) 2 ) sin 2 β + 2 i √ α cos 2 β  , L 2 = 1 2  ( c 2 − ( E + α ) 2 ) cos 2 β − 2 i √ α sin 2 β  , X = c. No w the pr escription β → t , c → − ∂ t leads to a qu an tum realization of L 1 , L 2 b y second order dif feren tial op erators. Ind eed L 1 = 1 2 (cos(2 t ) − 1) ∂ 2 t − 8 iξ sin(2 t ) ∂ t +  − E 2 + 64 ξ 2 + 8 iξ − 1 4 − α 2  cos(2 t ) + E − α 2 , 16 E.G. Kalnins, W. Miller Jr. and S. Post L 2 = 1 2 sin(2 t ) ∂ 2 t + 8 iξ cos(2 t ) ∂ t +  − E 2 + 64 ξ 2 + 8 iξ − 1 4 − α 2 ) cos(2 t  + E − α 2 , X = ∂ t . Here ξ is arbitrary and can b e remo v ed v ia a gauge transformation. The change of v ariable τ = e 2 it reduces th is mo d el to the form (13). This sh o ws that the f lexibilit y we had in constru c- ting dif feren tial op erator mo dels from the abstract repr esentati on theory by renormalizing our basis vect ors f n is replaced in the classical mo del case by appropriate canonical tr an s formations c → c , β → β + g ( c ). In either case there is essentia lly only one dif feren tial op erator mo d el that can b e tran s formed in v arious wa ys. It is clear that mo del I I cannot pro duce f inite order dif feren tial op erator realizations of the quantum quadr atic algebra, d ue to the inte rt wining of square ro ot dep endence for c and exp onenti al d ep enden ce for β . Ho w ev er, it will pro duce a dif ference op erator realization via T aylo r’s theorem: e a∂ t f ( t ) = f ( t + a ).T o show this exp licitly we make a co ordinate change suc h that 2 √ c + α∂ c = ∂ C in (20), which suggests r ealizations of the quantum op erators in the form L 1 f ( t ) = ( t 2 − α ) f ( t ) , X f ( t ) = h ( t ) f ( t + i ) + m ( t ) f ( t − i ) , L 2 f ( t ) = − i 2 ( i + 2 t ) h ( t ) f ( t + i ) + i 2 ( − i + 2 t ) m ( t ) f ( t − i ) . (23) A straigh tforw ard computation sho ws that the quantum alg ebra str ucture equations are s atisf ied if and only if h ( t ) m ( t + i ) = 1 4 ( α − t 2 − it )( t 2 + it − E ) t ( t + i ) . (24) Since α = − a 2 + 1 4 and E = − ( µ − 1 + a ) 2 + 1 4 for b ounded b elo w repr esen tatio ns, w e can factor (24) simply to obtain h ( t ) m ( t + i ) = − 1 4 t ( t + i )  t + i 2 + ia   t + i 2 − ia  ×  t + i 2 + iµ + ia   t + 3 i 2 − iµ − ia  . (25) Note that only the pro du ct (24) is determined, not the individ u al factors. Th us w e can c ho ose h ( t ), say , as an arbitrary nonzero function and then determine m ( t ) from (24). All these mo dif ications of the factors are accomplished b y gauge transform ations on the represen- tation space: ˜ f ( t ) = ρ ( t ) f ( t ) where ρ ( t ) is th e gauge function. If we c ho ose the factors in the form h ( t ) = i ( 1 2 − a − it )( µ + a − 1 2 − it ) 2 t , m ( t ) = − i ( 1 2 − a + it )( µ + a − 1 2 + it ) 2 t , then w e we get exactly the m o del (15). The f inite dimensional mo del is related by the simple c hange of v ariables t → i ( t − a + 1 / 2), µ = − m . In an y case, there is only a single solution of these equations, up to a gauge transformation. 5.3 The classical mo del for S9 This is the system on the complex sphere, with n ondegenerate p otent ial V = a 1 s 2 1 + a 2 s 2 2 + a 3 s 2 3 , Mo dels for Q u adratic Algebras 17 where s 2 1 + s 2 2 + s 2 3 = 1. T he classical S9 system has a basis of sym metries L 1 = J 2 1 + a 2 s 2 3 s 2 2 + a 3 s 2 2 s 2 3 , L 2 = J 2 2 + a 3 s 2 1 s 2 3 + a 1 s 2 3 s 2 1 , L 3 = J 2 3 + a 1 s 2 2 s 2 1 + a 2 s 2 1 s 2 2 , (26) where H = L 1 + L 2 + L 3 + a 1 + a 2 + a 3 and th e J i are def ined by J 3 = s 1 p s 2 − s 2 p s 1 and cyclic p ermutatio n of ind ices. The classical stru ctur e r elations are {L 1 , R} = 8 L 1 ( H + a 1 + a 2 + a 3 ) − 8 L 2 1 − 16 L 1 L 2 − 16 a 2 L 2 + 16 a 3 ( H + a 1 + a 2 + a 3 − L 1 − L 2 ) , {L 2 , R} = − 8 L 2 ( H + a 1 + a 2 + a 3 ) + 8 L 2 2 + 16 L 1 L 2 + 16 a 1 L 1 − 16 a 3 ( H + a 1 + a 2 + a 3 − L 1 − L 2 ) , with {L 1 , L 2 } = R and R 2 − 16 L 1 L 2 ( H + a 1 + a 2 + a 3 ) + 16 L 2 1 L 2 + 16 L 1 L 2 2 + 16 a 1 L 2 1 + 16 a 2 L 2 2 + 16 a 3 ( H + a 1 + a 2 + a 3 ) 2 − 32 a 3 ( H + a 1 + a 2 + a 3 )( L 1 + L 2 ) + 16 a 3 L 2 1 + 32 a 3 L 1 L 2 + 16 a 3 L 2 2 − 64 a 1 a 2 a 3 = 0 . T aking L 1 = c , H = E with c , β as conjugate v ariables, w e f in d the mo del L 2 = 1 2 ( a 1 + 2 a 2 + E − c ) − ( a 2 − a 3 )( a 1 + 2 a 2 + 2 a 3 + E ) 2( c + a 2 + a 3 ) + p (4 a 1 a 2 + 4 a 1 a 3 + 2 c ( E + a 1 + a 2 + a 3 ) + 4 ca 1 − ( E + a 1 + a 2 + a 3 ) 2 − c 2 )(4 a 2 a 3 − c 2 ) 2( a 2 + a 3 + c ) × cos(4 β √ a 2 + a 3 + c ) . (27) This suggests a dif ference op erator realization of the qu an tum mo del. In the quant um case the symmetry op erators L 1 , L 2 , L 3 are obtained from the corresp ond in g classical constan ts of th e m otion (26) through the replacemen ts J k → J k where the angular momen tum op erators J k are def ined b y J 3 = x 1 ∂ x 2 − x 2 ∂ x 1 and cyclic p ermutation of ind ices. Here H = L 1 + L 2 + L 3 + a 1 + a 2 + a 3 . The quantum structur e r elations can b e put in the symmetric form [ L i , R ] = 4 { L i , L k } − 4 { L i , L j } − (8 + 16 a j ) L j + (8 + 16 a k ) L k + 8( a j − a k ) , R 2 = 8 6 { L 1 , L 2 , L 3 } + − (16 a 1 + 12) L 2 1 − (16 a 2 + 12) L 2 2 − (16 a 3 + 12) L 2 3 + 52 3 ( { L 1 , L 2 } + { L 2 , L 3 } + { L 3 , L 1 } ) + 1 3 (16 + 176 a 1 ) L 1 + 1 3 (16 + 176 a 2 ) L 2 + 1 3 (16 + 176 a 3 ) L 3 + 32 3 ( a 1 + a 2 + a 3 ) + 48( a 1 a 2 + a 2 a 3 + a 3 a 1 ) + 64 a 1 a 2 a 3 . Here i , j , k are chosen such that ǫ ij k = 1 where ǫ is the pure sk ew-symmetric tensor, R = [ L 1 , L 2 ] and { L 1 , L j } = L i L j + L j L i with an analogous def inition of { L 1 , L 2 , L 3 } as a sum of 6 terms. In practice we will s ubstitute L 3 = H − L 1 − L 2 − a 1 − a 2 − a 3 in to these equations. Pro ceeding exactl y as in the S 3 case (23), (24), (25), we f ind that the dif ference op erator analogy of (27) for the qu an tum quadratic algebra is L 1 = 4 t 2 − 1 2 + β 2 + γ 2 , 18 E.G. Kalnins, W. Miller Jr. and S. Post L 2 = h ( t ) T i + m ( t ) T − i + ℓ ( t ) = [ − 4 α 2 − 8 α − 4 + 4 E 2 + 16 i ( α + 1 ) t + 16 t 2 ]( β + 1 + γ − 2 it )( β − 1 − γ + 2 it ) 1024 t ( t + i )(2 t + i ) 2 × [ − 4 α 2 − 4 + 8 α + 4 E 2 + 16 i (1 − α ) t + 16 t 2 ]( β + 1 − γ − 2 it )( β − 1 + γ + 2 it ) T i + T − i +  − 2 t 2 − 1 2 E 2 − 1 2 β 2 + 1 2 α 2 + 1 2 γ 2 + ( γ 2 − β 2 )( − 4 α 2 + 4 E 2 ) 8(1 + 4 t 2 )  , where a 1 = 1 4 − α 2 , a 2 = 1 4 − β 2 , a 3 = 1 4 − γ 2 , H = 1 4 − E 2 . The qu adratic terms factor into simple lin ear terms, and ju st as in the S3 case, it is only ℓ ( t ) and the pro du ct h ( t ) m ( t + i ) that is uniquely determined . W e can change the ind ivid ual fac- tors h ( t ), m ( t ) by a gauge tran s formation. With the c hange of v ariable t = iτ and a gauge transformation to an op erator with maximal s ymmetry in τ , we obtain the s tandard mo d el h ( t ) = ˜ h ( τ ) = ( A + τ )( B + τ )( C + τ )( D + τ ) 4 τ ( τ + 1 / 2) , m ( t ) = ˜ m ( τ ) = ( A − τ )( B − τ )( C − τ )( D − τ ) 4 τ ( τ − 1 / 2) , A = E + α + 1 2 , B = E − α + 1 2 , C = β + γ + 1 2 , D = β − γ + 1 2 . It follo ws that L 2 = ˜ h ( τ ) E +1 + ˜ m ( τ ) E − 1 + ˜ ℓ ( τ ) is a linear combinatio n of L 1 and th e dif ference op erator wh ose eigenfunctions are the Wilson p olynomials, ju st as found in [39]. Here E s f ( τ ) = f ( τ + s ). 6 Conclusions and prosp ects This pap er consists of t w o related parts. In th e f irst part we hav e studied the representati on theory for the quadratic algebra asso ciated with a 2D second order quantum sup erinteg rable system with degenerate p otent ial, namely S 3. W e hav e classif ied the p ossible f inite-dimensional represent ations and inf inite dimensional b ounded b elo w repr esentati ons, i.e., th ose with a lo w est w eigh t vect or. Then we hav e constructed the p ossible Hilb ert space mo d els for these represen- tations, in terms of dif ferential op erators or of dif ference op erators acting on sp aces of functions of one complex v ariable. Th ese m o dels mak e it easy to f ind raising and lo w ering op erators for the representa tions and to unco v er relatio nships b et w een the algebras and families of orthogonal p olynomials. Here S3 has b een treated as an example of a degenerate p oten tial su p erintegrable system. The example S9 of a n ondegenerate p oten tial wa s treated in [39]. In 2D th ere are 13 equiv alence classes of sup erinteg rable systems with n on trivial p otent ials: 7 nondegenerate and 6 d egenerate. Results for all of th ese cases will b e includ ed in the thesis of the th ird author. In the s econd part of this w ork w e ha v e tak en up the stud y of mo dels of the quadratic algebras asso ciated with th e classical second order sup er integrable systems. In eac h mo del there is only a sin gle pair of canonically conjugate v ariables, rather than the 2 pairs in th e original classical system. W e show ed, b ased on classical Hamilton–Jac obi theory , that such m o dels alw a ys exist. Then w e describ ed a pro cedur e to derive the one v ariable mo d els for the quantum quadr atic algebras from the mo dels for the classical quadratic algebras. S ince it is kno wn that there is a 1-1 relationship b et w een classica l and quan tum second order sup erinteg rable s ystems (ev en though the algebras are not the same), it is not too su rprising that one should b e able to compute the quan tum m o dels from the classica l m o dels. Ho w ev er, w e hav e made this explicit. Mo dels for Q u adratic Algebras 19 W e app lied this pr o cedure n ot only to obtain the d if feren tial and dif ference op erator mo d els for system S3, b u t also for the generic system S9. F or S9 we sh o w ed that there is a dif ference op erator mo del asso ciated w ith general Wilson p olynomials, but no dif ferent ial op erator mo d el. This construction demonstrates that the theory of general Wilson p olynomials is imb edded in classical mechanics in a manner quite dif ferent fr om the usual group theory (Racah p olynomial) approac h. There is m uc h m ore work to b e done. Once the mo dels are w ork ed out and the corresp onding functional Hilb ert spaces are constru cted, usually Hilb er t spaces with k ernel f unction, then on e needs to f ind intert win in g op erators th at map the mo del sp ace to the space on wh ic h the quan tum S c hr¨ odinger op erator is def ined. Also,w e ha v e demonstr ated how to determine the classical mo d els and sho w h o w they quan tize in a unique fashion. A puzzle here is that we are f inding classical mo dels corresp onding to n on-h yp ergeometric typ e v ariable separation. These classical mo d els t ypically in v olv e elliptic fu nctions. W e do not ye t u nderstand ho w they can b e quan tized. Th ey clearly d o not lead to d if feren tial or ord inary d if ference op erator qu an tum mo dels. Another part of our ef fort is to stud y th e structure of qu adratic alg ebras corresp onding to 3D nondegenerate sup erin tegrable sys tems, and to f ind tw o v ariable mo dels for them. This is a m uc h more dif f icult problem than in 2D, where it led to g eneral Wilson and Racah p olynomials, among other m o dels. Th e qu adratic algebra still closes at order 6 but now there are 6 linearly second order symmetries, r ather than 3, and they are fu nctionally dep endent, satisfying a p olynomial relation of order 8. 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