New Quasi Exactly Solvable Difference Equation

Journal of Nonlinear Mathematical P hysics V olume *, Numb er * (20**), 1–12 Ar ticle New Quasi Exactly So l v able Difference Equation Ryu Sasaki Y ukaw a Institute for The or etic al Physics, Kyoto University, Kyoto 606-850 2, Jap an E-mail: ryu@yukawa.kyoto-u.ac.jp R e c eive d Mo nth *, 200*; R evise d Month *, 200* ; A c c epte d Month *, 200 * Abstract Exact solv ability of tw o t ypica l examples o f the d iscrete quantum mec hanics, i.e. the dynamics of the Meixner-Pollaczek and the contin uous Hahn p olynomials with ful l p ar ameters , is newly demo ns trated b oth a t the Schr¨ odinger and Heisenberg picture levels. A new qua si exactly solv able differe nc e equation is constructed by crossing these tw o dynamics, that is, the quadra tic p otent ial function of the contin uous Hahn po lynomial is multiplied by the constant phase fac to r of the Meixner-Pollaczek type. Its ordinary quantum mec hanical co un terpart, if exists, do es not seem to b e known. 1 In tro du ction As shown recen tly , Q uasi Exact S olv abilit y (QES) is very closely r elated to exact s olv abil- it y [22, 17, 21]. If all the eigen v alues of a quantum mec hanical sys tem are known together with the corresp onding eigenfunctions, the system is exactly s olv able in the Sc h r¨ odinger picture. In contrast, a s y s tem is QES if only a finite num b er (usually the lo west lying ones) of exact eigen v alues and eigenfunctions are kno wn [26, 25, 24]. Amon g v arious char- acterisatio n of quasi e xact solv abilit y [26, 2 5, 24, 12, 8], the existe nce of the in v ariant p olynomial subsp ace is conceptually simple. T he metho d to obtain a QES system, adv o- cated by th e present author [22, 17, 21], by deforming an exactly solv able s ystem with an addition/m u ltiplicat ion of a higher ord er in teraction term together with a comp ensation term, exemplifies the stru cture of the in v arian t p olynomial subspace rather clearly through the action of the similarity transformed Hamiltonian ˜ H (3.5)–(3.7) in terms of the pseudo groundstate wa v efun ction φ 0 . This m ethod w as applied to the exactly solv able ordin ary quan tum mec hanics [21] of one degree of freedom and m u lti-particle sys tem of Calogero- Sutherland t yp e [4, 23]. Recen tly new QES difference equ atio ns of one degree of freedom [22] and multi- particle systems [17] are obtained by the application of the s ame method to the discrete quan tum m ec hanics [14] for the Ask ey-sc heme of hyp ergeometric orth ogonal p olynomials [1, 11] and for the Ru ij s enaars-Sc hneider-v an Diejen sys tems [20, 27]. Tw o of the exactly solv able d iscrete quan tum m ec hanics discussed in [14, 15, 22], th e Meixner-P ollaczek and th e con tin u ous Hahn p olynomials, are of sp ecial t yp es in the sense that their parameters are a s u bset of the allo w ed ones. The purp ose of the present pap er is three-fold. Firstly , to d emonstrate the exact solv- abilit y of the full dynamics of the Meixner-P ollaczek and th e con tinuous Hahn p olynomials Copyrigh t c  200* by Ryu S asaki 2 Ryu Sasaki in the Schr¨ odinger picture through shap e in v ariance [7 , 14]. The exact Heisen b erg op er- ator solutions are also constru cted through the closure relations (2.12), (2.26), (2.38). The structure of the inv ariant p olynomial su b space is sho w n explicitly by the action of ˜ H for eac h d egree m onomial (2.22), (2.33). Secondly , to obtain a new Q ES d ifference equation b y crossing the ab o ve men tioned exactly solv able d ynamics. T he new s ystem h as the qu adratic p otentia l with tw o complex parameters (2.29) coming from the con tinuous Hahn p olynomial and a constan t multiplica tiv e p hase factor e − iβ (2.17) coming f r om the Meixner-P ollaczek p olynomial. Thirdly , to giv e comments on exact Heisen b erg op erator solutions. The third part is closely related to the presenta tion in NEEDS 2007 W orkshop b y the presen t author, “Heisen b erg op erator s olutions for the C alog ero systems” [16]. This p ap er is organised as follo ws. In section t w o th e exact solv abilit y of the f ull dynamics of the Meixner-P ollaczek and con tin u ous Hahn p olynomials is demonstrated after brief r eview of the general setting of the d iscr ete quan tum mec h anics appr opriate for the Ask ey-sc heme of hyp ergeometric orthogonal p olynomials. Sectio n th ree is dev oted to the new QES difference equation obtained by crossing the dynamics of the fu ll Meixner- P ollaczek and con tinuous Hahn p olynomials. Section four is for the commen ts on the exact Heisen b erg op erator solutions. T heir d y n amical roles, algebraic in terpretation and the connection to a ‘ quantum Liouvil le the or em ’ are explained. 2 Hamiltonian F orm ulation for Dynamics of Hyp ergeomet- ric Orthogonal P olynomials It is well kn o wn that the classical orthogonal p olynomials, the Hermite, Laguerre and J a- cobi p olynomials with v arious degenerations (Gegen bauer, Legendr e, etc) constitute the eigenfunctions of exactly s olv able quan tum mechanics, f or example, the harmonic oscil- lator w ithout/with the centrifugal p otentia l, the P¨ oschl-T eller p oten tial etc. Thus it is quite natural to exp ect that the Ask ey-scheme of hyp ergeometric orthogonal p olynomi- als toget her with their q -analogues, whic h are generalisation/deformation of the classical orthogonal p olynomials, also constitute the eigenfunctions of certain quan tum mechanics- lik e systems, so that the orthogonalit y has p rop er explanation/in terp retatio n. In ‘discrete’ quan tum mec h anics [14], a Hamiltonian form ulation was in tro du ced for the d ynamics of sev eral t ypical examples of the Askey-sc h eme of h yp ergeometric orthogonal p olynomials. Since these p olynomials ob ey difference equations instead of different ial equatio ns, th e Hamiltonians con tain the momentum op erators in the exp onentiate d forms in con trast to the second order p olynomials in ordinary quantum mec hanics. Th ese examples of dis- crete quan tum mechanics are exactly solv able in the S c hr¨ od in ger picture due to the sh ap e in v ariance prop erties [7, 14] and th eir exact Heisenberg solutions are giv en in [15]. In this section we discuss t wo examples, the Me ixner-P ollac zek p olynomial and the con tinuous Hahn polynomial, in their full generalit y . In our p revious w ork on discrete quan tum mec hanics [14, 15], only the sp ecial case of the Meixner-Poll aczek polynomial with the ph ase angle φ = π / 2 and the sp ecial case of th e con tin u ou s Hahn p olynomial with t w o real parameters a 1 and a 2 are discussed, partly b ecause these sp ecial cases of th e t wo p olynomials app ear in sev eral other dynamical con texts [6, 3, 2] and, in particular, they app ear as describing the equilibrium p ositions [19, 13, 14] of the classica l Ruijsenaars- Sc h neider v an Diejen systems [20, 27]. New QES Difference Equ ation 3 W e sho w that these t wo p olynomials i n their full generalit y , that is, w ith a ge neral phase angle φ for the Meixner-Po llacze k p olynomial and with tw o complex parameters a 1 and a 2 for the con tin uou s Hahn p olynomial, a re exactly s olv able in the Schr¨ odinger as w ell as in the Heisenberg picture. Lat er in section 3 we sh o w that a new quasi exactly solv able system is ob tained by crossing these general Meixner-P ollacz ek a nd con tin u ous Hahn p olynomials, th at is, b y m u ltiplyin g the extra phase f actor to the p oten tial of the general cont in u ous Hahn p olynomial. The resu lting system is no longer exactly solv able but it is quasi exactly solv able by addin g a comp ensation term. 2.1 General Setting Here w e recapitulate the basic notation and rudimenta ry facts of discrete quan tum me- c hanics o f one d egree of freedom. F or details w e refer to [14, 18]. The Hamiltonian of a discr ete qu an tum mec h anical system of one degree of freedom to b e discuss ed in this pap er h as the follo wing general str ucture H def = p V ( x ) e p p V ( x ) ∗ + p V ( x ) ∗ e − p p V ( x ) − V ( x ) − V ( x ) ∗ (2.1) = p V ( x ) e − i∂ x p V ( x ) ∗ + p V ( x ) ∗ e + i∂ x p V ( x ) − V ( x ) − V ( x ) ∗ , (2.2) in which th e p oten tial function V ( x ) = V ( x ; λ ) dep ends, in general, on a s et of parameters λ . Th e exp onent iated momentum op erators cause a fin ite shift of the w av efunction in the imaginary direction: e ± i∂ x φ ( x ) = φ ( x ± i ). As in the sup ersym metric quan tum mec h anics [9, 5], the Hamiltonian is alwa ys factorised H = A † A, (2.3) A † def = p V ( x ) e − i 2 ∂ x − p V ( x ) ∗ e i 2 ∂ x , A def = e − i 2 ∂ x p V ( x ) ∗ − e i 2 ∂ x p V ( x ) , (2.4) whic h sho w s the (formal) hermiticit y and p ositiv e semi-definiteness of the Hamiltonian. See the discus sion in § 4 of [22] for deta iled realisation of hermiticit y . The groundstate w a v efunction φ 0 ( x ) is annihilated by the A op erator Aφ 0 ( x ) = 0 = ⇒ H φ 0 ( x ) = 0 , (2.5) whic h can b e c h osen r eal φ 0 ( x ) ∈ R . The eigenfun ctions of the Ha miltonian φ n ( x ) = φ n ( x ; λ ) h a v e the follo w ing general structur e: H φ n ( x ) = E n φ n ( x ) ( n = 0 , 1 , 2 , . . . ) , 0 = E 0 < E 1 < E 2 < · · · , (2.6) φ n ( x ; λ ) = φ 0 ( x ; λ ) P n ( η ( x ) ; λ ) . (2.7) Here P n is a p olynomial in η ( x ), whic h is called a sinusoidal coordin ate [15]. T h e orthogo- nalit y th eorem for the eigenfunctions b elonging to different eigenv alues imp lies th at { P n } are orthogonal p olynomials with resp ect to the weig h tfu nction φ 2 0 ( x ): Z φ 2 0 ( x ; λ ) P n ( η ( x ) ; λ ) ∗ P m ( η ( x ) ; λ ) dx ∝ δ n m . (2.8) 4 Ryu Sasaki Shap e In v ariance If t he reve rsed o rder Hamiltonian AA † has the same form as the original Hamiltonian A † A A ( λ ) A ( λ ) † = A ( λ + δ ) † A ( λ + δ ) + E 1 ( λ ) , (2.9) the system is called shap e in v ariant [7, 14]. Here δ denotes the shift of the parameters and an add itiv e constan t E 1 ( λ ) is to b e identified as the energy of the first excited level . Com b ined with th e basic fac t o f th e sup ersymmetric quantum mechanics that the t wo Hamiltonians A † A and A A † are iso-sp ectral ( except for the groun dstate), shap e inv ari- ance determines the entire energy sp ectrum and th e excited state eigenfun ctions from the groundstate wa vefunction: E n ( λ ) = n − 1 X s =0 E 1 ( λ + s δ ) , (2.10) φ n ( x ; λ ) ∝ A ( λ ) † A ( λ + δ ) † A ( λ + 2 δ ) † · · · A ( λ + ( n − 1) δ ) † φ 0 ( x ; λ + n δ ) . (2.11) This establishes the exact solv abilit y in the S c hr¨ odin ger picture. Heisen b erg Op erator Solution T he sinusoidal co ordinate η ( x ) has a remark able prop- ert y [15] that the multiple commutat ors with the Hamiltonian can b e reduced to η ( x ) itself and the first comm utator [ H , η ] through the closure relation [ H , [ H , η ] ] = η R 0 ( H ) + [ H , η ] R 1 ( H ) + R − 1 ( H ) . (2.12) Here R 0 ( H ) and R − 1 ( H ) are in general qu ad r atic p olynomials in H , whereas R 1 ( H ) is lin- ear in H . Th is leads to the exact Heisenb er g operator solution for the sinusoidal co ordinate η ( x ): e it H η ( x ) e − it H = a (+) e iα + ( H ) t + a ( − ) e iα − ( H ) t − R − 1 ( H ) /R 0 ( H ) , (2.13) α ± ( H ) def = 1 2  R 1 ( H ) ± p R 1 ( H ) 2 + 4 R 0 ( H )  , (2.14) a ( ± ) def =  ± [ H , η ( x )] ∓  η ( x ) + R − 1 ( H ) /R 0 ( H )  α ∓ ( H )    α + ( H ) − α − ( H )  . (2.1 5) The en tire sp ectrum {E n } can also b e d etermined from (2. 13) by starting from E 0 = 0 [15], as done by Heisenberg and Pa uli for the h armonic oscillator and the hydrogen atom. The p ositiv e and negativ e energy parts a ( ± ) of the Heisenberg op erator solution e it H η ( x ) e − it H are the annihilation-creation op erators: a (+) † = a ( − ) , a (+) φ n ( x ) ∝ φ n +1 ( x ) , a ( − ) φ n ( x ) ∝ φ n − 1 ( x ) . (2.16) The general theory of exact Heisen b erg op erator solutions f or exactly solv able m ulti- particle systems is yet to b e constru cted. F or th e sp ecial case of the C alog ero systems [4, 10], the totalit y of Heisenberg op erators are derived for any ro ot systems [16]. F or the classical ro ot sys tems A , B C and D , the num b er of particles can b e as large as w anted. See section 4 for commen ts on exact Heisen b erg op erator solutions in general. New QES Difference Equ ation 5 2.2 Meixner-P ollaczek p olynomial The p oten tial function V ( x ) for the Meixner-Poll aczek p olynomial is linear in x : V ( x ; λ ) def = e − iβ ( a + ix ) , λ def = a, (2.17) 0 < a ∈ R , φ ∈ R , β def = φ − π 2 , 0 < φ < π . (2.18) The sp ecial case discussed in [14, 6, 2, 3] is β = 0 or φ = π / 2. The ground s tate wa v efun c- tion φ 0 , as annih ilated by the op erator A , Aφ 0 = 0, is give n by φ 0 ( x ; a ) def = e β x | Γ( a + ix ) | = e ( φ − π 2 ) x p Γ( a + ix )Γ( a − ix ) . (2.19) The similarit y transformed Hamiltonian ˜ H in terms of th e grounds tate wa v efu nction φ 0 , ˜ H def = φ − 1 0 ◦ H ◦ φ 0 = V ( x )  e − i∂ x − 1  + V ( x ) ∗  e i∂ x − 1  (2.20) = ( a + ix ) e − iβ  e − i∂ x − 1  + ( a − ix ) e iβ  e i∂ x − 1  (2.21) acts on the p olynomial part of th e w a vefunction. It is obvi ous that ˜ H maps a p olynomial in to another and it is easy to v erify ˜ H x n = 2 n cos β x n + lo wer ord er terms , n ∈ Z + . (2.22) Th us we can fin d a degree n p olynomial eigenfunction P n ( x ) of the similarity transf orm ed Hamiltonian ˜ H ˜ H P n ( x ) = E n P n ( x ) , E n = 2 n cos β = 2 n sin φ, n = 0 , 1 , 2 , . . . , (2.23) whic h is called the Mei xner-P ollacz ek p olynomial [1 1]. I t is expressed in terms of the h yp ergeometric series P ( a ) n ( x ; φ ) = (2 a ) n n ! e inφ 2 F 1  − n, a + ix 2 a    1 − e − 2 iφ  , (2.24) in whic h ( b ) n is the standard Poc hhammer’s symb ol ( b ) n def = n Y k =1 ( b + k − 1) = b ( b + 1) · · · ( b + n − 1) . Shap e inv ariance is also easy to v erif y : A ( x ; a ) A ( x ; a ) † = A ( x ; a + 1 2 ) † A ( x ; a + 1 2 ) + E 1 ( λ ) , E 1 ( λ ) = 2 sin φ. (2.25) The parameter a is increased by 1 2 whereas th e new parameter φ is inv arian t. Since E 1 is indep en den t of th e shifted p arameter a , it is trivial to obtain the linear sp ectrum E n = 2 n sin φ , whic h is the same as (2.23). The sin usoidal co ordinate is η ( x ) = x . The closure relation (2.12) r eads simp ly [ H , [ H , x ]] = x 4 s in 2 φ + 2 cos φ H + 2 a sin 2 φ, α ± ( H ) = ± 2 sin φ, (2.26) 6 Ryu Sasaki indicating that x undergo es a sin usoidal motion with the frequ en cy 2 sin φ . Th e Heisen b erg op erator solution is e it H x e − it H = x cos[2 t sin φ ] + i [ H , x ] sin[2 t sin φ ] 2 sin φ + cos φ 2 sin 2 φ ( H + 2 a sin φ )(cos[2 t sin φ ] − 1) . (2.27) The annihilation-creation op erators are: a ( ± ) = ± [ H , x ] / (4 sin φ ) + 1 2  x + cos φ 2 sin 2 φ ( H + 2 a sin φ )  . (2.28) Ob viously th e expressions (2.26)–(2.28) are dr astically simplified f or the s p ecial case of φ = π/ 2 whic h were discussed in previous wo rk [14, 15]. 2.3 Con tinuous Hahn p olynomial The p oten tial function V ( x ) for the contin uous Hahn p olynomial is qu adratic in x : V ( x ; λ ) def = ( a 1 + ix )( a 2 + ix ) , λ = ( a 1 , a 2 ) , a 1 , a 2 ∈ C , Re( a 1 ) > 0 , Re( a 2 ) > 0 . (2.29) The sp ecial case discussed in [14, 6, 2, 3] is for real a 1 and a 2 . The ground state w a ve func- tion φ 0 , as annih ilated by the op erator A , Aφ 0 = 0, is give n by φ 0 ( x ; λ ) def = q Γ( a 1 + ix )Γ( a 2 + ix )Γ( a ∗ 1 − ix )Γ( a ∗ 2 − ix ) . (2.30) The similarit y transformed Hamiltonian ˜ H in terms of th e grounds tate wa v efu nction φ 0 , ˜ H def = φ − 1 0 ◦ H ◦ φ 0 = V ( x )  e − i∂ x − 1  + V ( x ) ∗  e i∂ x − 1  (2.31) = ( a 1 + ix )( a 2 + ix )  e − i∂ x − 1  + ( a ∗ 1 − ix )( a ∗ 2 − ix )  e i∂ x − 1  (2.32) acts on the p olynomial part of th e w a vefunction. It is obvi ous that ˜ H maps a p olynomial in to another and it is easy to v erify ˜ H x n = n ( n + a 1 + a ∗ 1 + a 2 + a ∗ 2 − 1) x n + lo wer order terms , n ∈ Z + . (2.33) Th us we can fin d a degree n p olynomial eigenfunction P n ( x ) of the similarity transf orm ed Hamiltonian ˜ H ˜ H P n ( x ) = E n P n ( x ) , E n = n ( n + a 1 + a ∗ 1 + a 2 + a ∗ 2 − 1) , n = 0 , 1 , 2 , . . . , (2.34) whic h is called th e con tin u ous Hahn p olynomial [11]. It is expressed in terms of the h yp ergeometric series p n ( x ; a 1 , a 2 , a ∗ 1 , a ∗ 2 ) (2.35) = i n ( a 1 + a ∗ 1 ) n ( a 1 + a ∗ 2 ) n n ! 3 F 2  − n, n + a 1 + a 2 + a ∗ 1 + a ∗ 2 − 1 , a 1 + ix a 1 + a ∗ 1 , a 1 + a ∗ 2    1  . New QES Difference Equ ation 7 Shap e inv ariance is also easy to v erif y : A ( x ; a 1 , a 2 ) A ( x ; a 1 , a 2 ) † = A ( x ; a 1 + 1 2 , a 2 + 1 2 ) † A ( x ; a 1 + 1 2 , a 2 + 1 2 ) + E 1 ( a 1 , a 2 ) , (2.36) δ def = ( 1 2 , 1 2 ) , E 1 ( a 1 , a 2 ) = b 1 , b 1 def = a 1 + a 2 + a ∗ 1 + a ∗ 2 = 2Re( a 1 + a 2 ) . (2.37) Here w e ha v e in tro duced an abbreviation b 1 for con venience. The p arameter a 1 and a 2 are increased by 1 2 . Since E 1 is linearly d ep endent on the shifted p arameters a 1 , and a 2 , it is trivial to obtain the quadratic sp ectrum E n = n ( n + a 1 + a 2 + a ∗ 1 + a ∗ 2 − 1) = n ( n + b 1 − 1), whic h is the s ame as (2.34). The sin usoidal co ordinate is η ( x ) = x . The closure relation (2.12) r eads simp ly [ H , [ H , x ]] = x (4 H + b 1 ( b 1 − 2)) + 2[ H , x ] + b 2 H + b 3 ( b 1 − 2) , (2.38) in whic h abbr eviations b 2 def = 2Im( a 1 + a 2 ) and b 3 def = 2Im( a 1 a 2 ) are used. The frequencies α ± ( H ) are α ± ( H ) def = 1 ± 2 √ H ′ , H ′ def = H + ( b 1 − 1) 2 / 4 , H ′ φ n = ( n + ( b 1 − 1) / 2) 2 φ n . (2.39) The Heisen b erg op erator solution reads e it H xe − it H = x − α − ( H ) e iα + ( H ) t + α + ( H ) e iα − ( H ) t 4 √ H ′ + [ H , x ] e iα + ( H ) t − e iα − ( H ) t 4 √ H ′ + b 2 H + b 3 ( b 1 − 2) 4( H + b 1 ( b 1 − 2)) − α − ( H ) e iα + ( H ) t + α + ( H ) e iα − ( H ) t 4 √ H ′ − 1 ! . (2.40) The annihilation and creation op erators are: a ′ ( ± ) def = a ( ± ) 4 √ H ′ = ± [ H , x ] ∓  x + b 2 H + b 3 ( b 1 − 2) 4( H + b 1 ( b 1 − 2))  α ∓ ( H ) . (2.41) Ob viously the expressions (2.38)–(2.41) b ecome drastically s imple for the sp ecial case of b 2 = b 3 = 0, w hic h were discu s sed in p revious works [14, 15]. 3 New QES Difference Equation Here w e will discuss th e discrete quantum mec hanics obtained by crossing the Meixner- P ollaczek and the con tin u ous Hahn p olynomials, that is, with the quadratic p otent ial function of th e con tin uou s Hahn p olynomial (2.29) multiplied by a constan t phase f acto r e − iβ of the Meixner-P ollaczek t yp e. As v aguely exp ected, the exact solv abilit y is n ot realised. W e will sh o w, instead, that the system is quasi exactly solv able b y add ing a comp ensation term whic h is linear in x : H def = p V ( x ) e − i∂ x p V ( x ) ∗ + p V ( x ) ∗ e + i∂ x p V ( x ) − V ( x ) − V ( x ) ∗ + α M x (3.1) = A † A + α M x, α M def = − 2 M sin β , M ∈ Z + , (3.2) V ( x ) def = ( a 1 + ix )( a 2 + ix ) e − iβ , a 1 , a 2 ∈ C , Re( a 1 ) > 0 , Re( a 2 ) > 0 . (3.3) 8 Ryu Sasaki It s hould b e noted that th e Hamiltonian is no longer p ositiv e semi-defin ite b ut the her- miticit y is pr eserv ed. T he main part, that is without the comp en sation term, is f actorised as b efore (2.4). The zero mo de of the A op erator Aφ 0 = 0 = ⇒ φ 0 ( x ) def = e β x q Γ( a 1 + ix )Γ( a 2 + ix )Γ( a ∗ 1 − ix )Γ( a ∗ 2 − ix ) , (3.4) is no longer the groun d state w av efunction. It is called the p s eudo groundstate w a vefunction [22]. The similarity transformed Hamiltonian ˜ H in terms of the pseudo groundstate wa v e- function φ 0 , ˜ H def = φ − 1 0 ◦ H ◦ φ 0 = V ( x )  e − i∂ x − 1  + V ( x ) ∗  e i∂ x − 1  + α M x (3.5) = ( a 1 + ix )( a 2 + ix ) e − iβ  e − i∂ x − 1  + ( a ∗ 1 − ix )( a ∗ 2 − ix ) e iβ  e i∂ x − 1  − 2 M sin β x, (3.6) acts on the p olynomial part of th e w a vefunction. It is obvi ous that ˜ H maps a p olynomial in to another and it is easy to v erify ˜ H x n = 2( −M + n ) sin β x n +1 + lo w er ord er terms , n ∈ Z + . (3.7) This means that the sys tem is not exactly solv able without the comp ensation term, but it is quasi exactly solv able, since ˜ H has an inv ariant p olynomial su bspace of degree M : ˜ H V M ⊆ V M , (3.8) V M def = Span  1 , x, x 2 , . . . , x M  , dim V M = M + 1 . (3.9) The Hamiltonian H (3.2 ) is obviously hermitian (self-adjoint) and all the eigen v alues are real and eige nfunctions can b e c h osen real. W e can obtain a finite n um b er ( M + 1) of exac t eigen v alues and eigenfunctions for eac h given M . T he oscillation theorem linking the num- b er of eigen v alues (from the groun dstate) to the zeros of eigenfun ctions do es not hold in the difference equations. The square in tegrabilit y of all the eigenfunctions R ∞ −∞ φ 2 ( x ) dx < ∞ is ob v ious . See [22 ] for other examples of quasi exactly solv able difference equations of one degree of freedom and [17] of many degrees of freedom. It is easy to demonstrate that multiplying an extra constan t ph ase factor e − iβ to the other exactly solv able p oten tial functions [14] V ( x ) def = ( a 1 + ix )( a 2 + ix )( a 3 + ix ) 2 ix (2 ix + 1) , con tinuous d ual Hahn , (3.10) V ( x ) def = ( a 1 + ix )( a 2 + ix )( a 3 + ix )( a 4 + ix ) 2 ix (2 ix + 1) , Wilson , (3.11) V ( x ) def = (1 − a 1 z )(1 − a 2 z )(1 − a 3 z )(1 − a 4 z ) (1 − z 2 )(1 − q z 2 ) , z = e ix , Ask ey-Wilson , (3.12) do es not p ro vide either exactly solv able or quasi exactly solv able d ynamical sys tems. The situation is the same for v arious r estrictions of the Askey-Wil son p olynomial. New QES Difference Equ ation 9 It is also easy to see that the q u asi exact solv abilit y of the systems discussed in [22] with the p otent ials V ( x ) def = ( a 1 + ix )( a 2 + ix )( a 3 + ix ) , (3.13) V ( x ) def = ( a 1 + ix )( a 2 + ix )( a 3 + ix )( a 4 + ix ) , (3.14) V ( x ) def = ( a 1 + ix )( a 2 + ix )( a 3 + ix )( a 4 + ix )( a 5 + ix ) 2 ix (2 ix + 1) , (3.15) V ( x ) def = ( a 1 + ix )( a 2 + ix )( a 3 + ix )( a 4 + ix )( a 5 + ix )( a 6 + ix ) 2 ix (2 ix + 1) , (3.16) V ( x ) def = (1 − a 1 z )(1 − a 2 z )(1 − a 3 z )(1 − a 4 z )(1 − a 5 z ) (1 − z 2 )(1 − q z 2 ) , z = e ix , (3.17) is d estro y ed if a constan t ph ase f acto r e − iβ is multiplied. 4 Commen ts on Exact Heisen b erg Op erator Solutions Let us start with a rather naiv e question; “What can w e learn m ore fr om the exact Heisen- b erg op erator solutions when we know the complete s p ectrum and the corresp ond ing eigen- functions?” A small digression on the w ell known relationship b et wee n the Schr¨ odinger and Heisenberg pictures would b e usefu l. Supp ose we hav e a complete set of s olutions of the Shr ¨ odinger equation H φ n = E n φ n . F or any observ able A , one can construct a (usually infinite) matrix ˆ A , ˆ A n m = h φ n | A | φ m i , satisfying the Heisen b erg equation of motion ∂ ˆ A ∂ t = i [ H , ˆ A ] . Ob viously s uc h an exact Heisenberg op er ator solution d o es n ot teac h us anyt hing more. But for a sp ecial c hoice of the observ ables, called the ‘sin us oidal co ordinates’ { η j } , j = 1 , . . . , r , with r b eing the degree of fr eedom, th e op erators { e i H t η j e − i H t } , j = 1 , . . . , r, can b e expr essed explicitly in terms of the fun d amen tal op erators { η j } , H and a finite n um b er of m ultiple commutato rs of { η j } with the Hamiltonian [ H , [ H , [ · · · , η j ] .. ]. These are the Heisen b erg op er ator solutions found by O dak e-Sasaki for a wide class of exactly solv able deg ree one quantum mec hanics including the discrete ones [15] and for typical m u lti-particle dyn amics of Caloge ro t yp e for an y ro ot system [16]. I t should b e stressed that the existence of sin usoidal co ordinates is not guarantee d a t all. Ther e are several exactly solv able d egree one quantum mec hanical s y s tems f or whic h our construction of the Heisen b erg op erator solutions do es not apply . V arious reduced Kepler problems and the Rosen-Morse p otenti als are the t ypical examples. See [15] for more details. F or m ulti- particle sy s tems, the exact Heisen b er g op erator solutions are kno wn [16] only for th e 10 Ryu Sasaki Calogero sys tems for any ro ot system [4, 10]. Th ere are other well -kno wn exactly s olv able m u lti-particle systems; the Sutherland sy s tems [23] and the Ru ijsenaars-Sc hn eider-v an Diejen systems [20, 27]. The name ‘sin usoidal’ implies that they all u n dergo sin usoidal motion but n ot harmonic. In classical mec han ics te rms, the frequencies dep end on the initial conditions. F rom the analysis p oin t of view, the sinusoidal co ordinates g enerate the p olynomial eigenfunction { P n } , φ n = φ 0 P n ( φ 0 is the groundstate wa v efun ction). In other w ords { P n } are orthogonal p olynomials in { η j } . Th e exact Heisenber g op erator solutions for { η j } p ro vide the complete set of m u lti-v ariable generalisatio n of the thr e e term r e curr enc e r elations , whic h characte rise orthogonal p olynomials in one v ariable. As stressed in [15, 16], the positive and negativ e frequency parts of the He isen b erg op erator solutions are the sets of annihilation-cr e ation op er ators . They generate the en tire eigenfunctions alg ebraically , and th u s form a dynamic al symmetry algebr a to gether with the Hamiltonian and p ossibly with the h igher conserv ed quantitie s (Hamiltonians). The structure of th ese dynamical symmetry algebras is iden tifi ed only for a few sp ecial cases of degree one, for example, su (1 , 1). It is a go o d c hallenge to identify the dynamical symmetry alge bra and its irreducible represen tations for ea c h kno w n exa ct Heisen b erg op erator solution, single an d m ulti-degree of freedom. F rom th e algebra p oint of view, the three term recur r ence relations for s ingle v ariable orthogonal p olynomials corresp ond to th e Clebsc h-Gordan d ecomp ositio n rules for rank one algebras. T h e m u lti-particle v ers ion w ould simply corresp ond to the higher rank count erparts of the Clebsc h-Gordan decomp osition rules. F rom a m ore basic dynamics p oint of view, one could consider the exact Heisen b erg op erator solutions and the asso ciated annihilation-creation op erators as an explicit but partial realisation of ‘ quantum Liouvil le the or em ’. T he classical Liouville theorem asserts that one can constru ct by quadrature only from the complete set of inv olutiv e conserv ed quan tities the generating function of a canonical transf ormation wh ic h brings the system to the action-angle form. In con trast, the u sual form ulation of qu an tum Liouville theorem do es n ot sa y an ything ab out the second half; quan tum mec hanical coun terp art of ‘bringing to the action-angle form’. Th e complete s et of the creation-annihilation op erators pla y the corresp onding role; ‘generating the en tire eigenfunctions from th e groun dstate w av e- function’. I f such generated eigenstates w ere the sim ultaneous eigenstates of the complete set of inv olutiv e conserv ed quantitie s, one could sa y that the quant um Liouville th eorem is f ully realised. It seems that there is s till s ome w ay to go for that goal . Ac kno wledgemen ts R. S. thanks Satoru Od ak e for u seful comments. This work is sup p orted in part by Gran ts- in-Aid for Scient ific Researc h from the Ministr y of Education, Culture, S p orts, Science and T ec hnology , No.1834006 1 and No.19540179 . References [1] G. E. Andrews, R. Askey and R. Ro y , Sp e cial F unct ions , Encyclop edia of mathematics and its applications, Cambridge, (1999). New QES Difference Equ ation 11 [2] N. M. A takis hiy ev and S. Suslov, The Hahn and Meixner p olynomials of an ima ginary ar gu- men t and some of their applica tions, J. Phys. A18 (1985), 158 3-1596; Difference a nalogs of the ha rmonic oscillato r, Theor. Math. Phys. 85 (1990), 1 055-106 2. [3] V. V. B orzov and E. V. 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