Exactly solvable `discrete quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states

Yukawa Institute Kyoto DPSU-08-1 YITP-08- 1 F eb rua ry 2008 Exactly solv able ‘discrete’ quan tum mec hanics; sh ap e in v ariance, Heis en b erg soluti ons, annihil ati on-creatio n op erators and cohere nt states Satoru Odak e a and Ryu Sasaki b a Departmen t of Ph ysics, Shin sh u Univ ersit y , Matsumoto 3 90-8621 , Japan b Y uk a w a Institute for The oretical Ph ysics, Ky oto Univ ersit y , Ky oto 606-8502, Japan Abstract V arious examples of exac tly solv able ‘discrete’ quant um mec hanics are explored ex- plicitly with emphasis on shap e in v ariance, Heisen b erg op erator solutions, annihilation- creation op erators, the dyn amical symmetry algebras and coheren t stat es. The eigen- functions are the ( q -)Askey- scheme of hyp ergeometric orthogonal p olynomials satis- fying difference equation v ersions of the Schr¨ odinger equation. V arious reductions (restrictions) o f the symmetry algebra of the Ask ey-Wilson sys tem are explored in detail. 1 In t ro duction General theory of exactly solv able ‘discrete’ quan tum mec hanics of one degree of freedom systems is presen ted with all know n examples . The ‘disc rete’ quan tum mec hanics is a simple extension or deformation of quantum mec hanics in whic h the momen tum o p erator p ap- p ears in the Hamiltonian in the exp onen tiated forms e ± γ p , γ ∈ R , in stead of p olynomials in ordinary quan tum mec hanics. The corresp o nding Sc hr¨ odinger equations are difference equations with imaginary shifts, in stead of differen tial. The eigenfunctions of the exactly solv able ‘discrete’ quan tum mechanics of one degree of freedom systems consist of the ( q - )Ask ey-sc heme of hypergeometric o rthogonal p o lynomials [1, 2], whic h are deformations of the classical orthogonal p olynomials, lik e the Hermite, Laguerre, Jacobi po lynomials, etc [3], constituting the eigenfunctions of exactly solv able ordinary quan tum mec hanics [4, 5]. These eigenp olynomials a re orthogonal with resp ect to absolutely contin uous measure functions, whic h are just the s quare of the ground state w av efunctions; a familiar s ituation in quantum mec hanics. F or another t yp e of orthogonal p olynomials with discrete measures [1, 2, 6], se e [7] fo r a unified theory . Lik e most exactly solv able quan tum mec hanics, ev ery example of exactly solv able ‘discrete’ quan tum mec hanics is endo w ed with dynamical symmetry , shap e invarianc e [8], whic h a llo ws to determine the en tire energy sp ectrum and the corresp ond- ing eigenfunctions when com bined with Crum’s theorem [9] or the factorisation metho d [4, 5]. In other w ords, shap e in v aria nce guaran tees exact solv ability in the Sc hr¨ odinger pic- ture [10, 11, 12]. As exp ected, exact solv ability in the Heisen b erg picture also holds for all these examples. The explic it forms of Heisen b erg op erator solutio ns give rise to the ex plicit expressions of annihilation/creation operators as the p ositive /negative frequency parts [13]. The annihilation/creation op erators together with the Hamiltonian constitute the dynamical symmetry alg ebra. In some cases, the algebras are simple and tangible, lik e the oscillator algebra and its q -deformations [14], or su (1 , 1). The presen t pap er is to supplemen t or to complete some results in previous publications [10, 11, 12, 13]. The ‘discrete’ quan t um mec hanics of t he Meixner-P ollaczek, the con tinuous Hahn, the con tinuous dual Hahn, the Wilson and the Aske y-Wilson polynomials discussed in [10, 11 , 12, 13] are only for restricted parameter ranges; f or example the angle w as φ = π / 2 for the Meixner-P ollaczek p olynomial and all the parameters w ere restricted real for the con tin uous Hahn, the c ontin uous dual Hahn, the Wilson and the Ask ey-Wilson p olynomials. This is due to a historical reason that these p olynomials with the restricted parameter ranges w ere first re cognised b y the pre sen t authors as describing the clas sical eq uilibrium positions [15, 16, 10, 11, 12, 17] of multi-particle exactly solv able dynamical systems o f Ruijsenaars- Sc hneider-v an Diejen ty p e [18 , 19]. It is a deformatio n of the classical results dating as far bac k as Stieltjes [20], [21, 22] that t he classical equilibrium p ositions of m ulti-particle exactly solv able dynamical systems of Calogero-Sutherland t yp e [23, 24] are describ ed by the zeros of the classical orthogonal p olynomials (the Hermite, Laguerre and Jacobi). The 2 ‘discrete’ quan tum mec hanics w as constructed [10, 11, 12] based o n the analogy that these orthogonal p olynomials w ould constitute the eigenfunctions of certain quantum mec hanical systems in the same wa y as the class ical orthogonal polynomials (the Hermite, Laguerre and Jacobi) do. As will b e sho wn in detail in the main text, these ort ho gonal p olynomials enjoy the ex act solv ability and re lated prop erties for the full ranges of the parameters. A ttempts to further deform these exactly solv able systems hav e yielded sev eral examples [25, 2 6, 27] of the so-called quasi-exactly solv a ble systems [28, 29]. Another ob jectiv e of the presen t pap er is to explore in detail the prop erties of the systems obtained b y restricting the Aske y- Wilson system, t r eat ed in § 5.2– § 5.8.2. Some of these hav e in teresting and use ful forms o f the dynamical symmetry alg ebras o r the explicit forms of coheren t state, etc, as evidenced by the q -oscillator algebras realised b y the contin uous (big) q -Hermite po lynomial [1 4]. Asp ects of ordinar y theory of orthogonal p o lynomials are not particularly emphasised. This pap er is organised as follow s. In section tw o, the general setting of the ‘discrete’ quan tum mec hanics is recapitulated with appropriate notation. Starting with the parameters in the p oten tial function and the Hamiltonian, v ar ious concepts and solution metho ds are briefly surv ey ed. Sections three to fiv e a re the main b o dy of the pap er, discussing v arious examples of exactly solv a ble ‘discrete’ quantum mec hanics. They are divided in to three groups according t o the sin usoidal co ordinate η ( x ). Section three is for the p olynomials in η ( x ) = x . Section four is for the p olynomials in η ( x ) = x 2 . Section fiv e is for the p o lynomials in η ( x ) = cos x . V ery roughly sp eaking, p olynomials in section three are the deformation of the Hermite polynomial; tho se in section four are the deformation of the La g uerre p o lynomial and those in sections fiv e are the deforma t io n of the Jacobi p o lynomial from the p o in t o f view of the sin usoidal co ordinates, but not from t he energy sp ectrum. Sec tion six is for a summary a nd commen ts. App endix A provide s a diag r a mmatic pro of of the hermiticity (self- adjoin tness) of the Hamiltonians of ‘discrete’ quantum mec hanics. App endix B is a collection of the definition of basic sym b ols and functions used in this pap er for self-con tainedness. 2 General setting The dynamical v ariables are the co ordinate x ( x ∈ R ) and the conjugate momen tum p , whic h is realised as a differen tial o p erator p = − id/dx . The other parameters are symbolically denoted as λ = ( λ 1 , λ 2 , . . . ) on top of q (0 < q < 1) and φ ( φ ∈ R ). F or the q -systems, the parameters are denoted a s q λ = ( q λ 1 , q λ 2 , . . . ). Complex conjuga tion is denoted b y 3 ∗ and the absolute v alue | f ( x ) | is | f ( x ) | = p f ( x ) f ( x ) ∗ . Here f ( x ) ∗ means ( f ( x )) ∗ and f ( x ) ∗ | x → x + a = f ( x + a ∗ ) ∗ , since x is real. Hamiltonian The H amiltonian has a general form H def = p V ( x ) e γ p p V ( x ) ∗ + p V ( x ) ∗ e − γ p p V ( x ) − V ( x ) − V ( x ) ∗ , (2.1) in whic h γ is a real constan t. It is either 1 or log q . The p oten tial function V dep ends on the parameters, V ( x ) = V ( x ; λ ), whereas the q a nd φ dep endence is not explicitly indicated. The parameter dep endence of the Hamiltonian H = H ( λ ) is not explicitly indicated in most cases. The eigen v alue problem or the time-indep enden t Sc hr¨ odinger equation is a difference equation in stead of differen tial in or dina r y quan tum mec hanics: H φ n ( x ) = E n φ n ( x ) ( n = 0 , 1 , 2 , . . . ) , E 0 < E 1 < E 2 < · · · , (2.2) in whic h φ n ( x ) = φ n ( x ; λ ) is the eigenfunction b elonging to the energy eigenv alue E n = E n ( λ ). The differenc e equation has inheren t non-uniqueness of s olutions; if φ ( x ) is a solution so is φ ( x ) Q ( x ) when Q ( x ) is an y p erio dic function with the p erio d i γ . This non- uniqueness problem is resolv ed when t he Hilb ert space of the state vec tors is sp ecified. See App endix A. F actorisation F actorisation of the Hamiltonian is an imp ortan t prop ert y H = T + + T − − V ( x ) − V ( x ) ∗ = ( S † + − S † − )( S + − S − ) = A † A , (2.3) in whic h v arious quantities S ± = S ± ( λ ), T ± = T ± ( λ ), A = A ( λ ) are defined as ( † denote the hermitian conjugation with resp ect to the c hosen inner product (2.75) and (A.1)–(A.3)): S + def = e γ p/ 2 p V ( x ) ∗ , S − def = e − γ p / 2 p V ( x ) , S † + def = p V ( x ) e γ p/ 2 , S † − def = p V ( x ) ∗ e − γ p / 2 , (2.4 ) T + def = S † + S + = p V ( x ) e γ p p V ( x ) ∗ , T − def = S † − S − = p V ( x ) ∗ e − γ p p V ( x ) , (2.5) A def = i ( S + − S − ) , A † def = − i ( S † + − S † − ) . (2.6) Ground state wa v efunction The ground state w av efunction φ 0 ( x ) = φ 0 ( x ; λ ) is annihi- lated by the A op erator A φ 0 ( x ) = 0 ⇒ H φ 0 ( x ) = 0 ⇒ E 0 = 0 , (2.7) 4 whic h is a zero mo de of the Hamiltonian. The abov e equation reads explic itly as q V ( x + iγ 2 ) ∗ φ 0 ( x − iγ 2 ) = q V ( x + iγ 2 ) φ 0 ( x + iγ 2 ) . (2.8) Among possible solutions, w e c ho ose a real and no deless φ 0 . As will b e sho wn in App endix A, the requiremen t of the hermiticit y (self-adjoin tness) of the Hamiltonian H selects a unique solution φ 0 , which is give n explicitly in eac h subsection ( 3 .10), (3 .2 5), (4.7), (4 .2 3), (5.11), (5.38), (5.59), (5.79), (5.97), (5.127) and (5.146). Similarit y transformed Hamiltonian The similarity tra nsfor med Hamiltonian e H = e H ( λ ) in terms of the gro und state w av efunction φ 0 (2.8) is e H def = φ 0 ( x ) − 1 ◦ H ◦ φ 0 ( x ) = e T + + e T − − V ( x ) − V ( x ) ∗ = V ( x ) e γ p + V ( x ) ∗ e − γ p − V ( x ) − V ( x ) ∗ , (2.9) in which e T ± are defined as e T + def = φ 0 ( x ) − 1 ◦ T + ◦ φ 0 ( x ) = V ( x ) e γ p , e T − def = φ 0 ( x ) − 1 ◦ T − ◦ φ 0 ( x ) = V ( x ) ∗ e − γ p . (2.10) It acts on the p olynomial part of the e igenfunction. Let us write the excited state eigenfunc- tion φ n ( x ) = φ n ( x ; λ ) as φ n ( x ; λ ) = φ 0 ( x ; λ ) P n ( η ( x ) ; λ ) , (2.11) in whic h P n ( η ) = P n ( η ; λ ) is a p olynomial in the sinusoi dal c o o r dinate η ( x ) [1 3 ]. Here η ( x ) is a real function of x . The sin usoidal co ordinate η ( x ) discussed in this pap er has no λ -dep endence in con trast to the case s studied in [7]. Then e H acts on P n ( η ): e H ( λ ) P n ( η ( x ) ; λ ) = E n ( λ ) P n ( η ( x ) ; λ ) . (2.12) F or all the examples discus sed in this pap er, e H is lower triangular in the sp ecial ba sis 1 , η ( x ) , η ( x ) 2 , . . . , η ( x ) n , . . . , (2.13) spanned b y the sin usoidal co ordinate η ( x ) ( η ( x ) = x, x 2 , cos x ) [10, 11, 12, 13]: e H ( λ ) η ( x ) n = E n ( λ ) η ( x ) n + lo wer orders in η ( x ) . (2.14) 5 Shape in v ariance The fa ctorised Hamiltonian (2.3) has the dynamical symmetry called shap e invarianc e [8] if the fo llo wing relation holds: A ( λ ) A ( λ ) † = κ A ( λ + δ ) † A ( λ + δ ) + E 1 ( λ ) , (2.15) in which κ is a real p ositiv e parameter and δ denotes the shift of the parameters and E 1 ( λ ) is the eigen v alue o f the first excited state. This relation is satisfied by all the examples discusse d in this pap er. Shap e in v ariance means that the original Hamiltonian H ( λ ) and the asso c iate d Hamilto nia n A ( λ ) A ( λ ) † in Crum’s [9 ] sense (or the susy partner Hamiltonian in the so-called sup ersymmetric quan tum mec ha nics [4, 5]) ha v e the same shap e up to a m ultiplicativ e factor κ and an additiv e constant E 1 ( λ ). In terms of the p otential function V ( x ; λ ), t he ab o v e relation reads ex plicitly as V ( x − iγ 2 ; λ ) V ( x + iγ 2 ; λ ) ∗ = κ 2 V ( x ; λ + δ ) V ( x + iγ ; λ + δ ) ∗ , (2.16) V ( x + iγ 2 ; λ ) + V ( x + iγ 2 ; λ ) ∗ = κ  V ( x ; λ + δ ) + V ( x ; λ + δ ) ∗  − E 1 ( λ ) . (2.17) Among man y consequences of shape in v ariance, w e list the mos t s alient ones. All the eigen- v alues are generated b y E 1 ( λ ) and the c orresp onding eigenfunctions are g enerated from the kno wn form of the g round state eigenfunction φ 0 (2.7) together with the multiple action of the successiv e A † op erator [10, 11, 12 ]: E n ( λ ) = n − 1 X s =0 κ s E 1 ( λ + s δ ) , (2.18) φ n ( x ; λ ) ∝ A ( λ ) † A ( λ + δ ) † A ( λ + 2 δ ) † · · · A ( λ + ( n − 1) δ ) † φ 0 ( x ; λ + n δ ) . (2.19) The latter is related t o a Ro drigues type form ula for the eigenpolynomials. W e illustrate the shap e in v ariance and Crum’s sc heme in Fig.1 at the end of this section. The Hilb ert space b elonging to the Ha milto nian H ( λ ) is denoted as H λ . Closure relation Another imp o rtan t symmetry concept of exactly solv able quantum me- c hanics is the clos ur e r elation [13, 7]: [ H , [ H , η ] ] = η R 0 ( H ) + [ H , η ] R 1 ( H ) + R − 1 ( H ) . (2.20) Here η ( x ) is the sin usoidal co ordinate and R i ( H ) is a p olynomial in H . A t the classical mec hanics leve l, it is easy to see that the closure relatio n means that η ( x ) undergo es a 6 sin usoidal mo t io n with f r equency p R 0 ( E ). The closure relation (2.20) is satisfied b y all the examples discuss ed in this pap er a nd the explicit forms o f R i ( H ), i = − 1 , 0 , 1 and E n ( λ ) a re giv en in eac h subsection. The closure relation (2.2 0) enables us to express any m ultiple comm utator [ H , [ H , · · · , [ H , η ( x )] · · · ]] as a linear com binatio n of the op era t o rs η ( x ) and [ H , η ( x )] with co efficien ts dep ending on the Hamiltonian H only . As w e will see shortly , the exact Heisen b erg operator solution a nd t he annihilation/creation op erators are obtained as a conseq uence [13, 7]. Let us consider the closure relatio n (2.20) as an algebraic constraint on η ( x ) a nd the Hamiltonian, for giv en constan ts { r ( j ) i } . The l.h.s. consists of e 2 γ p , e γ p , 1 , e − γ p , e − 2 γ p , then R i can b e pa r a metrised as R 0 ( y ) = r (2) 0 y 2 + r (1) 0 y + r (0) 0 , R 1 ( y ) = r (1) 1 y + r (0) 1 , R − 1 ( y ) = r (2) − 1 y 2 + r (1) − 1 y + r (0) − 1 . (2.21) The s imilarity transformatio n of (2.20) [ e H , [ e H , η ] ] = η R 0 ( e H ) + [ e H , η ] R 1 ( e H ) + R − 1 ( e H ) (2.22) giv es rise to the follo wing five conditions: η ( x − 2 iγ ) − 2 η ( x − iγ ) + η ( x ) = r (2) 0 η ( x ) + r (2) − 1 + r (1) 1  η ( x − iγ ) − η ( x )  , (2.23) η ( x + 2 iγ ) − 2 η ( x + iγ ) + η ( x ) = r (2) 0 η ( x ) + r (2) − 1 + r (1) 1  η ( x + iγ ) − η ( x )  , (2.24)  η ( x − iγ ) − η ( x )  V ( x − iγ ) + V ( x + iγ ) ∗ − V ( x ) − V ( x ) ∗  = −  r (2) 0 η ( x ) + r (2) − 1  V ( x − iγ ) + V ( x + iγ ) ∗ + V ( x ) + V ( x ) ∗  − r (1) 1  η ( x − iγ ) − η ( x )  V ( x − iγ ) + V ( x + iγ ) ∗  + r (1) 0 η ( x ) + r (1) − 1 + r (0) 1  η ( x − iγ ) − η ( x )  , (2.25)  η ( x + iγ ) − η ( x )  V ( x − iγ ) ∗ + V ( x + iγ ) − V ( x ) ∗ − V ( x )  = −  r (2) 0 η ( x ) + r (2) − 1  V ( x − iγ ) ∗ + V ( x + iγ ) + V ( x ) ∗ + V ( x )  − r (1) 1  η ( x + iγ ) − η ( x )  V ( x − iγ ) ∗ + V ( x + iγ )  + r (1) 0 η ( x ) + r (1) − 1 + r (0) 1  η ( x + iγ ) − η ( x )  , (2.26) 2  η ( x ) − η ( x − i γ )  V ( x ) V ( x + iγ ) ∗ + 2  η ( x ) − η ( x + iγ )  V ( x ) ∗ V ( x + iγ ) =  r (2) 0 η ( x ) + r (2) − 1  V ( x ) V ( x + iγ ) ∗ + V ( x ) ∗ V ( x + iγ ) +  V ( x ) + V ( x ) ∗  2  + r (1) 1  η ( x − iγ ) − η ( x )  V ( x ) V ( x + iγ ) ∗ + r (1) 1  η ( x + iγ ) − η ( x )  V ( x ) ∗ V ( x + iγ ) −  r (1) 0 η ( x ) + r (1) − 1  V ( x ) + V ( x ) ∗  + r (0) 0 η ( x ) + r (0) − 1 . (2.27) 7 F or real { r ( j ) i } ( this is indeed the case for all the examples discussed in this pap er), (2.24) and (2.26) are the complex conjugate of (2.23) and (2.25), re sp ectiv ely . In con trast to the cases of t he ortho gonal polynomials with discrete measure s discussed in section 4 of [7], the determination of η ( x ) and the p ossible forms of V ( x ) is not s traight- forw ard due to the am biguities of perio dic functions with iγ p erio d. Here w e men tion only the basic results. It is easy t o see that (2.23)–( 2.26) require r (2) 0 = r (1) 1 and r (1) 0 = 2 r (0) 1 , whic h is consisten t with the hermitian conjuga t io n of (2.20). With these constrain ts, the first condition (2.23) reads with x → x + iγ η ( x − iγ ) − (2 + r (1) 1 ) η ( x ) + η ( x + iγ ) = r (2) − 1 . (2.28) F ollow ing the argumen ts give n in section 4 and app endix A of [7], we deduce from (2.25) and (2.27) the general relationship  η ( x − iγ ) − η ( x )  η ( x + iγ ) − η ( x )  ( V ( x ) + V ( x ) ∗ ) = − r (0) 1 η ( x ) 2 − r (1) − 1 η ( x ) − C 1 ( x ) , (2.29)  η ( x − 2 iγ ) − η ( x )  η ( x − iγ ) − η ( x + iγ )  V ( x ) V ( x + iγ ) ∗ =  r (0) 1 η ( x − iγ ) η ( x ) + r (1) − 1 η ( x − iγ ) + C 1 ( x )  r (0) 1 η ( x − iγ ) η ( x ) + r (1) − 1 η ( x ) + C 1 ( x )   η ( x − iγ ) − η ( x )  2 − r (0) 0 η ( x − iγ ) η ( x ) − r (0) − 1  η ( x − iγ ) + η ( x )  + C 2 ( x ) , (2.30) in whic h C j ( x ) ( j = 1 , 2) is an arbitra ry function satisfying the p erio dicit y C j ( x + iγ ) = C j ( x ). The hermiticit y of the Hamiltonian H w ould restrict C j ( x ) sev erely . F urther analysis of the closure re lation (2.23)–(2.27) will b e published elsewhe re. Lik e the cases o f discrete measures [7], the dual clos ure relation [ η , [ η , H ] ] = H R dual 0 ( η ) + [ η , H ] R dual 1 ( η ) + R dual − 1 ( η ) (2.31) holds and R dual i are giv en by R dual 1 ( η ( x )) =  η ( x − iγ ) − η ( x )  +  η ( x + iγ ) − η ( x )  , (2.32) R dual 0 ( η ( x )) = −  η ( x − iγ ) − η ( x )  η ( x + iγ ) − η ( x )  , (2.33) R dual − 1 ( η ( x )) =  V ( x ) + V ( x ) ∗  R dual 0 ( η ( x )) . (2.34) Eqs. (2.2 8) and (2.29) imply R dual 1 ( y ) = r (1) 1 y + r (2) − 1 and R dual − 1 ( η ( x )) = r (0) 1 η ( x ) 2 + r (1) − 1 η ( x ) + C 1 ( x ). 8 Auxiliary fun ction ϕ In all the ex amples discussed in this paper, the ground state w av e- function with shifted x and parameters φ 0 ( x − iγ 2 ; λ + δ ) is related to its original v alue φ 0 ( x ; λ ) via a re al auxiliary function ϕ : φ 0 ( x − iγ 2 ; λ + δ ) = p V ( x ; λ ) ϕ ( x − iγ 2 ) φ 0 ( x ; λ ) . (2.35) The aux iliary function ϕ ( x ) discus sed in this pap er has no λ -dep endence in con trast to the cases studie d in [7]. It is easy to see that (2.3 5) implie s (2 .8). The explicit fo rms of ϕ ( x ) are giv en at the b eginning of eac h section (3 .1), (4.1 ) , ( 5 .1). ‘Similarit y’ transformation I I ‘Similarit y’ transformed Hamiltonian or that of S ± , S † ± op erators (2.4) tak e simple r forms w ith the help of the auxiliary function ϕ (2.35): φ 0 ( x ; λ + δ ) − 1 ◦ S ± ( λ ) ◦ φ 0 ( x ; λ ) = ϕ ( x ) − 1 e ± γ p / 2 , (2.36) φ 0 ( x ; λ ) − 1 ◦ S ± ( λ ) † ◦ φ 0 ( x ; λ + δ ) = ( V ( x ; λ ) e γ p/ 2 ϕ ( x ) , V ( x ; λ ) ∗ e − γ p / 2 ϕ ( x ) . (2.37) Note t ha t the parameter shifts ± δ are prop erly incorp orated. F orw ard/Backw ard shift op erators With (2.36)–(2.37) the ‘s imilarity’ transformed A and A † op erators are obtained. They are called the forw ard/backw ard shift op erators: e H ( λ ) = B ( λ ) F ( λ ) , (2.38) F ( λ ) def = φ 0 ( x ; λ + δ ) − 1 ◦ A ( λ ) ◦ φ 0 ( x ; λ ) = i ϕ ( x ) − 1  e γ p/ 2 − e − γ p / 2  , (2.39) B ( λ ) def = φ 0 ( x ; λ ) − 1 ◦ A ( λ ) † ◦ φ 0 ( x ; λ + δ ) = − i  V ( x ; λ ) e γ p/ 2 − V ( x ; λ ) ∗ e − γ p / 2  ϕ ( x ) . (2.40) The action of the fo r ward shift op erator F ( λ ) and the backw ard shift op erator B ( λ ) on the p olynomial P n ( η ; λ ) are: F ( λ ) P n ( η ; λ ) = f n ( λ ) P n − 1 ( η ; λ + δ ) , (2.41) B ( λ ) P n ( η ; λ + δ ) = b n ( λ ) P n +1 ( η ; λ ) , (2.42) in which f n ( λ ) and b n ( λ ) are real constants related to E n ( λ ): f n ( λ ) b n − 1 ( λ ) = E n ( λ ) . (2.43) 9 F or the cases studied in [7] b n ( λ ) is actually indep enden t of n , but here it dep ends on n . In terms of the forw ard and bac kw ard shift op erators, the shap e in v ariance condition (2.15) reads F ( λ ) B ( λ ) = κ B ( λ + δ ) F ( λ + δ ) + E 1 ( λ ) . (2.44) Corresp onding to (2.19), a Ro drigues t yp e form ula for the eigenp olynomials is P n ( η ; λ ) = B ( λ ) b n − 1 ( λ ) B ( λ + δ ) b n − 2 ( λ + δ ) B ( λ + 2 δ ) b n − 3 ( λ + 2 δ ) · · · B ( λ + ( n − 1) δ ) b 0 ( λ + ( n − 1) δ ) · P 0 ( η ; λ + n δ ) , (2.45) where P 0 ( η ; λ + n δ ) = 1 f or all the ex amples giv en in this pap er. With these quan tities the action of A ( λ ) and A ( λ ) † on the eigenfunction φ n can b e simply express ed as A ( λ ) φ n ( x ; λ ) = f n ( λ ) φ n − 1 ( x ; λ + δ ) , (2.46) A ( λ ) † φ n ( x ; λ + δ ) = b n ( λ ) φ n +1 ( x ; λ ) . (2.47) Three term r ecurrence relation The p olynomial part of the eigenfunction P n ( η ) is an orthogonal p olynomial with the measure φ 0 ( x ) 2 . It satisfies three term recurrence relations [1, 2]. Let us first write the relation for the monic p olynomial P monic n ( η ) = η n + lo w er degree in η : P n ( η ) = c n P monic n ( η ) , (2.48) P monic n +1 ( η ) − ( η − a rec n ) P monic n ( η ) + b rec n P monic n − 1 ( η ) = 0 ( n ≥ 0) , (2.49) with P monic − 1 ( η ) = 0. F or P n ( η ) it reads η P n ( η ) = A n P n +1 ( η ) + B n P n ( η ) + C n P n − 1 ( η ) , (2.50) A n = c n c n +1 , B n = a rec n , C n = c n c n − 1 b rec n . (2.51) Sometimes w e write t he parameter dependence explicitly as P n ( η ) = P n ( η ; λ ), a rec n = a rec n ( λ ), b rec n = b rec n ( λ ), c n = c n ( λ ), A n = A n ( λ ), B n = B n ( λ ), C n = C n ( λ ), f n ( λ ) and b n ( λ ). They are giv en in eac h subsection. Heisen b erg op erator and Annihilation-Creation op erators The exact Heisen b erg op erator solution for η ( x ) is easily obtained [13] from the closure relation (2.20 ): e it H η ( x ) e − it H = a (+) e iα + ( H ) t + a ( − ) e iα − ( H ) t − R − 1 ( H ) R 0 ( H ) − 1 , (2.52) 10 α ± ( H ) def = 1 2  R 1 ( H ) ± p R 1 ( H ) 2 + 4 R 0 ( H )  , (2.53) R 1 ( H ) = α + ( H ) + α − ( H ) , R 0 ( H ) = − α + ( H ) α − ( H ) , (2.54) a ( ± ) def = ±  [ H , η ( x )] −  η ( x ) + R − 1 ( H ) R 0 ( H ) − 1  α ∓ ( H )   α + ( H ) − α − ( H )  − 1 (2.55) = ±  α + ( H ) − α − ( H )  − 1  [ H , η ( x )] + α ± ( H )  η ( x ) + R − 1 ( H ) R 0 ( H ) − 1   . (2.56) The po sitiv e/negative frequency parts of the Heisen b erg op era t o r s olution, a ( ± ) are the an- nihilation cre ation op erators a (+) † = a ( − ) , a (+) φ n ( x ) = A n φ n +1 ( x ) , a ( − ) φ n ( x ) = C n φ n − 1 ( x ) . (2.57) Since α ± ( E n ) = E n ± 1 − E n , (2.58) w e obtain a ( ± ) φ n ( x ) = ± 1 E n +1 − E n − 1  [ H , η ( x )] + ( E n − E n ∓ 1 ) η ( x ) + R − 1 ( E n ) E n ± 1 − E n  φ n ( x ) . (2.59) Comm ut ation relations of a ( ± ) and H Simple comm utation relations [ H , a ( ± ) ] = a ( ± ) α ± ( H ) (2.60) follo w from (2.55) and (2.20). When applied to φ n , we obtain with the help of (2.58), [ H , a ( ± ) ] φ n = ( E n ± 1 − E n ) a ( ± ) φ n . (2.61) Comm ut a tion relations o f a ( ± ) are expressed in terms of the co efficien ts of t he three term recurrence relation b y (2.57): a ( − ) a (+) φ n = A n C n +1 φ n = b rec n +1 φ n , a (+) a ( − ) φ n = C n A n − 1 φ n = b rec n φ n , (2.62) ⇒ [ a ( − ) , a (+) ] φ n = ( b rec n +1 − b rec n ) φ n . (2.63) These relation simply mean the op erato r relations a ( − ) a (+) = f ( H ) , (2.64) a (+) a ( − ) = g ( H ) , (2.65) in whic h f and g are ana lytic functions of H . In other words, H and a ( ± ) form a so- called quasi-linear algebra [3 0]. This is b ecause the definition of the annihilation/creatio n 11 op erators depend only on the closure relation (2 .20), without any other inputs. The situation is quite differen t from those of the wide v ariet y of prop osed a nnihilation/creation operat o rs for v arious quan tum systems [31], most of whic h we re introduced within the framework of ‘algebraic theory of coheren t states’. In all these cases there is no g ua r an tee for symmetry relations like (2.64), (2.65). In man y cases it is conv enien t to in tro duce the ‘n um b er op erator’ (or the ‘lev el op erator’) N N φ n def = nφ n . (2.66) F or t he following t yp es of energy spectra, the n um b er o p erator N can b e expresse d as a function o f the Hamiltonian H : E n = an ( a > 0) ⇒ N = a − 1 H , (2.67) E n = n ( n + b ) ( b > 0) ⇒ N = q H + 1 4 b 2 − 1 2 b, (2.68) E n = q − n − 1 ⇒ q N = ( H + 1) − 1 , (2.69) E n = ( q − n − 1)(1 − bq n ) (0 < b < 1) ⇒ q N = 1 2 b  H + b + 1 − p ( H + b + 1) 2 − 4 b  . (2.70) Ob viously the Hamiltonian is expressed as H = E N . Then (2.63) can b e expres sed s imply as [ a ( − ) , a (+) ] = b rec N +1 − b rec N (2.71) and (2.61) is rewritten as [ H , a ( ± ) ] = E N a ( ± ) − a ( ± ) E N = a ( ± ) ( E N ± 1 − E N ) . (2.72) With a deformed commutator [ A, B ] α def = AB − αB A, (2.73) w e ha v e [ a ( − ) , a (+) ] α = b rec N +1 − αb rec N . (2.74) Orthogonalit y and normalisation The scalar pro duct for the elemen ts of the Hilb ert space b elonging to the Hamilto nian H is ( g , f ) def = Z dx g ( x ) ∗ f ( x ) , (2.75) 12 in which the in tegration range dep ends on the specific Hamiltonian or the p olynomial. The orthogonality of the eigen ve ctors { φ n ( x ) } , φ n ( x ) = φ 0 ( x ) P n ( η ( x )) is: ( φ n , φ m ) = Z dx φ 0 ( x ; λ ) 2 P n ( η ( x ) ; λ ) ∗ P m ( η ( x ) ; λ ) = h n ( λ ) δ nm , (2.76) in which h n ( λ ) > 0 . The c onstants h n , c n and b rec n are related as b rec n = c 2 n − 1 c 2 n h n h n − 1 ( n ≥ 1) , h n = h 0 c 2 n n Y j =1 b rec n ( n ≥ 0) . (2.77) Let us denote the n -th normalised e igenfunction as ˆ φ n ( x ; λ ) = N n ( λ ) P n ( η ( x ) ; λ ) ˆ φ 0 ( x ; λ ) , ˆ φ 0 ( x ; λ ) = φ 0 ( x ; λ ) p h 0 ( λ ) , N n ( λ ) = s h 0 ( λ ) h n ( λ ) . (2.7 8 ) These normalisation constan ts are giv en for eac h polynomial. Coheren t states There are man y different nonequiv alen t definitions of coheren t states. Here w e adopt the most conv en tional one, as the eigen v ector of the annihilation op erator a ( − ) , ( 2 .57): a ( − ) ψ ( α , x ) = αψ ( α, x ) , α ∈ C . (2.79) It is ex pressed in terms of the co efficien t C n of the three term recurrence relation (2.50) and (2.51) as [13] ψ ( α , x ) = ψ ( α, x ; λ ) = φ 0 ( x ; λ ) ∞ X n =0 α n Q n k =1 C k P n ( η ( x ) ; λ ) . (2.80) Th us w e obtain one new coheren t state for each p olynomial; (3 .19), (3.39), (4.16), (4.37), (5.20), (5.51), (5.71), (5.91), (5.118), (5.1 37) and (5.158). If the sum on the r.h.s. is ex pressed b y a simple function, it is a g enerating function of the p olynomial P n ( η ). In most explicit examples to b e discussed in later sections, the p oten tial functions, the Hamiltonians and th us the p olynomials themselv es are totally symmetric in the parameters, see f or example, the Ask ey-Wilson p olynomial § 5.1. The abov e coheren t state, b eing totally symmetric, giv es the b est candidate for a sy mmetric generating function. F or the p olynomials to b e discussed in later sections, how ev er, most of t he known generating functions are not totally symmetric. 13 λ -shift operators Let us fix an orthonormal basis { ˆ φ n ( x ; λ ) } and define a unitary op erator U ( U † ) as U ˆ φ n ( x ; λ ) def = ˆ φ n ( x ; λ + δ ) , U † ˆ φ n ( x ; λ + δ ) = ˆ φ n ( x ; λ ) . (2.81) Then we can d efine another set of annihilation-creation op erators ˆ a , ˆ a † : ˆ a def = U † A , ˆ a † = A † U . (2.82) They satisfy H = ˆ a † ˆ a and their action on φ n are deriv ed from (2.46) and (2.47), ˆ aφ n ( x ; λ ) ∝ φ n − 1 ( x ; λ ), ˆ a † φ n ( x ; λ ) ∝ φ n +1 ( x ; λ ). Although this kind of annihilation-creation op erator s ha v e b een considered in man y literatur e [31], it should b e stressed that they are formal b ecause U and U † are formal op erators. On the ot her hand, a ( ± ) obtained from t he Heisen b erg solution are explicitly expressed in terms of difference op erators (differential op erato rs, in ordinary quan tum mec hanics), (2.55). Note that the construction metho d of ˆ a and ˆ a † is based on the shap e inv ariance but that of a ( ± ) is not. The latter is ba sed on the closure relation. The ke y p oint of the construction of ˆ a and ˆ a † is t he prop er shift o f the par a meters λ , whic h is achiev ed by the formal op erators U a nd U † . W e introduce another set of λ -shift op erators X and X † explicitly in terms of difference op erators through the follow ing relations: a (+) = A † X , a ( − ) = X † A . (2.83) By using the shap e in v ariance (2.15), w e ha ve A a (+) = AA † X =  κ A ( λ + δ ) † A ( λ + δ ) + E 1  X =  κ H ( λ + δ ) + E 1  X . (2.84) Since κ H ( λ + δ ) + E 1 is a p ositiv e opera t or, we obtain X =  κ H ( λ + δ ) + E 1  − 1 A a (+) =  κ H ( λ + δ ) + E 1  − 1 A ×  [ H , η ( x )] −  η ( x ) + R − 1 ( H ) R 0 ( H ) − 1  α − ( H )   α + ( H ) − α − ( H )  − 1 . (2.85) Similarly X † is expressed as X † = a ( − ) A †  κ H ( λ + δ ) + E 1  − 1 . (2.86) Their action on φ n are X φ n ( x ; λ ) = A n ( λ ) b n ( λ ) φ n ( x ; λ + δ ) , (2.87) 14 (v) (vi) (iv) (iii) (ii) (i) (i): a ( − ) ( λ ), ˆ a ( λ ) (ii): a (+) ( λ ), ˆ a ( λ ) † (iii): A ( λ ) (iv): A ( λ ) † (v): X ( λ ), U ( λ ) (vi): X ( λ ) † , U ( λ ) † E 0 E 1 E 2 E 3 . . . φ 1 φ 2 φ 3 φ 0 ( x ; λ ) φ 0 ( x ; λ + δ ) φ 0 ( x ; λ +2 δ ) φ 0 ( x ; λ +3 δ ) H ( λ ) H ( λ + δ ) H ( λ +2 δ ) H ( λ +3 δ ) . . . H λ H λ + δ H λ +2 δ H λ +3 δ . . . Figure 1: Shap e inv ariance and Crum’s sc heme. X † φ n ( x ; λ + δ ) = C n +1 ( λ ) f n +1 ( λ ) φ n ( x ; λ ) , (2.88) and the λ -shift without c hanging the lev el n is ac hiev ed, as exp ected. The λ -shift operat o rs for the p olynomials P n ( η ( x ) ; λ ) are giv en b y φ 0 ( x ; λ + δ ) − 1 ◦ X ◦ φ 0 ( x ; λ ) and φ 0 ( x ; λ ) − 1 ◦ X † ◦ φ 0 ( x ; λ + δ ). The expression of X and X † ma y b e simplified for some particular cases, see § 3.2 , § 4.2, § 5.5. Finally w e illustrate the shap e inv ariance and Crum’s sc heme in Fig.1. The Hilb ert space b elonging to the Hamiltonian H ( λ ) is denoted as H λ . The action of v arious op erat o rs and their doma ins and images are also illustrated in Fig.1: H ( λ ) , a ( ± ) ( λ ) , ˆ a ( λ ) , ˆ a ( λ ) † : H λ → H λ , (2.89) A ( λ ) , X ( λ ) , U ( λ ) : H λ → H λ + δ , (2.90) A ( λ ) † , X ( λ ) † , U ( λ ) † : H λ + δ → H λ . (2.91) 15 3 η ( x ) = x F rom this section to section 5, we presen t v ario us formu las a nd results sp ecific to each example of the exactly solv able ‘discrete’ quantum mec hanics. These examples are divided in to three gro ups according to the form of the sinus oidal coordinat e; η ( x ) = x in this se ction, η ( x ) = x 2 in section 4, η ( x ) = cos x in section 5 . The na mes of the subsections are taken from the name of the corresp onding o rthogonal p olynomial and the n um b er, for example, [KS1.4] indicates the corresp onding subs ection of the revie w of Ko ek o ek and Swarttou w [6 ]. In all the examples in this s ection, w e ha v e η ( x ) = x, −∞ < x < ∞ , γ = 1 , κ = 1 , ϕ ( x ) = 1 . (3.1) 3.1 con tin uous Hahn [KS1.4] In previous w o r ks [10, 11, 12, 13], the parameters a 1 and a 2 w ere res tricted to real, po sitiv e v alues. Now they are complex w ith po sitive real parts. parameters and p oten tial functions λ def = ( a 1 , a 2 ) , δ = ( 1 2 , 1 2 ); Re a i > 0; V ( x ; λ ) def = ( a 1 + ix )( a 2 + ix ) . (3.2) shape in v ariance and closure relation E n ( λ ) = n ( n + b 1 − 1) , (3.3) R 1 ( y ) = 2 , R 0 ( y ) = 4 y + b 1 ( b 1 − 2) , (3.4) R − 1 ( y ) = − i ( a 1 + a 2 − a 3 − a 4 ) y − i ( b 1 − 2)( a 1 a 2 − a 3 a 4 ) , (3.5) b 1 def = 4 X j =1 a j , ( a 3 , a 4 ) def = ( a ∗ 1 , a ∗ 2 ) or ( a ∗ 2 , a ∗ 1 ) . (3.6) These can be rewritten a s E n ( λ ) = n ( n + 2Re( a 1 + a 2 ) − 1) , (3.7) R 0 ( y ) = 4 y + 4Re( a 1 + a 2 )  Re( a 1 + a 2 ) − 1  , (3.8) R − 1 ( y ) = 2Im( a 1 + a 2 ) y + 4  Re( a 1 + a 2 ) − 1  Im( a 1 a 2 ) . (3.9) 16 eigenfunctions φ 0 ( x ; λ ) def = | Γ( a 1 + ix )Γ( a 2 + ix ) | = p Γ( a 1 + ix )Γ( a 2 + ix )Γ( a 3 − ix )Γ( a 4 − ix ) , (3.10) P n ( η ; λ ) = p n ( x ; a 1 , a 2 , a 3 , a 4 ) def = i n ( a 1 + a 3 ) n ( a 1 + a 4 ) n n ! 3 F 2  − n, n + a 1 + a 2 + a 3 + a 4 − 1 , a 1 + ix a 1 + a 3 , a 1 + a 4    1  , (3.11) whic h are sy mmetric under a 1 ↔ a 2 and a 3 ↔ a 4 separately . c n = ( n + b 1 − 1) n n ! , (3.12) a rec n = i  a 1 − ( n + b 1 − 1)( n + a 1 + a 3 )( n + a 1 + a 4 ) (2 n + b 1 − 1)(2 n + b 1 ) + n ( n + a 2 + a 3 − 1)( n + a 2 + a 4 − 1) (2 n + b 1 − 2)(2 n + b 1 − 1)  , (3.13) b rec n = n ( n + b 1 − 2) Q 2 j =1 Q 4 k =3 ( n + a j + a k − 1) (2 n + b 1 − 3)(2 n + b 1 − 2) 2 (2 n + b 1 − 1) , (3.14) f n ( λ ) = n + b 1 − 1 , b n ( λ ) = n + 1 . (3.15) annihilation/creation op erators and commutation relations α ± ( H ) = 1 ± 2 √ H ′ , H ′ def = H + 1 4 ( b 1 − 1) 2 , (3.16) N = √ H ′ − 1 2 ( b 1 − 1) (for b 1 > 1) , (3.17) [ H , a ( ± ) ] = a ( ± ) (1 ± 2 √ H ′ ) . (3.18) The annihilation/creation op erators (2.55) and t heir comm uta tion relation (2.63) are not so simplifi ed b ecause b rec n +1 − b rec n = (quartic p olynomial in n ) / (cubic p olynomial in n ) has a length y expre ssion. coheren t state ψ ( α , x ; λ ) = φ 0 ( x ; λ ) ∞ X n =0 ( b 1 ) 2 n α n Q 2 j =1 Q 4 k =3 ( a j + a k ) n P n ( η ( x ) ; λ ) . (3.19) The r.h.s is symmetric under a 1 ↔ a 2 and a 3 ↔ a 4 separately . W e a re not aw are if a concise summation fo rm ula exists or no t . Seve ral non-symmetric generating functions for the contin uous Hahn p olynomial are giv en in [6]. 17 orthogonalit y Z ∞ −∞ φ 0 ( x ; λ ) 2 P n ( η ; λ ) P m ( η ; λ ) dx = 2 π Q 2 j =1 Q 4 k =3 Γ( n + a j + a k ) n ! (2 n + b 1 − 1)Γ( n + b 1 − 1) δ nm , (3.20) 1 h 0 ( λ ) = Γ( b 1 ) 2 π Q 2 j =1 Q 4 k =3 Γ( a j + a k ) , h 0 ( λ ) h n ( λ ) = b 1 + 2 n − 1 b 1 + n − 1 n ! ( b 1 ) n Q 2 j =1 Q 4 k =3 ( a j + a k ) n . (3.21) 3.2 Meixner-P ollaczek [KS1.7] In previous works [10, 13, 32], the pa r ameter φ w as fixed to π / 2. Here w e treat the most general case 0 < φ < π . parameters and p oten tial function λ def = a, δ = 1 2 , φ (0 < φ < π ); a > 0; V ( x ; λ ) def = e i ( π 2 − φ ) ( a + ix ) . (3.22) shape in v ariance and closure relation E n ( λ ) = 2 n sin φ, (3.23) R 1 ( y ) = 0 , R 0 ( y ) = 4 sin 2 φ, R − 1 ( y ) = 2 y c os φ + 2 a sin 2 φ. (3.24) eigenfunctions φ 0 ( x ; λ ) def = e ( φ − π 2 ) x | Γ( a + i x ) | , (3.25) P n ( η ; λ ) = P ( a ) n ( x ; φ ) def = (2 a ) n n ! e inφ 2 F 1  − n, a + ix 2 a    1 − e − 2 iφ  , (3.26) c n = (2 sin φ ) n n ! a rec n = − n + a tan φ , b rec n = n ( n + 2 a − 1) (2 sin φ ) 2 , (3.27) f n ( λ ) = 2 sin φ, b n ( λ ) = n + 1 . (3.28) The polynomial has the following symmetry P ( a ) n ( x ; − φ ) = P ( a ) n ( − x ; φ ). annihilation/creation op erators and commutation relations α ± ( H ) = ± 2 sin φ, N = 1 2 sin φ H , (3.29) a ( ± ) = ± 1 4 sin φ [ H , η ] + 1 2 η + cos φ 4 sin 2 φ ( H + 2 a sin φ ) , (3.30) b rec n +1 − b rec n = n + a 2 sin 2 φ , (3.31) 18 [ H , a ( ± ) ] = ± 2 sin φ a ( ± ) , (3.32) [ a ( − ) , a (+) ] = 1 4 sin 3 φ ( H + 2 a sin φ ) . (3.33) su (1 , 1) algebra : J ± = 2 sin φ a ( ± ) , J 3 = 1 2 sin φ ( H + 2 a sin φ ) , [ J 3 , J ± ] = ± J ± , [ J − , J + ] = 2 J 3 . (3.34) The s u (1 , 1) or sl (2 , R ) algebra report ed b efore [13 , 32] is a sp ecial case of the presen t one. λ -shift operators F o r the sp ecial case of φ = π / 2 the annihilation/creation op erato rs a r e closely related to the A and A † op erators: a (+) = A † X , X = 1 4 ( S + + S − ) , (3.35) a ( − ) = X † A , X † = 1 4 ( S † + + S † − ) , (3.36) φ 0 ( x ; λ + δ ) − 1 X ( λ ) φ 0 ( x ; λ ) · P n ( η ; λ ) = 1 2 P n ( η ; λ + δ ) , (3.37) φ 0 ( x ; λ ) − 1 X ( λ ) † φ 0 ( x ; λ + δ ) · P n ( η ; λ + δ ) = 1 4 ( n + 2 a ) P n ( η ; λ ) . (3.38) coheren t st ate The coheren t state give s a simple generating f unction, whic h generalises the previous result [1 3]: ψ ( α , x ; λ ) = φ 0 ( x ; λ ) ∞ X n =0 (2 sin φ ) n α n (2 a ) n P n ( η ( x ) ; λ ) = φ 0 ( x ; λ ) e iα (1 − e 2 iφ ) 1 F 1  a + ix 2 a    − 4 iα sin 2 φ  . (3.39) orthogonalit y Z ∞ −∞ φ 0 ( x ; λ ) 2 P n ( η ; λ ) P m ( η ; λ ) dx = 2 π Γ( n + 2 a ) n ! (2 sin φ ) 2 a δ nm , (3.40) 1 h 0 ( λ ) = (2 sin φ ) 2 a 2 π Γ(2 a ) , h 0 ( λ ) h n ( λ ) = n ! (2 a ) n . (3.41) The exact solv a bilit y of the con tinuous Hahn and Meixne r-Pollaczek p olynomials for the full parameters are discuss ed in [27] in connection with their further deformation to give another e xample of quasi e xactly solv able system. 19 4 η ( x ) = x 2 In all the examples in this s ection, w e ha v e η ( x ) = x 2 , 0 < x < ∞ , γ = 1 , κ = 1 , ϕ ( x ) = 2 x. (4.1) 4.1 Wilson [KS1.1] The Wilson p olynomial is the most general one in this category . The parameters a 1 ,. . . , a 4 w ere restricted to real p ositive v alues in previous w orks [10, 11, 12, 13 ]. The generic situation to b e discussed in this paper is { a ∗ 1 , a ∗ 2 , a ∗ 3 , a ∗ 4 } = { a 1 , a 2 , a 3 , a 4 } (as a s et ) , Re a i > 0 (1 ≤ i ≤ 4) . (4.2) parameters and p oten tial function λ def = ( a 1 , a 2 , a 3 , a 4 ) , δ = ( 1 2 , 1 2 , 1 2 , 1 2 ); V ( x ; λ ) def = ( a 1 + ix )( a 2 + ix )( a 3 + ix )( a 4 + ix ) 2 ix (2 ix + 1 ) . (4 .3) shape in v ariance and closure relation E n ( λ ) = n ( n + b 1 − 1) , (4.4) R 1 ( y ) = 2 , R 0 ( y ) = 4 y + b 1 ( b 1 − 2) , R − 1 ( y ) = − 2 y 2 + ( b 1 − 2 b 2 ) y + (2 − b 1 ) b 3 , (4.5) b 1 def = 4 X j =1 a j , b 2 def = X 1 ≤ j 1) , (4.14) [ H , a ( ± ) ] = a ( ± ) (1 ± 2 √ H ′ ) . (4.15) The annihilation/creation op erators (2.55) and their comm utatio n relation (2.63) are not so simplified b ecause the expression b rec n +1 − b rec n = (10-th de gree po lynomial in n ) / (7 -th degree p olynomial in n ) is quite complicated. coheren t state ψ ( α , x ; λ ) = φ 0 ( x ; λ ) ∞ X n =0 ( − 1) n ( b 1 ) 2 n α n n ! Q 1 ≤ j 0. This is dual to the con tin uous Hahn § 3.1 in the sense that the roles of η ( x ) and E n are in terc hanged. F or the con tin uous Hahn, η ( x ) = x and E n is quadratic in n , whereas η ( x ) is quadratic in x and E n = n for the dual Hahn. The dualit y 21 has sharper meaning for p olynomials with discrete orthogona lity measures, see for example [7]. parameters and p oten tial function λ def = ( a 1 , a 2 , a 3 ) , δ = ( 1 2 , 1 2 , 1 2 ); V ( x ; λ ) def = ( a 1 + ix )( a 2 + ix )( a 3 + ix ) 2 ix (2 ix + 1 ) . (4.19) shape in v ariance and closure relation E n ( λ ) = n, (4.20) R 1 ( y ) = 0 , R 0 ( y ) = 1 , R − 1 ( y ) = − 2 y 2 + (1 − 2 b 1 ) y − b 2 , (4.21) b 1 def = a 1 + a 2 + a 3 , b 2 def = a 1 a 2 + a 1 a 3 + a 2 a 3 . (4.22) eigenfunctions φ 0 ( x ; λ ) def =     Q 3 j =1 Γ( a j + ix ) Γ(2 ix )     , (4.23) P n ( η ; λ ) = S n ( x 2 ; a 1 , a 2 , a 3 ) def = ( a 1 + a 2 ) n ( a 1 + a 3 ) n 3 F 2  − n, a 1 + ix, a 1 − ix a 1 + a 2 , a 1 + a 3    1  , (4.24) whic h are sy mmetric under the p ermutations of ( a 1 , a 2 , a 3 ). c n = ( − 1) n , (4.25) a rec n = ( n + a 1 + a 2 )( n + a 1 + a 3 ) + n ( n + a 2 + a 3 − 1) − a 2 1 , (4.26) b rec n = n Y 1 ≤ j 0 , (5.162) in which the sup erscript ASC denotes the quan tit y of the Al-Salam-Chihara p olynomial. These tw o p olynomials are differen t only b y a m ultiplicativ e constan t: P ASC n ( η ) ( q ; q ) n = P q MP n ( η ) . (5.163 ) 6 Summary and C ommen t s Kno wn examples of exactly solv able ‘discrete’ quan tum mec hanics of o ne degree of free- dom are discussed in detail and in full generalit y . The shap e in v a riance property , the exact solutions in the Sc hr¨ odinger and Heisen b erg pictures, t he annihilation/creation op erators together with their symmetry algebra, the coheren t state as the eigen ve ctor of the anni- hilation op erator, the ground state w a v efunction giving the orthogo na lit y measure of the eigenp olynomial are giv en explicitly for eac h s ystem, whic h is named after the corresp onding orthogonal p olynomial. The presen t pap er supplem en ts the earlier results [10, 11, 12 , 13, 7]. The main fo cus is the p olynomials obtained b y res tricting the Ask ey-Wilson p o lynomials. In general, they hav e simple and tractable symmetry algebras, some of t hem are the q -o scillator algebra [14]. Another main featur e is the coheren t states. As many as elev en new and ex- act coheren t states are presen ted (3.19), (3.39), (4.16), (4.37 ) , (5.20), (5.51), (5.71), (5 .91), (5.118), (5.137), (5.15 8) a s the eigen v ectors of the annihilation op erators for the ‘discrete’ quan tum mec hanical systems. These coheren t states are by construction totally symmetric in the symmetric parameters of the Hamiltonians. In other w o r ds, they realise the dynamically 38 fa v ourable generating functions of the e igenp o lynomials. Lik e the standard c oheren t state of the harmonic oscillator, these new coherent s tates are ex p ected to find v a rious applications in man y branc hes of ph ysical sciences, in particular, quantum optics a nd quan tum informa- tion. It w ould b e in teresting to in ve stigate if and to what exten t these new coheren t states share the remark able prop erties of the standard coheren t state of the harmonic oscillator. One in teresting future task is to solv e the closure relation (2 .2 3)–(2.27) algebraically to determine all the p ossible for ms of the sin usoidal coordinate η ( x ) and the potential function V ( x ). F or the ordinary quan tum mec hanics and for the orthogo na l polynomials of discrete measures, this task was done in App endix A of [13] and App endix A of [7]. The presen t case is more complicated than these due to the prese nce of arbitrary p erio dic functions w ith p erio d iγ . It is in teresting to see if difference equation v ersions of the soliton p oten tial, i.e. 1 / cosh 2 x p oten tial in o rdinary quan tum mec hanics, ( see, for example, § 3.1.3 of [1 3]) with η ( x ) = sinh x , and the Morse p oten tial with η ( x ) = e − x (see § 3.1.4 of [13]) are con t ained as solutions or not. Ac kno wledge ments This work is supp orted in part b y Grants -in-Aid for Scien tific R esearc h from the Ministry of Education, Culture, Sp o rts, Science and T ec hnology , No.18340061 and No.1954017 9. App endix A: Diagrammatic pro of of the her mit i c it y of the Hamiltonian Here w e give a diagrammatic pro o f of the hermiticit y (self-adjoin tness) of the Hamiltonian (2.1) for the three differen t cases of t he sinus oidal co ordinates corr esp o nding to sections 3 – 5. A less detailed pro o f of the hermiticity can b e found in [2 6]. The hermiticity or self- adjoin tness of the Hamilto nia n H means ( g , H f ) = ( H g , f ) for a giv en inner pro duct ( g , f ) (2.75) for arbitrary elemen ts f and g of the appro priate Hilb ert space. It is necessary and sufficien t to sh ow that in a certain dense subspace of the Hilbert space. The o bvious ch oice for suc h a s ubspace is s panned b y the ground state w av efunction φ 0 , whic h is giv en in eac h subsection (3.10), (3.25), (4.7), (4.23), (5.11), (5.38), (5.59), (5.7 9), (5.97), (5.1 2 7), (5.146), times the eigenp o lynomials P n ( η ( x )). The types of the polynomials are: ( a ) : polynomials in η ( x ) = x for the Ha miltonians in section 3 , 39 ( g , f ) = Z ∞ −∞ g ( x ) ∗ f ( x ) d x, f ( x ) = φ 0 ( x ) P ( x ) , g ( x ) = φ 0 ( x ) Q ( x ) , (A.1) ( b ) : p olynomials in η ( x ) = x 2 for the Hamiltonians in section 4 , ( g , f ) = Z ∞ 0 g ( x ) ∗ f ( x ) d x, f ( x ) = φ 0 ( x ) P ( x 2 ) , g ( x ) = φ 0 ( x ) Q ( x 2 ) , (A.2) ( c ) : polynomials in η ( x ) = cos x for the Hamilto nia ns in section 5 , ( g , f ) = Z π 0 g ( x ) ∗ f ( x ) d x, f ( x ) = φ 0 ( x ) P (cos x ) , g ( x ) = φ 0 ( x ) Q (cos x ) . (A.3) This clearly remo v es the non-uniqueness of the eigenfunctions, whic h was men tioned in section tw o. F or the Hamiltonian (2.3) H = T + + T − − V ( x ) − V ( x ) ∗ , it is ob vious that the function pa rt − V ( x ) − V ( x ) ∗ is hermitian b y itse lf. When T + = p V ( x ) e γ p p V ( x ) ∗ acts on f , the argumen t o f f is shifted fro m x to x − iγ . With the comp ensating c ha nge of inte grat io n v ariable from x to x + iγ one can formally show ( g , T + f ) = ( T + g , f ) in a straightforw a r d w a y . Similarly w e ha v e ( g , T − f ) = ( T − g , f ) by another change of in tegration v ariable x to x − iγ . This is the ‘formal hermiticit y .’ In realit y , the shift of in tegr a tion v ariable, to b e realised b y the Cauc h y integral, w ould in v olv e additional in tegratio n con tours: ( a ) : ( − ∞ , ± i − ∞ ) , (+ ∞ , ± i + ∞ ) for the Hamiltonians in section 3 , (A.4) ( b ) : ( 0 , ± i ) , (+ ∞ , ± i + ∞ ) for the Hamiltonians in section 4 , (A.5) ( c ) : (0 , ± i log q ) , ( π , π ± i log q ) f or the Hamiltonians in sec tion 5 . (A.6) It is easy to v erify that all the singularities ar ising fr o m V and V ∗ in cases ( b ) and ( c ) are cancelled b y the zeros coming from the ground state wa v efunctions φ 0 and φ ∗ 0 , and the Cauc hy in tegr a tion form ula applies in all cases. As can be seen from the diag rams in Fig.2 the con tribution of the additiona l con tour in tegrals (A.4)–(A.6) cancel with eac h other and the shifts of integration v ariables are justified and the he rmiticit y is established. First, the con tribution from the con tours at infinit y in ( a ) v a nish iden tically due to the strong damping b y φ 0 and φ ∗ 0 , see (3.10) and (3.25). This establis hes the hermiticit y in the case ( a ). Next let us discuss the case ( b ) in detail. In this case γ = 1. The in tegrand of ( g , T ± f ) are g ∗ T + f = φ 0 ( x ) ∗ Q ( x 2 ) ∗ p V ( x ) p V ( x + i ) ∗ φ 0 ( x − i ) P (( x − i ) 2 ) def = F ( x ) , (A.7) g ∗ T − f = φ 0 ( x ) ∗ Q ( x 2 ) ∗ p V ( x ) ∗ p V ( x + i ) φ 0 ( x + i ) P (( x + i ) 2 ) def = G ( x ) . ( A.8) 40 0 iγ C ′ 2 − R (iii) 0 iγ C 1 R (i) 0 iγ C 1 + C ′ 2 − R R (iv) 0 − iγ C 2 R (ii) Figure 2: Integration con tours in complex x plane. The endpo in t R = ∞ for cases ( a ) and ( b ), R = π for case ( c ). (F or case ( c ), iγ is in the lo we r half plane b ecause of γ = log q < 0 .) Due to the ev enness of the e igenfunctions, φ 0 ( − x ) = φ 0 ( x ), P (( − x ) 2 ) = P ( x 2 ), Q (( − x ) 2 ) = Q ( x 2 ) and V ( x ) ∗ = V ( − x ∗ ), w e ha v e G ( x ) = φ 0 ( − x ) ∗ Q (( − x ) 2 ) ∗ p V ( − x ) p V ( − x + i ) ∗ φ 0 ( − x − i ) P (( − x − i ) 2 ) = F ( − x ) . ( A.9 ) On the other hand, the in tegrand of ( T ± g , f ) are ( T + g ) ∗ f = p V ( x ) ∗ p V ( x + i ) φ 0 ( x − i ) ∗ Q (( x − i ) 2 ) ∗ φ 0 ( x ) P ( x 2 ) = F ( x + i ) , (A.10) ( T − g ) ∗ f = p V ( x ) p V ( x + i ) ∗ φ 0 ( x + i ) ∗ Q (( x + i ) 2 ) ∗ φ 0 ( x ) P ( x 2 ) = G ( x − i ) = F ( − x + i ) , (A.11) in whic h (A.9) is used for the last equalit y . Since the in tegrands are analytic in x and there is no p ole within the contours, see Fig.2 , w e ha v e I C 1 F ( x ) dx = 0 , I C 2 G ( x ) dx = I C 2 F ( − x ) dx = I C ′ 2 F ( x ) dx = 0 . (A.12) Com bining them, w e obtain 0 = I C 1 F ( x ) dx + I C ′ 2 F ( x ) dx = I C 1 + C ′ 2 F ( x ) dx = Z ∞ −∞ F ( x ) dx − Z ∞ −∞ F ( x + i ) dx + Z ↑ at+ ∞ F ( x ) dx + Z ↓ at − ∞ F ( x ) dx. (A.13) The con tribution from the contours at infinity in the case ( b ) v anish iden tically due to the strong da mping b y φ 0 and φ ∗ 0 , see (4.7) and (4.2 3). Th us (A.13) implies R ∞ −∞ F ( x ) dx = 41 R ∞ −∞ F ( x + i ) dx . The l.h.s . is Z ∞ 0 F ( x ) dx + Z ∞ 0 G ( x ) dx = ( g , T + f ) + ( g , T − f ) . (A.14) The r.h.s . is Z ∞ 0 F ( x + i ) dx + Z ∞ 0 G ( x − i ) d x = ( T + g , f ) + ( T − g , f ) . (A.15) Th us the hermiticit y of the Hamiltonians for the case ( b ) is pro ved. The hermiticit y of the Hamiltonians for the case ( c ) is pro v ed in a similar w ay together with the ev enness and the 2 π p erio dicity of the ground state w a v efunction φ 0 ( x ), the sin usoidal co ordinate η ( x ) = cos x and the p otential function V ( x ); φ 0 ( − x ) = φ 0 ( x ), η ( − x ) = η ( x ), V ( x ) ∗ = V ( − x ∗ ), φ 0 ( x + 2 π ) = φ ( x ), η ( x + 2 π ) = η ( x ), V ( x + 2 π ) = V ( x ). App endix B: Some defin itions related to the hyp ergeo- metric and q -hyp ergeo met ric functions F or self-con tainedness w e collect sev eral definitions r elat ed to the ( q -)hypergeometric func- tions [6 ]. ◦ P o c hhammer sym b ol ( a ) n : ( a ) n def = n Y k =1 ( a + k − 1) = a ( a + 1) · · · ( a + n − 1) = Γ( a + n ) Γ( a ) . (B.1) ◦ q -P o c hhammer s ym b ol ( a ; q ) n : ( a ; q ) n def = n Y k =1 (1 − aq k − 1 ) = (1 − a )(1 − aq ) · · · (1 − aq n − 1 ) . (B.2) ◦ hy p ergeometric series r F s : r F s  a 1 , · · · , a r b 1 , · · · , b s    z  def = ∞ X n =0 ( a 1 , · · · , a r ) n ( b 1 , · · · , b s ) n z n n ! , (B.3) where ( a 1 , · · · , a r ) n def = Q r j =1 ( a j ) n = ( a 1 ) n · · · ( a r ) n . ◦ q -h yp ergeometric series (the basic hypergeometric series ) r φ s : r φ s  a 1 , · · · , a r b 1 , · · · , b s    q ; z  def = ∞ X n =0 ( a 1 , · · · , a r ; q ) n ( b 1 , · · · , b s ; q ) n ( − 1) (1+ s − r ) n q (1+ s − r ) n ( n − 1) / 2 z n ( q ; q ) n , (B.4) where ( a 1 , · · · , a r ; q ) n def = Q r j =1 ( a j ; q ) n = ( a 1 ; q ) n · · · ( a r ; q ) n . ◦ q -gamma function Γ q ( z ): Γ q ( z ) def = ( q ; q ) ∞ ( q z ; q ) ∞ (1 − q ) 1 − z , lim q ր 1 Γ q ( z ) = Γ( z ) . (B.5) 42 References [1] G. E. Andrews, R. Ask ey and R. R o y , Sp e cial F unctions , Encyclop edia of mathematics and its applications, Cambridge, (199 9 ). [2] M. E. H. 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