Unified theory of exactly and quasi-exactly solvable `Discrete quantum mechanics: I. Formalism

Yukawa Institute Kyoto DPSU-09-2 YITP-09- 14 Ma rch 2009 Unified theory of exactly and quasi-exactly solv able ‘Discre t e’ quan tum mec hanics: I. F ormalism Satoru Odak e a and Ryu Sasaki b a Departmen t of Ph ys ics, Shinsh u Univ ersit y , Matsumoto 390-8 6 21, Japan b Y uk aw a Institute for Theoretical Phy sics, Ky ot o Univ ersity , Ky o to 606- 8502, Japan Abstract W e present a sim p le recip e to construct exa ctly and quasi-exactly solv able Hamilto- nians in one-dimensional ‘discrete’ quan tum mechanics, in which the S c h r¨ odinger equa- tion is a d ifference equation. It repr o duces all the kn own ones w hose eigenfunctions consist of th e Ask ey sc h eme of hypergeometric orthogonal p olynomials of a con tin u ous or a discrete v ariable. The r ecip e also predicts several new ones. An essentia l role is play ed b y the sinusoidal co ordinate, whic h generates the closure relation and the Ask ey-Wilson alge bra together w ith the Hamiltonian. The relationship b et w een the closure rela tion a nd the Askey-Wi lson algebra is clarified. 1 In tro duc t ion F or one dimensional quantum mec hanical systems, tw o sufficien t conditions for exact solv- ability are kno wn. The first is the shap e invarian c e [1], whic h guarantees exact solv abil- it y in t he Sc h¨ odinger picture. The whole set of energy eigenv alues and the correspo nding eigenfunctions can b e obtained explicitly thro ugh shap e inv ariance com bined with Crum’s theorem [2 ], or the factorisation metho d [3 ] or the sup ersymmetric quantum mec hanics [4]. The second is the closur e r e lation [5]. It a llows to construct the exact Heisen b erg op erator solution of the sinusoidal c o or din ate η ( x ), whic h g enerates the closure relation to g ether with the Hamiltonian. The p ositiv e/negative energy par t s o f the Heisen b erg op erator solution giv e the annihila tion / cr e ation op er ators , in terms of whic h ev ery eigenstate can b e built up algebraically starting from the g roundstate. Th us exact solv ability in the Heisen b erg picture is realised. It is in teresting to note that these tw o sufficien t conditions apply equally w ell in the ‘discrete’ quan t um mec hanics (QM) [6, 7, 5, 8, 9], whic h is a simple extension or deformation of QM. In discrete Q M the dynamical v aria bles are, as in t he ordinary QM, the co ordinate x and the conjugate momen tum p , whic h is realised as p = − i∂ x . The Hamiltonian con tains the momen tum op erator in exp onentiated forms e ± β p , whic h acts on w av efunctions as finite shift op er ators , either in the pure imaginary directions or t he real directions. Th us the Sc hr¨ o dinger equation in discrete QM is a differ e n c e e quation instead of differen tial in ordinary QM. V arious examples of exactly solv able discrete quan tum mec hanics are know n for b oth of the t w o t yp es of shifts [6, 7 , 10, 5, 8, 9 ], and the eigenfunctions consist o f the Ask ey-sc heme of the h yp ergeometric orthog onal p olynomials [11, 12, 13] of a contin uous (pure imaginary shifts) and a discrete (real shifts) v ariable. It should b e stressed, how eve r, that t hese tw o sufficien t conditions do not tell ho w to build exactly solv able mo dels. In t his pap er w e presen t a simple theory of constructing exactly solv able Hamiltonians in discrete QM. It co v ers a ll the kno wn examples of exactly solv able discrete QM with b oth pure imaginary a nd real shifts [8, 9] and it predicts sev eral new ones to b e explored in a subsequen t publication [1 4]. Moreo v er, the theory is general enough to generate quasi-e x actly s o lvable Hamiltonians in the same manner. The quasi- exact solv ability means, in con trast t o the exact solv ability , that o nly a finite num b er of energy eigen v alues a nd the corresp onding eigenfunctions can b e obta ined exactly [15]. This unified theory a lso incorp o rates the kno wn examples of quasi-exactly solv able Hamiltonians [16, 17]. A new t yp e of quasi-exactly solv able Hamiltonians is constructed in this pap er and its explicit examples will b e surve y ed in a subseque n t publication [14]. One of the merits of the pr esen t approac h is that it reve als the common structure underlying the exactly and quasi-exactly solv able theories. In ordinary QM, the corresp onding theory w as already given in the App endix A of [5], althoug h it do es not cov er the quasi-exact solv ability . The prese n t pap er is organised as follows . In section t w o the general setting of the 2 discrete quantum mec hanics is briefly review ed and in § 2.1 the Hamiltonians f or the pure imaginary shifts and for the real shifts cases are giv en and the g eneral strategy of working in the v ector space of p olynomials in the sin usoidal co ordinate is explained. In § 2.2, based on a few p ostulates, v arious prop erties of the sinus oidal co ordinate η ( x ), whic h is t he essen tial ingredien t of the presen t theory , are presen t ed in some detail. The main result of the pap er, the unified form of the exactly and quasi-exactly solv able ‘Hamiltonians, ’ is giv en in § 2.3. The action of the Hamiltonian o n the p olynomials of the sinus oidal co ordinate is explained in § 2.4. It simply maps a degree n p olynomial in to a degree n + L − 2 p o lynomial. Here L is the degree of a certain p olynomial constituting the p oten tial function in the Hamiltonian. The exactly solv able case ( L = 2) is discussed in section three. In § 3.1, the closure relation, whic h used to b e v erified f or eac h given Hamiltonian, is show n to b e satisfied once and fo r all by the prop osed exactly solv able Ha milto nian. The nature of the dual closure relation, whic h plays an imp orta nt ro le in the theory of discrete Q M with real shifts and the corresp o nding theory of orthogo nal p olynomials of a discrete v ariable, is examined and compared with that o f the closure relatio n in § 3.2 . The relat io nship b et wee n the closure plus dual closure relations and the Ask ey-Wilson algebra [1 8, 19, 20, 21] is elucidated in § 3.3. In § 3.4, shap e in v ariance is explained and shown to b e satisfied for the pure imaginar y shifts case § 3.4.1 and for the real shifts case § 3.4.2. The quasi-exactly solv able ‘Hamiltonians’ are discusse d in section fo ur . The QES case with L = 3 is a chiev ed in § 4.1 b y adjusting the comp ensation term whic h is linear in η ( x ). A new t yp e of QES with L = 4 is in t ro duced in § 4.2, whic h has quadratic in η ( x ) comp ensation terms. It is sho wn that QES is not p ossible for L ≥ 5 in § 4.3. The issue of returning from the ‘Hamiltonian’ in the p olynomial space to the original Hamiltonian H is discussed in section fiv e. This is related to the pro p erties of the (pseudo-)groundstate φ 0 . The final section is for a summary , con ta ining the simple recip e to construct exactly and quasi-exactly solv able Hamiltonians. App endix A provides the explicit forms of the sin usoidal co or dinat es with which the actual exactly and quasi-exactly solv able Hamiltonians are constructed. There are eight differen t η ( x ) for the contin uous v a riable x and fiv e fo r the discrete x . App endix B g iv es the pro of o f the hermiticity of the Hamiltonian, whic h is sligh tly more in volv ed than in the ordinary QM. App endix C recapitulates the elemen tary form ulas for the eigen v alues and eigen v ectors of an upp er-triangular matrix, to whic h the exactly solv a ble ( L = 2) ‘Hamiltonian’ in the p olynomial space r educes. 3 2 ‘Discre t e’ Quan tum Mec hanics Throughout this pap er w e consider ‘discrete’ quan tum mec hanics of one degree of free- dom. Discrete quan tum mec ha nics is a generalisation of quan tum mec hanics in whic h the Sc h¨ odinger equation is a difference equation instead of differen tial in ordinary QM [6, 7, 10, 5, 8, 9]. In other w ords, the Hamiltonian contains the momentum op erator p = − i∂ x in exp onen tia ted forms e ± β p whic h w or k as shift op erators on the w av efunction e ± β p ψ ( x ) = ψ ( x ∓ iβ ) . (2.1) According to the tw o choices of the par ameter β , either r e a l or pur e imaginary , we ha v e t wo types of discrete QM; with (i) pure imaginary shifts, or (ii) real shifts, resp ectiv ely . In the case of pure imag inary shifts, ψ ( x ∓ iγ ), γ ∈ R 6 =0 , w e require the w av efunction to b e an analytic function of x with its domain including the real a xis or a part of it on whic h the dynamical v ariable x is defined. F or the real shifts case, the diff erence equation giv es constrain ts on w av efunctions only on equally spaced lattice p oints. T hen we c ho ose, after prop er rescaling, the v ariable x to b e an inte ger , with the tota l n umber either finite ( N + 1) or infinite. T o sum up, the dynamical v ariable x of the o ne dimensional discrete quan tum mech anics tak es con tin uous or discrete v alues: imaginary shifts : x ∈ R , x ∈ ( x 1 , x 2 ) , (2.2) r e al shifts : x ∈ Z , x ∈ [0 , N ] or [0 , ∞ ) . (2.3) Here x 1 , x 2 ma y b e finite, −∞ or + ∞ . Corresp ondingly , the inner pro duct of the w av efunc- tions has the followin g form: imaginary shifts : ( f , g ) = Z x 2 x 1 f ∗ ( x ) g ( x ) d x, (2.4) r e al shi f ts : ( f , g ) = N X x =0 f ( x ) ∗ g ( x ) or ∞ X x =0 f ( x ) ∗ g ( x ) , (2.5) and the norm of f ( x ) is | | f | | = p ( f , f ). In the case of imaginary shifts, other functions app earing in the Hamiltonian need to b e ana lytic in x within the same domain. Let us in tr o duce the ∗ -op eration on an analytic f unction, ∗ : f 7→ f ∗ . If f ( x ) = P n a n x n , a n ∈ C , then f ∗ ( x ) def = P n a ∗ n x n , in whic h a ∗ n is the complex conjug ation of a n . Ob viously f ∗∗ ( x ) = f ( x ) 4 and f ( x ) ∗ = f ∗ ( x ∗ ). If f is an analytic function, so is g ( x ) def = f ( x − a ), a ∈ C . The ∗ -op eration on this analytic function is g ∗ ( x ) =  f ( x ∗ − a )  ∗ = f ∗ ( x − a ∗ ). If a function satisfies f ∗ = f , then it takes real v a lues on the r eal line. The ‘absolute v a lue’ of an analytic function to b e used in this pap er is defined b y | f ( x ) | def = p f ( x ) f ∗ ( x ), whic h is again ana lytic and real non-negativ e on t he real axis. Note that the ∗ -op eration is used in the inner pro duct for the pure ima g inary shifts case (2 .4 ) so tha t the entire in tegrand is an analytic function, to o. This is essen tial for the pro of of hermiticit y to b e presen ted in App endix B. In quan tum mec hanics, the eigen v alue pro blem of a give n Hamiltonian is the cen tral issue. In this pap er, w e will consider the Hamiltonians having finite or semi-infinite discrete energy lev els only: 0 = E (0) < E (1) < E (2) < · · · . (2.6) Here w e ha ve c hosen the additiv e constan t of the Hamiltonian so that the groundstate energy v anishes. In other words, the Hamiltonian is p os i tive semi -definite . It is a w ell kno wn theorem in linear algebra that any p ositive semi-definite hermitian matrix can b e factorised as a pro duct of a certain matrix, say A , and its hermitian conjugate A † . As w e will see shortly , the Hamiltonians of discrete quantum mec hanics hav e the same prop ert y , b oth with the imaginary and real shifts. 2.1 Hamiltonian and Strategy The Hamiltonian of one dimensional discrete quan tum mec ha nics has a simple fo rm H def = ε  p V + ( x ) e β p p V − ( x ) + p V − ( x ) e − β p p V + ( x ) − V + ( x ) − V − ( x )  . (2.7) Corresp onding to the imaginary/real shifts cases, the pa rameter β , the p oten tial functions V ± ( x ) and a sign factor ε are imaginary shifts : β = γ , ε = 1 , V + ( x ) = V ( x ) , V − ( x ) = V ∗ ( x ) , r e al shifts : β = i, ε = − 1 , V + ( x ) = B ( x ) , V − ( x ) = D ( x ) , (2.8) with γ ∈ R 6 =0 . The p oten tia l function B ( x ) and D ( x ) a re p ositiv e and v a nish at b oundaries: B ( x ) > 0 , D ( x ) > 0 , D (0) = 0 ; B ( N ) = 0 for the finite case . (2.9) As men tioned ab ov e, e ± β p are shift op erators e ± β p f ( x ) = f ( x ∓ iβ ), and the Sc hr ¨ odinger equation H φ n ( x ) = E ( n ) φ n ( x ) , n = 0 , 1 , 2 , . . . , (2.10) 5 is a difference equation. The hermiticit y of the Hamiltonian is manifest for the real shifts case b ecause the Hamiltonian is a real symmetric matrix. F or the imaginary shifts case, see App endix B. This p ositiv e semi-definite Hamiltonian (2.7) can b e factorized: H = A † A . (2.11) Corresp onding to the imagina ry/real shifts cases, A and A † are A = i  e γ p/ 2 p V ∗ ( x ) − e − γ p / 2 p V ( x )  , A † = − i  p V ( x ) e γ p/ 2 − p V ∗ ( x ) e − γ p / 2  , (2.12) A = p B ( x ) − e ∂ p D ( x ) , A † = p B ( x ) − p D ( x ) e − ∂ . (2.13) The groundstate w a v efunction φ 0 ( x ) is determined as a zero mo de of A , A φ 0 ( x ) = 0 . (2.14) The similarity transformed Hamiltonian e H in terms of the groundstate wa v efunction φ 0 has a m uch simpler form than the original Hamilto nia n H : e H def = φ 0 ( x ) − 1 ◦ H ◦ φ 0 ( x ) (2.15) = ε  V + ( x )( e β p − 1) + V − ( x )( e − β p − 1)  . (2.16) In the second equation w e ha v e used (2.14). In the following w e will tak e e H instead of H as the starting p oin t. That is, we reve rse t he argumen t and construct directly the ‘Hamiltonian’ e H (2.16) based on a certain function η ( x ) to b e called the sinusoidal c o or dinate . The necessary prop erties of the sin usoidal co ordinate will b e in tro duced in the next subsection § 2.2. The general strategy is to construct the ‘Hamiltonian’ e H in suc h a w ay that it maps a p olynomial in η ( x ) in to ano ther: e HV n ⊆ V n + L − 2 ⊂ V ∞ . Here V n ( n ∈ Z ≥ 0 ) is defined b y V n def = Span  1 , η ( x ) , . . . , η ( x ) n  , V ∞ def = lim n →∞ V n . (2.17) The g oal is ac hiev ed by choosing v ery sp ecial fo rms of V ± ( x ) as give n in (2.30)–(2.31), that is V ( x ) and V ∗ ( x ) or B ( x ) and D ( x ) are p olynomials of degree L in the sin usoidal co o rdinate η ( x ) and it s shifts η ( x ∓ iβ ) divided by sp ecial quadratic p olynomials in them. This provides a unified theory of exactly solv able and quasi-exactly solv able discrete Q M. Exactly solv able 6 QM are realised by choosing e H in suc h a w ay ( L = 2) that e HV n ⊆ V n is satisfied for all n . Then the existence of a n eigenfunction, o r to b e more precise, a degree n eigenp o lynomial, of e H is guarante ed for each in teger n . On the other hand, quasi-exact solv ability is atta ined b y adjusting the parameters o f e H in suc h a w ay ( L = 3 , 4) that e H ′ V M ⊆ V M is r ealised for an in teger M . Here e H ′ is a mo dificatio n of e H b y the addition of the comp ensation terms. Then the ‘Hamiltonian’ e H ′ has an M + 1 = dim( V M )-dimensional in v ariant space, prov iding M + 1 eigenp olynomials of e H ′ . After obtaining suc h (quasi-)exactly solv a ble ‘Hamilto nia n’ e H ( e H ′ ), w e hav e to find the (quasi-)groundstate w av efunction φ 0 in order to return to the true Hamiltonian H ( H ′ ) by (2.1 5). It should b e noted that the existence of suc h a (quasi- )groundstate w a v efunction is not gua ran teed a priori since w e hav e star t ed with e H instead o f H . In the case of quasi-exactly solv a ble QM, the p ositive semi-definiteness of the Hamiltonian (2.6) is in general lost due to the inclusion o f the comp ensation terms to e H ′ . 2.2 sin usoidal co ordinate Motiv a t ed b y the study in [5, 8, 9], let us define a sin usoidal co ordinate η ( x ) as a real (or ‘real’ analytic η ∗ ( x ) = η ( x ) in the case of pure imaginary shifts) function of x satisfying the follo wing symmetric shift-addition prop ert y: η ( x − iβ ) + η ( x + iβ ) = (2 + r (1) 1 ) η ( x ) + r (2) − 1 . (2.18) Here r (1) 1 and r (2) − 1 are real parameters and we a ssume r (1) 1 > − 4. These t wo, r (1) 1 and r (2) − 1 , are fundamen tal para meters app earing in b oth exactly and quasi-exactly solv able dynamical systems . F or the exactly solv able systems , these tw o para meters also manifest themselv es (3.4) in the clo s ur e r elation , another c haracterisation of exact solv a bilit y , to b e discuss ed in § 3.1. Since a p olynomial in η ( x ) is a lso a p olynomial in aη ( x ) + b ( a, b : real constan ts), w e imp ose t w o conditions 1 (w e assume 0 ∈ [ x 1 , x 2 ]) η (0) = 0 and η ( x ) : monoto ne increasing function , (2.19) whic h are no t ess en tial for (quasi-)exact solv ability but importa n t for expressing v ario us form ulas in a unified w ay . W e imp ose another condition, to b e called the symm etric shift- m ultiplication prop ert y: η ( x − iβ ) η ( x + iβ ) =  η ( x ) − η ( − iβ )  η ( x ) − η ( iβ )  , (2.20) 1 F or the real shifts case , suc h η ( x ) satisfying (2.1 8) and (2.1 9) can b e classified int o five types (A.9)–(A.13) [8] and they a lso sa tisfy the condition (2.2 0). 7 together with η ( x ) 6 = η ( x − iβ ) 6 = η ( x + iβ ) 6 = η ( x ). The t wo conditions (2 .18) and (2 .2 0) imply that an y symmetric p olynomial in η ( x − iβ ) and η ( x + iβ ) is express ed as a p olynomial in η ( x ). Esp ecially w e ha ve ( n ≥ − 1) g n ( x ) def = η ( x − iβ ) n +1 − η ( x + iβ ) n +1 η ( x − iβ ) − η ( x + iβ ) =  a p olynomial of degree n in η ( x )  = n X k =0 g ( k ) n η ( x ) n − k . (2.21) The co efficien t g ( k ) n is real b ecause g ∗ n ( x ) = g n ( x ). W e set g ( k ) n = 0 except for 0 ≤ k ≤ n . Since g n ( x ) satisfies the following three term recurrence relation g n +1 ( x ) =  η ( x − iβ ) + η ( x + iβ )  g n ( x ) − η ( x − iβ ) η ( x + iβ ) g n − 1 ( x ) ( n ≥ 0) , (2.22) w e can write do wn g ( k ) n explicitly . Esp ecially g ( k ) n for k = 0 , 1 are g (0) n = [ n + 1] , (2.23) g (1) n =      1 6 n ( n + 1 )(2 n + 1) r (2) − 1 for r (1) 1 = 0 , n [ n + 1 ] − ( n + 1)[ n ] r (1) 1 r (2) − 1 for r (1) 1 6 = 0 . (2.24) Here w e ha v e defined [ n ] as [ n ] def =            n for r (1) 1 = 0 , e αn − e − αn e α − e − α for r (1) 1 > 0  ⇐ r (1) 1 = ( e α 2 − e − α 2 ) 2 ( α > 0)  , e iαn − e − iαn e iα − e − iα for − 4 < r (1) 1 < 0  ⇐ r (1) 1 = ( e i α 2 − e − i α 2 ) 2 (0 < α < π )  . (2.25) Note that r (1) 1 and r (2) − 1 are express ed as r (1) 1 = [2] − 2 , r (2) − 1 = η ( − iβ ) + η ( iβ ) . (2.26) F or n, m ∈ Z , n ≥ m − 1, we ha ve n X r = m g (1) r =        1 12 ( n + m + 1)( n − m + 1)( n 2 + 2 n + m 2 ) r (2) − 1 for r (1) 1 = 0 , ( n + 1)[ n + 1] − m [ m ] − [ 1 2 ] − 2 [ n + m +1 2 ][ n − m +1 2 ] r (1) 1 r (2) − 1 for r (1) 1 6 = 0 . (2.27) The follow ing prop erties of [ n ] are useful: [ a ][ a + c ] − [ b ][ b + c ] = [ a − b ][ a + b + c ] , (2.28) n X r = m [ r ] = [ n + m 2 ][ n − m +1 2 ] [ 1 2 ] ( n, m ∈ Z , n ≥ m − 1) . (2.29) 8 2.3 p oten tial functions The first goal is to construct a general form of the ‘Hamiltonian’ e H suc h that a p olynomial in η ( x ) is mapp ed in to another. It is achie v ed by the follo wing form of the p oten tial functions V ± ( x ): V ± ( x ) = e V ± ( x )  η ( x ∓ iβ ) − η ( x )  η ( x ∓ iβ ) − η ( x ± iβ )  , (2.30) e V ± ( x ) = X k,l ≥ 0 k + l ≤ L v k ,l η ( x ) k η ( x ∓ iβ ) l , (2.31) where L is a na t ural n um b er roug hly indicating the degree of η ( x ) in e V ± ( x ) and v k ,l are real constan ts, with the constraint P k + l = L v 2 k ,l 6 = 0. It is imp o rtan t that the same v k ,l app ears in b oth e V ± ( x ). As w e will see in the next subsection § 2.4 , the ‘Hamiltonian’ e H with the ab o ve V ± ( x ) maps a degree n p olynomial in η ( x ) to a degree n + L − 2 p olynomial (2.36), (2.3 9). The essen tial part of the formu la (2.30) is the denominators. They hav e the same form as the generic f orm ula, derived b y the presen t authors, for the co efficien ts of the three t erm recurrence relations of the ortho g onal p olynomials, (4.52) and (4.53) in [8]. The translation rules are the duality corresp ondence itself, (3.14)–(3 .18) in [8]: E ( n ) → η ( x ) , − A n → V + ( x ) , − C n → V − ( x ) , α +  E ( n )  → η ( x − iβ ) − η ( x ) , α −  E ( n )  → η ( x + iβ ) − η ( x ) . (2.32) Some of the parameters v k ,l in (2.31) are redundan t. F rom (2.18) and (2.20), we hav e η ( x ∓ iβ ) 2 = ( 2 + r (1) 1 ) η ( x ) η ( x ∓ iβ ) − η ( x ) 2 + r (2) − 1  η ( x ) + η ( x ∓ iβ )  − η ( − iβ ) η ( iβ ) . (2.33) By using this rep eatedly , a monomial η ( x ∓ iβ ) l can b e reduced to a p olynomial of degree one in η ( x ∓ iβ ) whose co efficien ts are p olynomials in η ( x ). Therefore it is sufficien t to k eep v k ,l with l = 0 , 1. The remaining 2 L + 1 parameters v k ,l ( k + l ≤ L , l = 0 , 1) are indep enden t, with one of whic h corresp onds to the ov erall no rmalization of the Hamiltonian. In fact, if t wo sets of parameters { v k ,l } and { v ′ k ,l } ( k + l ≤ L , l = 0 , 1) give the same V ± ( x ), namely , P L k =0 ( v k , 0 − v ′ k , 0 ) η ( x ) k + P L − 1 k =0 ( v k , 1 − v ′ k , 1 ) η ( x ) k η ( x ∓ iβ ) = 0, then w e obtain v k ,l = v ′ k ,l . Therefore there is no more redundancy in v k ,l ( k + l ≤ L , l = 0 , 1). Note that w e ha v e not y et imp o sed the b oundary condition D (0 ) = 0 (2.9) . The sin usoidal co ordinate η ( x ) itself ma y hav e extra parameters. 9 2.4 e H on the p olynomial space The action of e H (2.16) o n η ( x ) n b ecomes with (2.30) and (2.31): e H η ( x ) n = ε  V + ( x )  η ( x − iβ ) n − η ( x ) n  + V − ( x )  η ( x + iβ ) n − η ( x ) n   = ε e V + ( x ) P n − 1 r =0 η ( x ) r η ( x − iβ ) n − 1 − r − e V − ( x ) P n − 1 r =0 η ( x ) r η ( x + iβ ) n − 1 − r η ( x − iβ ) − η ( x + iβ ) = ε n − 1 X r =0 η ( x ) r e V + ( x ) η ( x − iβ ) n − 1 − r − e V − ( x ) η ( x + iβ ) n − 1 − r η ( x − iβ ) − η ( x + iβ ) = ε n − 1 X r =0 η ( x ) r X k,l ≥ 0 k + l ≤ L v k ,l η ( x ) k η ( x − iβ ) l + n − 1 − r − η ( x + iβ ) l + n − 1 − r η ( x − iβ ) − η ( x + iβ ) = ε X k,l ≥ 0 k + l ≤ L v k ,l n − 1 X r =0 η ( x ) k + r g n + l − r − 2 ( x ) =  a p olynomial o f degree n + L − 2 in η ( x )  = ε X k,l ≥ 0 k + l ≤ L n − 1 X r =0 n + l − r − 2 X j =0 v k ,l g ( j ) n + l − r − 2 η ( x ) n + k + l − 2 − j = ε n + L − 2 X m =0 η ( x ) n + L − 2 − m m X j =max( m − L, 0) X k,l ≥ 0 k + l = L − m + j v k ,l n − 1 X r =0 g ( j ) n + l − r − 2 = n + L − 2 X m =0 η ( x ) n + L − 2 − m m X j =max( m − L, 0) e m,j,n . (2.34) Here e m,j,n (the L -dep endence is implicit) is defined b y e m,j,n def = ε L − m + j X l =0 v L − m + j − l,l n − 1 X r =0 g ( j ) n + l − r − 2 . (2.35) Therefore the matrix elemen ts of e H in the basis { η ( x ) n } n =0 , 1 ,... is giv en b y e H η ( x ) n = n + L − 2 X m =0 η ( x ) m e H η m,n , e H η m,n = n + L − 2 − m X j =max( n − 2 − m, 0) e n + L − 2 − m,j,n . (2.36) The co efficien ts e m, 0 ,n and e m, 1 ,n b ecome, by using (2.29) and (2.27) : e m, 0 ,n = ε [ n 2 ] [ 1 2 ] L − m X l =0 v L − m − l,l [ n +2 l − 1 2 ] , (2.37) 10 e m, 1 ,n = ε L − m +1 X l =0 v L − m +1 − l,l ×      1 12 n ( n + 2 l − 2)  ( n + l − 1) 2 + l 2 − 2 l  r (2) − 1 for r (1) 1 = 0 , ( n + l − 1)[ n + l − 1] − ( l − 1)[ l − 1] − [ 1 2 ] − 2 [ n +2 l − 2 2 ][ n 2 ] r (1) 1 r (2) − 1 for r (1) 1 6 = 0 . (2.38) So far the conditions v k ,l = 0 for l ≥ 2 are not used. W e hav e established e HV n ⊆ V n + L − 2 , (2.39) where V n is the p olynomial space defined in (2.17). F or L = 2, V n is e H -inv arian t. Therefore this case is exactly solv a ble; all the eigen v alues a nd eigenfunctions of e H can b e o btained explicitly and the eigenfunction is a p o lynomial of degree n in η ( x ) for each n . On the other hand, L ≥ 3 cases are not exactly solv able but some cases can b e made quasi-exactly solv able b y certain mo dification to b e discussed presen tly . F or L = 0 , 1 cases, the matrix e H η = ( e H η m,n ) 0 ≤ m,n ≤ K with finite K is not diagonalizable except for K = 0 , 1. In t he fo llo wing we will set v k ,l = 0 f o r l ≥ 2, see § 2.3. F or the real shifts case, the condition D (0) = 0 (2.9) is satisfied b y c ho osing v 0 , 0 as v 0 , 0 = − v 0 , 1 η ( − 1). 3 Exactly Solv able e H The L = 2 case is exactly solv able. Since the Hamilto nia n of the p olynomial space e H is an upp er triangular ma t r ix (2.36), its eigen v alues and eigenv ectors a re easily obtained explicitly , see App endix C. The eigenv alue E ( n ) is E ( n ) = e H η n,n = e 0 , 0 ,n = ε [ n 2 ] [ 1 2 ]  v 2 , 0 [ n − 1 2 ] + v 1 , 1 [ n +1 2 ]  , (3.1) and the cor r espo nding eigenpo lynomial P n  η ( x )  is expressed as a determinan t o f the fol- lo wing order n + 1 matrix, P n  η ( x )  ∝             1 η ( x ) η ( x ) 2 · · · η ( x ) n E (0) − E ( n ) e H η 0 , 1 e H η 0 , 2 · · · e H η 0 ,n E (1) − E ( n ) e H η 1 , 2 · · · e H η 1 ,n . . . . . . . . . 0 E ( n − 1) − E ( n ) e H η n − 1 ,n             . (3.2) F or a choice of the sinusoidal co ordinate among the p ossible forms (A.1)–(A.13) a nd the v alues of the fiv e parameters, v 0 , 0 , v 1 , 0 , v 0 , 1 , v 1 , 1 and v 2 , 0 , these t wo form ulas (3.1) and (3.2), 11 although clumsy , giv e the complete solutions of the ‘Sc hr¨ odinger equation’ e H P n ( η ( x )) = E ( n ) P n ( η ( x )) at the alg ebraic lev el. F or the solutions of a full quantum mec hanical problem, ho w ev er, o ne needs the square-in tegrable g roundstate w av efunction φ 0 ( x ) (2.14), whic h is essen tial for the existence of the Hamiltonian H and the verification of its hermiticit y . These conditions w ould usually restrict the r a nges of the parameters v 0 , 0 , . . . , v 2 , 0 . F or sp ecific problems, how ever, there are more p ow erful and systematic solution metho ds based on the shap e in v ariance [1, 6, 7, 8, 9] and the closure relation [5, 8, 9]. These t wo are indep enden t and sufficien t conditions for exact solv ability whic h are a pplicable to not only ordinary QM but also discrete QM. In our previous works [6, 7, 5, 8, 9 ] these conditions we re v erified f or eac h sp ecific problem. Here we will provide pro of s ba sed on the g eneric for m of the exactly solv able ( L = 2) ‘Hamiltonian’ e H , (2.16), (2.30), (2 .31). These pro of s apply to all the exactly solv able discrete Q M. In the rest of this section w e assume the existence of the ground state w a v efunction φ 0 ( x ) (2.14). 3.1 closure relation The closure relation is a comm utator relation b et w een the Hamiltonian H and the sinus oidal co ordinate η ( x ) [5, 8, 9]: [ H , [ H , η ] ] = η R 0 ( H ) + [ H , η ] R 1 ( H ) + R − 1 ( H ) . (3.3) Here R i ( z ) are p olynomials with real co efficien ts r ( j ) i , R 1 ( z ) = r (1) 1 z + r (0) 1 , R 0 ( z ) = r (2) 0 z 2 + r (1) 0 z + r (0) 0 , R − 1 ( z ) = r (2) − 1 z 2 + r (1) − 1 z + r (0) − 1 . (3.4) Reflecting the f a ct that the Hamiltonian H has shift op erators e ± β p , whereas η ( x ) has no ne, the function R 0 ( H ) and R − 1 ( H ) are quadratic in H and R 1 ( H ) is linear in H . By similarity transforming ( 3.3) in terms of the ground state w a v efunction φ 0 , it is rewritten as [ e H , [ e H , η ] ] = η R 0 ( e H ) + [ e H , η ] R 1 ( e H ) + R − 1 ( e H ) . (3.5) The closure relation (3.3) allow s us to o btain the exact Heisen b erg op erator solution for η ( x ), and the a nnihilation and creation op erators a ( ± ) are extracted fro m this exact Heisen b erg op erator solution [5]: 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 , (3.6) 12 α ± ( H ) def = 1 2  R 1 ( H ) ± p R 1 ( H ) 2 + 4 R 0 ( H )  , (3.7) R 1 ( H ) = α + ( H ) + α − ( H ) , R 0 ( H ) = − α + ( H ) α − ( H ) , (3.8) a ( ± ) def = ±  [ H , η ( x )] −  η ( x ) + R − 1 ( H ) R 0 ( H ) − 1  α ∓ ( H )   α + ( H ) − α − ( H )  − 1 (3.9) = ±  α + ( H ) − α − ( H )  − 1  [ H , η ( x )] + α ± ( H )  η ( x ) + R − 1 ( H ) R 0 ( H ) − 1   . (3.10) The energy sp ectrum is determined by t he ov er- determined r ecursion relations E ( n + 1) = E ( n ) + α +  E ( n )  and E ( n − 1) = E ( n ) + α −  E ( n )  with E (0) = 0, and the excited state w av efunctions { φ n ( x ) } ar e obtained by successiv e action of the creation op erato r a (+) on the groundstate w av efunction φ 0 ( x ). The closure relation (3.5) ( o r (3.3)) is equiv alent to the follo wing set of fiv e equations: η ( x − 2 iβ ) − 2 η ( x − iβ ) + η ( x ) = r (2) 0 η ( x ) + r (2) − 1 + r (1) 1  η ( x − iβ ) − η ( x )  , (3.11) η ( x + 2 iβ ) − 2 η ( x + iβ ) + η ( x ) = r (2) 0 η ( x ) + r (2) − 1 + r (1) 1  η ( x + iβ ) − η ( x )  , (3.12)  η ( x − iβ ) − η ( x )  V + ( x − iβ ) + V − ( x − iβ ) − V + ( x ) − V − ( x )  = −  r (2) 0 η ( x ) + r (2) − 1  V + ( x − iβ ) + V − ( x − iβ ) + V + ( x ) + V − ( x )  − r (1) 1  η ( x − iβ ) − η ( x )  V + ( x − iβ ) + V − ( x − iβ )  + ε − 1  r (1) 0 η ( x ) + r (1) − 1 + r (0) 1  η ( x − iβ ) − η ( x )  , (3.13)  η ( x + iβ ) − η ( x )  V + ( x + iβ ) + V − ( x + iβ ) − V + ( x ) − V − ( x )  = −  r (2) 0 η ( x ) + r (2) − 1  V + ( x + iβ ) + V − ( x + iβ ) + V + ( x ) + V − ( x )  − r (1) 1  η ( x + iβ ) − η ( x )  V + ( x + iβ ) + V − ( x + iβ )  + ε − 1  r (1) 0 η ( x ) + r (1) − 1 + r (0) 1  η ( x + iβ ) − η ( x )  , ( 3 .14) 2  η ( x ) − η ( x − iβ )  V + ( x ) V − ( x − iβ ) + 2  η ( x ) − η ( x + iβ )  V − ( x ) V + ( x + iβ ) =  r (2) 0 η ( x ) + r (2) − 1  V + ( x ) V − ( x − iβ ) + V − ( x ) V + ( x + iβ ) +  V + ( x ) + V − ( x )  2  + r (1) 1  η ( x − iβ ) − η ( x )  V + ( x ) V − ( x − iβ ) + r (1) 1  η ( x + iβ ) − η ( x )  V − ( x ) V + ( x + iβ ) − ε − 1  r (1) 0 η ( x ) + r (1) − 1  V + ( x ) + V − ( x )  + ε − 2  r (0) 0 η ( x ) + r (0) − 1  . (3.15) Ob viously (3.11) a nd (3.12 ) a r e equiv alen t and so are (3.1 3) a nd (3.1 4), under the condi- tion (3.16). By substituting our ch oice of V ± ( x ) (2.3 0)–(2.31) for L = 2 , it is straigh tforw ard to verify the other three equations (3.13)–(3.15). The co efficien ts r ( j ) i app earing in (3.4) are expresse d b y the par a meters v 1 , 0 , v 0 , 1 , v 1 , 1 and v 2 , 0 together with the t wo para meters r (1) 1 13 and r (2) − 1 whic h ha v e already app eared in the definition of η ( x ) (2 .1 8) (see also (2.26)): r (2) 0 = r (1) 1 , r (1) 0 = 2 r (0) 1 , (3.16) ε − 1 r (0) 1 = v 2 , 0 + v 1 , 1 , ε − 2 r (0) 0 = − v 2 , 0 v 1 , 1 , (3.17) ε − 1 r (1) − 1 = v 1 , 0 + v 0 , 1 , ε − 2 r (0) − 1 = − v 2 , 0 v 0 , 1 . (3.18) Note that v 0 , 0 do es not app ear. It implies that fo r the imagina r y shifts case the commutation relation b etw een the annihilat ion and creation op erators do es not dep end on v 0 , 0 . With these form ulas, the explicit forms of α ± ( H ) (3.7) can b e expressed in terms of r (1) 1 , v 2 , 0 and v 1 , 1 . It is straigh tforw ard to v erify the eigenv alue form ula (3.1 ). This concludes the unified pro of of the closure relation for all the discrete QM. 3.2 dual closure relation The dual closur e r e l a tion has the same fo rms as the closure relatio n (3.3) and ( 3 .5) with the roles of Hamiltonian H ( e H ) and the sin usoidal co o r dina t e η ( x ) exc hang ed: [ η , [ η , H ] ] = H R dual 0 ( η ) + [ η , H ] R dual 1 ( η ) + R dual − 1 ( η ) , (3.19) [ η , [ η , e H ] ] = e H R dual 0 ( η ) + [ η , e H ] R dual 1 ( η ) + R dual − 1 ( η ) , (3.20) where R dual i ( z ) are as ye t unkno wn p o lynomials. W e will sho w b elo w that the dual closure relation is the ch aracteristic f eature shared by all the ‘Hamiltonians’ e H whic h map a p o ly- nomial in η ( x ) into another. Therefore its dynamical con tents are not so constraining as the closure relation, exc ept fo r the real shifts (the discrete v ariable) exactly solv able ( L = 2) case, where the closure relation and the dua l closure relations are on the same fo oting as sho wn in [8]. By substituting the ‘Hamiltonian’ e H (2.16 ) without an y further sp ecification of V ± in to the a b o v e ( 3.20), w e find it is equiv alent to the following set of three equations:  η ( x ) − η ( x − iβ )  2 = R dual 0  η ( x − iβ )  +  η ( x ) − η ( x − iβ )  R dual 1  η ( x − iβ )  , (3.21)  η ( x ) − η ( x + iβ )  2 = R dual 0  η ( x + iβ )  +  η ( x ) − η ( x + iβ )  R dual 1  η ( x + iβ )  , (3.22) 0 = − ε  V + ( x ) + V − ( x )  R dual 0  η ( x )  + R dual − 1  η ( x )  . (3.23) These imply R dual 1  η ( x )  =  η ( x − iβ ) − η ( x )  +  η ( x + iβ ) − η ( x )  , (3.24) 14 R dual 0  η ( x )  = −  η ( x − iβ ) − η ( x )  η ( x + iβ ) − η ( x )  , (3.25) R dual − 1  η ( x )  = ε  V + ( x ) + V − ( x )  R dual 0 ( η ( x )) . (3.26) By using the defining prop erties of the sin usoidal co ordinate (2.18)–(2.20) , w e actually find that R dual 1 ( z ) is a degree 1 p olynomial in z and R dual 0 ( z ) is a quadratic p olynomial: R dual 1 ( z ) = r (1) 1 z + r (2) − 1 , (3.27) R dual 0 ( z ) = r (1) 1 z 2 + 2 r (2) − 1 z − η ( − iβ ) η ( iβ ) . (3.28) By using the explicit forms of V ± (2.30)–(2.31) (with an arbitrary L ) we obta in R dual − 1 ( z ) = ε  v 0 , 0 + L X k =1 ( v k , 0 + v k − 1 , 1 ) z k  . (3.29) Therefore all R dual i ( z ) a re p olynomials a nd the dual closure r elatio n is demonstrated in a unified fashion fo r an arbitrary L . Th us it do es not c haracterise the exact nor the quasi- exact solv ability . F or the exactly solv able L = 2 case, b y using (3.17) and (3 .18), R dual − 1 ( z ) can b e written as R dual − 1 ( z ) = r (0) 1 z 2 + r (1) − 1 z + εv 0 , 0 . (3.30) F or the real shifts case, in order to satisfy D (0) = 0 (2.9), we hav e to take v 0 , 0 = − η ( − 1) v 0 , 1 and this implies εv 0 , 0 = η (1) η ( − 1) B (0). See (4.10 4 )–(4.106) in [8]. 3.3 Ask ey-Wilson alge bra Here w e will fo cus o n the exactly solv able systems a nd will briefly commen t on the rela- tionship b etw een the closure plus the dual closure relations and t he so-called Ask ey-Wilson algebra [18, 19, 20, 2 1]. By simply expanding the double commutators in the closure (3.3 ) and the dual closure (3.19) relations, w e obtain t w o cubic relations generated b y the t wo op erators H a nd η : H 2 η − (2 + r (1) 1 ) H η H + η H 2 − r (0) 1 ( H η + η H ) − r (0) 0 η = r (2) − 1 H 2 + r (1) − 1 H + r (0) − 1 , (3.31) η 2 H − (2 + r (1) 1 ) η H η + H η 2 − r (2) − 1 ( η H + H η ) + η ( − iβ ) η ( iβ ) H = r (0) 1 η 2 + r (1) − 1 η + εv 0 , 0 . (3.32) F rom its structure, the closure relation is at most linear in η and a t most quadratic in H . So the l.h.s. o f (3.31) has terms con taining one f a ctor of η and the r.h.s, none. It is simply 15 R − 1 ( H ). Lik ewise, the l.h.s. of ( 3 .32) has terms con taining one fa ctor of H and the r.h.s, none. It is simply R dual − 1 ( H ). In (3.32), η ( − iβ ) η ( iβ ) is just a real n umber, not an op erator. These ha ve the same form as the so-called Ask ey-Wilson alg ebra, whic h ha s man y different expressions . The original one is due to Zhedanov [18]. Here w e presen t a sligh tly more general v ersion than the or ig inal one and is due to [19, 20]. It is generated b y three elemen ts K 1 , K 2 , K 3 : [ K 1 , K 2 ] = K 3 , (3.33) [ K 3 , K 1 ] = 2 ρK 1 K 2 K 1 + a 2 ( K 1 K 2 + K 2 K 1 ) + a 1 K 2 1 + c 2 K 2 + dK 1 + g 2 , (3.34) [ K 2 , K 3 ] = 2 ρK 2 K 1 K 2 + a 1 ( K 2 K 1 + K 1 K 2 ) + a 2 K 2 2 + c 1 K 1 + dK 2 + g 1 . (3.35) By expanding the comm utators and eliminating K 3 , they are reduced to K 2 1 K 2 + 2( ρ − 1) K 1 K 2 K 1 + K 2 K 2 1 + a 2 ( K 1 K 2 + K 2 K 1 ) + c 2 K 2 = − a 1 K 2 1 − dK 1 − g 2 , (3.36) K 2 2 K 1 + 2( ρ − 1) K 2 K 1 K 2 + K 1 K 2 2 + a 1 ( K 2 K 1 + K 1 K 2 ) + c 1 K 1 = − a 2 K 2 2 − dK 2 − g 1 . (3.37) Another v ersion due to T erwilliger [21] is generated b y t w o indep enden t elemen t s A and A × and it has only expanded fo r ms: A 2 A × − β T AA × A + A × A 2 − γ ( AA × + A × A ) − ρ T A × = γ × A 2 + ω A + η T , (3.38) A × 2 A − β T A × AA × + AA × 2 − γ × ( A × A + AA × ) − ρ × T A = γ A × 2 + ω A × + η × T . (3.39) Here is the list of corresp ondence of the generators a nd co efficien ts: ref. [19, 20] ref. [21] this pap er K 1 A H K 2 A × η 2(1 − ρ ) β T 2 + r (1) 1 − a 2 γ r (0) 1 = ε ( v 2 , 0 + v 1 , 1 ) − a 1 γ × r (2) − 1 = η ( − iβ ) + η ( iβ ) − c 2 ρ T r (0) 0 = − ε 2 v 2 , 0 v 1 , 1 − c 1 ρ × T − η ( − iβ ) η ( iβ ) − d ω r (1) − 1 = ε ( v 1 , 0 + v 0 , 1 ) − g 2 η T r (0) − 1 = − ε 2 v 2 , 0 v 0 , 1 − g 1 η × T εv 0 , 0 . (3.40) In [19] the Casimir o p erator Q commuting with all the generators of the algebra, [ K 1 , Q ] = [ K 2 , Q ] = [ K 3 , Q ] = 0 is giv en: Q = K 1 K 2 K 1 K 2 + K 2 K 1 K 2 K 1 − (1 − ρ )( K 1 K 2 2 K 1 + K 2 K 2 1 K 2 ) 16 + (2 − ρ )( a 1 K 1 K 2 K 1 + a 2 K 2 K 1 K 2 ) + (1 − ρ )( c 1 K 2 1 + c 2 K 2 2 ) (3.41) + ( d − a 1 a 2 )( K 1 K 2 + K 2 K 1 ) +  (2 − ρ ) g 1 − a 2 c 1  K 1 +  (2 − ρ ) g 2 − a 1 c 2  K 2 . With the ab ov e substitution (3.40), K 1 → H , K 2 → η , etc, the Casimir op erator turns out to b e a constan t [22]: Q = ε 2 ( v 1 , 1 v 0 , 0 − v 1 , 0 v 0 , 1 − r (2) − 1 v 2 , 0 v 0 , 1 ) . (3.42) Although this fact might a pp ear striking from the pure algebra p oin t of view (3.33)–(3.35), it is rather trivial in quan tum mec hanics. In o ne-dimensional quan tum mec hanics, there is no dynamical op erator which comm utes with the Hamiltonian. Therefore, if Q comm utes with H , it m ust b e a constant. No w here are some commen ts on the dissimilarit y . The first ob vious difference is the structure. While the Ask ey-Wilson algebra (3.3 3)–(3.35) or (3.38)–(3.39) has no inheren t structure, t he closure relation (3.3) has the right structure to lead to the Heisen b erg op erator solution for η ( x ), whose p ositiv e and negativ e energy parts are the annihilation-creation op erators [5, 8, 9]. It is the Hamiltonian a nd t he a nnihilat io n-creation op erato rs that f o rm the d yna m ic al symmetry algebr a of the system [8, 9], not the closure or dual- closure relations, nor the Ask ey-Wilson algebra relations. The q -oscillator algebra of [23] is t he typic al example of the dynamical symmetry alg ebra th us obtained. The next is the difference in c har a cter of the Ask ey-Wilson alg ebra itself for the tw o cases; the pure imagina ry shifts and the real shifts cases. The main scene of application of the Ask ey-Wilson algebra is the theory of the o rthogonal p olynomials of a discrete v ariable. The ( q -)Racah p olynomials are the typ ical example of this gro up [13, 8]. In o ur language, it is the theory of the eigenp olynomials of e H in discrete quan tum mec hanics with r eal shifts. One outstanding feature o f these p olynomials is the duality [8, 21 ]. F or t he eigenp olynomials of e H e H P n ( η ( x )) = E ( n ) P n ( η ( x )) , n = 0 , 1 , . . . , (3.43) there exist the dual p olynomials Q x ( E ( n )), satisfying the relation P n ( η ( x )) = Q x ( E ( n )) , x = 0 , 1 , . . . , n = 0 , 1 , . . . . (3.44) This dualit y x ↔ n , η ∼ η ( x ) ↔ E ( n ) ∼ H is reflected in the symmetry b et w een the pair of op erators (called the Leonard pair [24]) K 1 and K 2 or A and A × in the Ask ey-Wilson algebra. The Ask ey-Wilson algebra or the closure and dual closure r elat io ns ar e quite instrumen tal 17 in clarifying v arious prop erties o f the pair of orthogonal p olynomials of a discrete v ariable [8, 21]. No w let us consider the discrete quan tum mec hanics with t he pure imaginary shifts. In this case, the sin usoidal co ordinate η ( x ) t a k es the contin uous v alue (sp ectrum) for t he con tinuous range of x ∈ ( x 1 , x 2 ) (2.2), whic h is mark edly differen t from the spectrum of H p ostulated to take the semi-infinite discrete v a lues (2.6). The eigenp olynomials e H P n ( η ( x )) = E ( n ) P n ( η ( x )) dep end on the con tin uous parameter x and they hav e no dual p olynomials. The Ask ey-Wilson a nd the Wilson p olynomials a r e the t ypical examples [13, 9 ]. As shown in previous w ork [5, 9 ] and in § 3.2, the essen tial information on exact solv abilit y is con tained only in the closure relation (3.3). There is no evidence that the dual closure relation pla ys a comparable ro le t o the closure relation. There fore w e ma y conclude that the apparen t symmetry b etw een H and η , o r K 1 and K 2 or A and A × in the Ask ey-Wilson a lgebra is quite misleading for the pure imaginary shifts case. In other w ords, a part of the Ask ey- Wilson algebra is irrelev an t to the orthogo nal p olynomials of a con tin uous v ariable. 3.4 shap e in v ariance Let us briefly review the conditio n and the outcome of the shap e inv ariance [1 ] in our la n- guage. In many cases the Hamiltonian con tains some parameter(s), λ = ( λ 1 , λ 2 , . . . ). Here w e write para meter dep endence explicitly , H ( λ ), A ( λ ), E ( n ; λ ), φ n ( x ; λ ), etc, since it is the cen tral issue. The shap e inv ariance condition is [6, 7, 8, 9] A ( λ ) A ( λ ) † = κ A ( λ ′ ) † A ( λ ′ ) + E (1 ; λ ) , (3.45) where κ is a real p ositiv e parameter and λ ′ is uniquely determined b y λ . Let us write the mapping as a function, λ ′ = si( λ ). In concrete examples, if w e take λ appropriately , λ ′ has a simple additiv e f orm λ ′ = λ + δ . The energy sp ectrum and the excited state wa v efunction are determined by the data of the g roundstate wa v efunction φ 0 ( x λ ) and the energy of the first excited state E (1 ; λ ) as follo ws: E ( n ; λ ) = n − 1 X s =0 κ s E (1 ; λ [ s ] ) , (3.46) φ n ( x ; λ ) ∝ A ( λ [0] ) † A ( λ [1] ) † A ( λ [2] ) † · · · A ( λ [ n − 1] ) † φ 0 ( x ; λ [ n ] ) . (3.47) Here λ [ n ] is λ [0] = λ , λ [ n ] = si ( λ [ n − 1] ) ( n = 1 , 2 , . . . ). 18 3.4.1 pure imaginary shifts case Here is a unified pro of of the shap e in v aria nce for the discrete quantum mec hanics with pure imaginary shifts. The shap e inv ariance condition (3.45) is decomp o sed to t he following set of t wo equations: V ( x − i γ 2 ; λ ) V ∗ ( x − i γ 2 ; λ ) = κ 2 V ( x ; λ ′ ) V ∗ ( x − iγ ; λ ′ ) , (3.48) V ( x + i γ 2 ; λ ) + V ∗ ( x − i γ 2 ; λ ) = κ  V ( x ; λ ′ ) + V ∗ ( x ; λ ′ )) − E ( 1 ; λ ) . (3.49) W e a ssume that η ( x ) satisfies the relation η ( x ) = [ 1 2 ]  η ( x − i γ 2 ) + η ( x + i γ 2 ) − η ( − i γ 2 ) − η ( i γ 2 )  . (3.50) Moreo ver we assume that η ( x ; λ ′ ) = η ( x ; λ ), for example it is satisfied if η ( x ) is λ - indep enden t. Both ar e easily v erified in each of t he explicit examples listed in the Ap- p endix A, (A.1)–(A.8). When the forms of the p ot ential functions (2.30) and (2.31) (with L = 2) are substituted, the shap e inv ariance conditions (3.4 8)–(3.49) are satisfied. If w e tak e { v k , 0 ( k = 0 , 1 , 2) , v k , 1 ( k = 0 , 1 ) } as λ , then λ ′ and E (1 ; λ ) a re κv ′ 2 , 0 = − v 1 , 1 , (3.51) κv ′ 1 , 1 = v 2 , 0 + [2] v 1 , 1 , (3.52) κv ′ 1 , 0 = [ 1 2 ]( v 1 , 0 − v 0 , 1 ) + r (2) − 1  [ 1 4 ] 2 [ 1 2 ] v 2 , 0 + v 1 , 1  , (3.53) κv ′ 0 , 1 = [ 1 2 ] v 1 , 0 + [ 3 2 ] v 0 , 1 + r (2) − 1  [ 1 4 ] 2 [ 1 2 ] v 2 , 0 + [ 1 4 ][ 3 4 ] [ 1 2 ] 2 v 1 , 1  , (3.54) κv ′ 0 , 0 = v 0 , 0 + r (2) − 1  [ 1 4 ][ 3 4 ] [ 1 2 ] v 0 , 1 − [ 1 4 ] 2 [ 1 2 ] v 1 , 0  + [ 1 2 ] 2  [ 1 4 ] 4 [ 1 2 ] 4 r (2) 2 − 1 − η ( − iγ ) η ( iγ )  v 2 , 0 − [ 1 2 ]  [ 1 4 ] 3 [ 3 4 ] [ 1 2 ] 3 r (2) 2 − 1 + [ 3 2 ] η ( − iγ ) η ( iγ )  v 1 , 1 , (3.55) E (1 ; λ ) = v 1 , 1 . (3.56) Note t ha t the ab ov e f orm ula E (1 ; λ ) is consisten t with the general form ula ( 3.1). It is elemen tary t o ve rify that the quadratic recursion formula generated by ( 3 .51) and (3.5 2) coupled with the shap e in v aria nce energy formulas (3.46) and (3.56) r epro duces the energy eigen v alue form ula (3.1). How ev er, the ot her form ulas for the parameter shifts (3.53)–(3.55) seem to o complicated to b e practical. As sho wn in [6, 7 , 5, 9], the para meter shifts ar e m uch simpler for the kno wn examples. 19 remark In ordinary quantum mec hanics t here is a metho d for constructing a family of isosp ectral Hamiltonians, kno wn a s Crum’s theorem [2]. R ecently we hav e obtained its discrete quan tum mec hanics ve rsion, see [25]. If φ 1 ( x ) tak e a form φ 1 ( x ) = φ 0 ( x )(const + const · η ( x )), which o ccurs indeed in the setting of this pap er, t he p otential function of the first asso ciated Hamiltonia n is giv en b y V [1] ( x + i γ 2 ) = V ( x ) η ( x − iγ ) − η ( x ) η ( x ) − η ( x + iγ ) . (3.57 ) Therefore, if shap e in v aria nce holds, V ( x ) satisfies V ( x + i γ 2 ; λ ′ ) = κ − 1 V ( x ; λ ) η ( x − iγ ; λ ) − η ( x ; λ ) η ( x ; λ ) − η ( x + iγ ; λ ) , (3.58) in whic h the sin usoidal co o r dinate ma y depend on λ . 3.4.2 real shifts case The shap e inv ariance (3.45) is equiv alent to the following set of tw o equations: B ( x + 1 ; λ ) D ( x + 1 ; λ ) = κ 2 B ( x ; λ ′ ) D ( x + 1 ; λ ′ ) , (3.59) B ( x ; λ ) + D ( x + 1 ; λ ) = κ  B ( x ; λ ′ ) + D ( x ; λ ′ )) + E (1 ; λ ) . (3.60) F or the classified five t yp es of η ( x ), (i) ′ –(v) ′ in (A.9)–( A.1 3 ), the shap e in v aria nce holds. The b oundary condition D (0 ) = 0 (2.9) forces to c ho ose v 0 , 0 as v 0 , 0 = − v 0 , 1 η ( − 1). Th us we tak e the parameters { v k , 0 ( k = 1 , 2) , v k , 1 ( k = 0 , 1) } (and d for (ii) ′ and (v) ′ ) as λ , then λ ′ and E (1 ; λ ) ar e κv ′ 2 , 0 = − v 1 , 1 , (3.61) κv ′ 1 , 1 = v 2 , 0 + [2] v 1 , 1 , (3.62) κv ′ 1 , 0 = µ [ 1 2 ]( v 1 , 0 − v 0 , 1 ) + µ [ 1 2 ] η (1) v 2 , 0 + ν r [2] − 1 v 1 , 1 , (3.63) κv ′ 0 , 1 = µ  [ 1 2 ] v 1 , 0 + [ 3 2 ] v 0 , 1  + µ [ 1 2 ] η (1) v 2 , 0 +  ν r [2] − 1 + µ [ 1 2 ]  η (1) − η ( − 1)   v 1 , 1 , (3.64) E (1 ; λ ) = − v 1 , 1 , (3.65) d ′ = ( d + 1 for (ii) ′ dq for (v) ′ , (3.66) in whic h µ and ν ar e constants µ =      1 for (i) ′ –(ii) ′ q − 1 2 for (iii) ′ q 1 2 for (iv) ′ – (v) ′ , ν =    1 for (i) ′ –(iv) ′ 1 + dq 1 + d for (v) ′ . (3.67) 20 The quadratic recursion formu la (3 .6 1) and (3.62) a re exactly the same as those of the pure imaginary shifts case (3 .51) and (3.52). Therefore the shap e in v ariance energy formulas (3.46) and (3 .6 5) pro duce the same energy sp ectra (3.1). F or the finite dimensional case, the natural n umber N satisfying B ( N ; λ ) = 0 (2.9) is also coun ted as a v arying parameter. Then the shap e inv ariance including the conditions N ′ = N − 1 , B ( N ′ ; λ ′ ) = 0 (3.68) is satisfied. As sho wn in [8], the parameter shifts are muc h simpler for the kno wn examples. 4 Quasi-Exactly Solv able e H ′ Quasi-exact solv ability (QES) means that only a finite part of the sp ectrum and the cor- resp onding eigenfunctions can b e obtained exactly [15]. Usually suc h a theory con tains a finite dimensional v ector space [26] consisting of p o lynomials of a certain degree whic h forms an inv arian t subspace of the ‘Hamiltonian’ e H , or mor e precisely its mo dification e H ′ . There are many wa ys to accomplish QES. The metho d of t his pap er can b e considered as a simple generalisation of the one in [27]. That is, to add non-solv able higher o rder term(s) together with comp ensation term(s) to an exactly solv able theory . As is clear from the construction, the sin usoidal co ordinate pla ys an essen tial role. F or a given p ositiv e integer M , let us try to find a QES ‘Hamiltonian’ e H , or more precisely its mo dification e H ′ , ha ving an in v aria n t p olynomial space V M : e H ′ V M ⊆ V M . (4.1) F or L ≥ 3 , (2.34) is e H η ( x ) n = L − 3 X m =0 η ( x ) n + L − 2 − m m X j =0 e m,j,n +  a p olynomial of degree n in η ( x )  . (4.2) So let us define e H ′ b y adding comp ensation terms to e H as e H ′ def = e H − L − 3 X m =0 e m ( M ) η ( x ) L − 2 − m , e m ( M ) def = m X j =0 e m,j,M . ( 4 .3) Then w e hav e e H ′ η ( x ) M ∈ V M . F or 1 ≤ m ′ ≤ L − 3, we ha ve e H ′ η ( x ) M − m ′ = L − m ′ − 3 X m =0 η ( x ) M + L − m ′ − 2 − m  m X j =0 e m,j,M − m ′ − e m ( M )  21 +  a p olynomial of degree M in η ( x )  . (4.4) If w e could c ho ose v k ,l to satisfy all these conditions m X j =0 e m,j,M − m ′ − e m ( M ) = 0 (1 ≤ m ′ ≤ L − 3 , 0 ≤ m ≤ L − m ′ − 3) , (4.5) then w e w o uld obtain e H ′ V M ⊆ V M . 4.1 QES with L = 3 F or the L = 3 case, e H ′ is defined b y adding one comp ensation t erm o f degree one e H ′ def = e H − e 0 ( M ) η ( x ) , e 0 ( M ) def = e 0 , 0 ,M , (4.6) and w e ha v e a c hiev ed the quasi-exact solv a bilit y e H ′ V M ⊆ V M . The n um b er of exactly determined eigenstates is M + 1 = dim( V M ). In this case there is no extra conditions for v k ,l . The explicit form o f e 0 ( M ) is e 0 ( M ) = ε [ M 2 ] [ 1 2 ]  [ M − 1 2 ] v 3 , 0 + [ M +1 2 ] v 2 , 1  . (4.7) This Q ES theory has t w o more parameters v 3 , 0 and v 2 , 1 on top of those in the original exactly-solv able theory ( L = 2). Most kno wn examples of QES b elong to this category but those in ordinary quan tum mec hanics hav e only one extra para meter. 4.2 QES with L = 4 This t yp e of QES theory is new. F or L = 4 case, e H ′ is defined b y adding a linear and a quadratic in η ( x ) comp ensation terms to the Hamiltonian e H : e H ′ def = e H − e 0 ( M ) η ( x ) 2 − e 1 ( M ) η ( x ) , e 0 ( M ) def = e 0 , 0 ,M , e 1 ( M ) def = e 1 , 0 ,M + e 1 , 1 ,M , (4.8) and e H ′ η ( x ) M ∈ V M . By using (2.28) w e ha ve e H ′ η ( x ) M − 1 = η ( x ) M +1  e 0 , 0 ,M − 1 − e 0 ( M )  +  a p olynomial of degree M in η ( x )  = − εη ( x ) M +1  [ M − 1] v 4 , 0 + [ M ] v 3 , 1  +  a p olynomial of degree M in η ( x )  . (4.9) In order to eliminate the η ( x ) M +1 term, w e c ho ose v 3 , 1 as v 3 , 1 = − [ M − 1] [ M ] v 4 , 0 . (4.10) 22 W e ha v e ac hiev ed the quasi-exact solv ability e H ′ V M ⊆ V M . The explicit forms of e 0 ( M ) a nd e 1 ( M ) are e 0 ( M ) = − ε [4][ M 2 ][ M − 1 2 ] [ 1 2 ][ M + 3] v 4 , 0 , (4.11) e 1 ( M ) = ε [ M 2 ] [ 1 2 ]  [ M − 1 2 ] v 3 , 0 + [ M +1 2 ] v 2 , 1  − εr (2) − 1 v 4 , 0 ×    M ( M − 1)( M 2 +5 M +8) M +3 for r (1) 1 = 0 , 2[ M 2 ][ M − 1 2 ] r (1) 1 [ 1 2 ][ M +3]  [4] − 2[3] + 2[ 1 2 ] [2 M +5] [ 2 M +5 2 ]  for r (1) 1 6 = 0 . (4.12) The theory has three more free par a meters on top of those of the original exactly solv able theory ( L = 2). 4.3 non-QES for L ≥ 5 The higher L b ecomes, the num b er of conditions to b e satisfied (4.5) increases more rapidly than the n um b er of additional parameters. W e will sho w that L ≥ 5 case cannot b e made QES. The condition (4.5 ) with m = 0 giv es e 0 , 0 ,M = e 0 , 0 ,M − m ′ (1 ≤ m ′ ≤ L − 3), and by using ( 2.37) and (2.28) w e obtain [ M − m ′ +1 2 ] v L, 0 + [ M − m ′ − 1 2 ] v L − 1 , 1 = 0 (1 ≤ m ′ ≤ L − 3) . (4.13) F or L ≥ 5 case, these equations do not ha v e non-trivial solutions. F or m ′ = 1 , 2 w e obtain  [ M − 1] [ M ] [ M − 3 2 ] [ M − 1 2 ]   v L, 0 v L − 1 , 1  =  0 0  . (4.14) The determinant of this matrix is [ 1 2 ] whic h do es not v anish. Th us w e obt a in v L, 0 = v L − 1 , 1 = 0. Namely there is no v k ,l ( k + l = L ) term. Therefore L ≥ 5 case cannot b e made QES. 5 (Quasi-)Exactly Solv able Hamiltoni an If there exists a groundstate w av efunction φ 0 ( x ) whic h satisfies (2.14) (and | | φ 0 | | < ∞ , the hermiticit y of H ), w e can return to the Hamiltonian H from the ‘Hamiltonian’ e H b y the in vers e similarit y tra nsformation (2.15). In the same w ay the QES Hamiltonian H ′ is ob- tained from e H ′ b y the in v erse similarit y transformation in terms o f the pseudo-g r oundstate w av efunction φ 0 ( x ) satisfying A φ 0 = 0 (2.14), H ′ def = φ 0 ( x ) ◦ e H ′ ◦ φ 0 ( x ) − 1 . (5.1) 23 It should b e noted that φ 0 ( x ) is neither the groundstate nor an eigenstate of the t otal Hamiltonian e H ′ . Th us it is called the pseudo-gro undstate w a vefunc tion. F or the L = 3 , 4 cases w e ha v e L = 3 : H ′ def = H − e 0 ( M ) η ( x ) , (5.2) L = 4 : H ′ def = H − e 0 ( M ) η ( x ) 2 − e 1 ( M ) η ( x ) . (5.3) Let us note that the Hamiltonian H ′ do es not factorise and the semi p ositiv e-definite sp ec- trum is lost due to the comp ensation terms. F or the pure imaginary shifts case, the existence of (pseudo-)groundstate w av efunction φ 0 ( x ) strongly dep ends on the concrete form o f V ( x ) and its parameter r a nge. There is no general form ula to write down φ 0 ( x ) in terms of V ( x ). On the other hand, f or the real shifts case, the (pseudo-)groundstate wa v efunction φ 0 ( x ) is uniquely giv en b y [8] φ 0 ( x ) = v u u t x − 1 Y y = 0 B ( y ) D ( y + 1) . (5.4) The p ositivit y of B ( x ) and D ( x ) (2.9) restricts their parameter ra nge. F or the infinite case x ∈ [0 , ∞ ), the square-summabilit y | | φ 0 | | < ∞ restricts the asymptotic forms of B ( x ) and D ( x ). 6 Summary and th e Recip e Based on the sin usoidal co o rdinate η ( x ) w e ha v e systematically explored a unified theory of one-dimensional exactly a nd quasi-exactly solv able ‘discrete’ quantum mec hanical mo dels. The Hamiltonians of discrete quantum mec hanics hav e shift op erators as exponentiated forms of the momen tum op erator p = − i∂ x , e ± β p = e ∓ iβ ∂ x . This metho d applies to b o t h the pure imaginary shifts ( β = γ ∈ R 6 =0 ) and the real shifts cases ( β = i = √ − 1), whic h ha v e a con tinuous and a discrete dynamical v ariable x , resp ectiv ely . The main input is the sp ecial form of the p o ten tia l functions V ± ( x ) (2.30) and (2 .31), with which the ‘Hamiltonian’ e H (2.16) maps a p olynomial in η ( x ) in to ano t her. W e o bta in exactly solv able mo dels (degree n → n ) and quasi-exactly solv able mo dels (degree n → n + 1 , n + 2) by adding comp ensation terms, whic h a r e linear a nd quadratic in η ( x ), resp ectiv ely . Th e QES Hamiltonians based on the mapping (degree n → n + 2) are new. The correspo nding result in the ordinary QM can b e found in the App endix of [5]. This 24 early work, how ev er, do es not co v er the quasi-exactly solv a ble cases. In this connection, see the recen t dev elopmen ts [28]. The presen t pap er is for the presen tation of the ba sic formalism. Application and concrete examples will b e explored in a subseq uen t publication [14]. The explicit forms of v arious sin usoidal co ordinates are listed in App endix A. The fo rms of the (pseudo-)gro undstate w a vefunc tions φ 0 ( x ) for the pure imaginary shifts (the con t in uous v ariable) case dep end on the choice of the sin usoidal co or dinates. They are ‘gamma functions’ ha ving v arious shift prop erties; t he (Euler) gamma function for (i)–(ii) (A.1)–(A.2), the q - gamma function (or the q -P o c hhammer sym b ol) fo r (iii)–(iv) (A.3)–(A.4), the double gamma function (or the quan tum dilogar it hm f unction) for (v)–(viii) (A.5)–(A.8) . In a subsequen t publication w e will presen t explicit examples of new Hamiltonians based on (A.5)–(A.8). Their eigenfunctions con tain orthogonal p o lynomials and the double gamma functions as the or t hogonality measure functions. Eigenp olynomials P n ( η ( x )) f or exactly solv able QM b elong to the Ask ey-sche me of h ypergeometric o rthogonal p olynomials. V arious examples of exactly solv able QM w ere in ves tigated for (i)–(iv) (A.1)–(A.4) [9] and (i) ′ –(v) ′ (A.9)–(A.13) [8], and quasi-exactly solv able Q M w ere partially examined in [16, 17]. The simple recip e to construct an exactly o r quasi-exactly solv a ble Hamiltonian is as follo ws: (1) Cho ose the sin usoidal coo rdinate among (A.1)– (A.8) if the v a riable is con tinuous, among ( A.9)–(A.13 ) for the discrete v ariable. (2) Cho ose L = 2 for exact solv ability and L = 3 , 4 for quasi-exact solv ability and write do wn the ‘Hamiltonian’ e H in t he p olynomial space (2.30) and (2.31) with the free parameters v k ,l , k + l ≤ L , l = 0 , 1. F or the quasi-exactly solv able case, add the prop er comp ensation terms (4.6)– (4.7) for the L = 3 case and (4.8 )–(4.12) for the L = 4 case. (3) Determine the (pseudo-)gro undstate φ 0 as a zero mo de of A , (2.14), which can b e fo und among the v arious gamma functions listed as ab ov e o r (5.4) for the discrete v aria ble case. (4) Restrict the pa rameter rang es so that the square-in tegrabilit y of φ 0 and the hermiticit y is satisfied for the contin uous v ariable case. F or the discrete v ariable case the p o sitivit y B ( x ) > 0 a nd D ( x ) > 0 a nd the b oundary condition(s) D (0) = 0, ( B ( N ) = 0) (2.9) 25 are the conditions to restrict the parameters. F or the infinite dimensional case the square summabilit y P ∞ x =0 φ 2 0 ( x ) < ∞ mu st b e satisfied, to o. (5) Apply the inv erse similarit y t r a nsformation (5.1) in t erms of φ 0 on the ‘Hamiltonian’ in the p olynomial space to g et the Hamiltonian H or H ′ . Ac knowledgemen ts W e tha nk P aul T erwilliger for fruitful discussion. This w o rk is supp o rted in part by G r a n ts- in-Aid for Scien tific Researc h from the Ministry of Education, Culture, Sp orts, Science and T ech nology , No.1834 0061 and No .19540179 . A List of S i n uso idal Co ordinates Here w e list the explicit forms of the sinusoidal co ordinates, based o n whic h v arious concrete examples of exactly and quasi-exactly solv able theories are constructed. These are eigh t sinus oidal co ordinates fo r the pure imaginary shifts case (contin uous x ): (i) : η ( x ) = x, −∞ < x < ∞ , γ = 1 , (A.1) (ii) : η ( x ) = x 2 , 0 < x < ∞ , γ = 1 , (A.2) (iii) : η ( x ) = 1 − cos x, 0 < x < π , γ ∈ R 6 =0 , (A.3) (iv) : η ( x ) = sin x, − π 2 < x < π 2 , γ ∈ R 6 =0 , (A.4) (v) : η ( x ) = 1 − e − x , −∞ < x < ∞ , γ ∈ R 6 =0 , (A.5) (vi) : η ( x ) = e x − 1 , −∞ < x < ∞ , γ ∈ R 6 =0 , (A.6) (vii) : η ( x ) = cosh x − 1 , 0 < x < ∞ , γ ∈ R 6 =0 , (A.7) (viii) : η ( x ) = sinh x, −∞ < x < ∞ , γ ∈ R 6 =0 , (A.8) and fiv e sin usoidal co ordinat es for the real shifts case (in teger x ): (i) ′ : η ( x ) = x, (A.9) (ii) ′ : η ( x ) = ǫ ′ x ( x + d ) , ǫ ′ = n 1 for d > − 1 , − 1 for d < − N , (A.10) (iii) ′ : η ( x ) = 1 − q x , (A.11) (iv) ′ : η ( x ) = q − x − 1 , (A.12) 26 (v) ′ : η ( x ) = ǫ ′ ( q − x − 1)(1 − dq x ) , ǫ ′ = n 1 for d < q − 1 , − 1 for d > q − N , (A.13) where 0 < q < 1. As sho wn in detail in § 4C of [8], the ab ov e five sin usoidal co ordinates fo r the real shifts (A.9)–(A.13) exhaust a ll the solutions of (2.18)–(2.20) up to a multiplic ativ e factor. On the o ther hand, those for t he pure imaginar y shifts (i)–(viii) (A.1)–( A.8) are merely t ypical examples satisfying all the p ostulates for the sin usoidal co ordinate (2.1 8)–(2.20) and the extra one used fo r t he shap e in v aria nce (3.50). It is easy to see that η ( x ) = x + sinh(2 π x ), −∞ < x < ∞ , γ = 1 is a go o d sin usoidal co ordinate for the imaginary shifts but it f ails to fulfill the extra condition (3.50). B Hermiticit y of the Hamilto nian In this App endix we recapitulate the pro of of the hermiticity o f the Hamiltonian H (2.7) for the pure ima g inary shifts ( contin uous v ariable) case [16, 9]. F or the real shifts (discrete v ar ia ble) case the Hamiltonian H (2.7) is a hermitian matrix (real symmetric matrix) and there is no problem for the hermiticit y . Th us we consider only the con tin uous v ariables case, in whic h the wa v efunctions and the p oten tial functions ar e analytic function of x as explained in § 2. The ∗ - op eration on analytic functions is also defined there. By using the for mula ( AB ) † = B † A † , w e obtain H † = H but this is formal hermiticit y . In or der to demonstrate the true hermiticit y of H , we hav e to sho w ( g , H f ) = ( H g , f ) with resp ect to the inner pro duct (2.4). Since the eigenfunctions considered in this pa p er ha v e the fo r m φ 0 ( x ) P n  η ( x )  , it is sufficie n t to c hec k for f ( x ) = φ 0 ( x ) P  η ( x )  and g ( x ) = φ 0 ( x ) Q  η ( x )  , where P ( η ) and Q ( η ) are p olynomials in η . Since H is real H ∗ = H , namely H maps a ‘real’ f unction t o a ‘real’ function ( f ∗ ( x ) = f ( x ) ⇒ ( H f ) ∗ = ( H f )) , we can tak e φ 0 ( x ), P  η ( x )  and Q  η ( x )  to b e ‘real’ functions of x and w e do so in the follo wing. The (pseudo-)groundstate w a vefunc tion φ 0 ( x ) is determined as a zero mo de of A , A φ 0 = 0 (2.14). The equation reads q V ∗ ( x − i γ 2 ) φ 0 ( x − i γ 2 ) = q V ( x + i γ 2 ) φ 0 ( x + i γ 2 ) . (B.1) Let us define T + = p V ( x ) e γ p p V ∗ ( x ) and T − = p V ∗ ( x ) e − γ p p V ( x ) . Then the Hamiltonian (2.7) is H = T + + T − − V ( x ) − V ∗ ( x ). F or the QES case, the comp ensation terms are added. It is obv ious that the function part − V ( x ) − V ∗ ( x ) (plus p ossible comp ensation t erms) is 27 hermitian b y itself. Let us define tw o analytic functions F ( x ) and G ( x ) a s follo ws: g ∗ ( x ) T + f ( x ) = φ ∗ 0 ( x ) Q ∗  η ∗ ( x )  p V ( x ) p V ∗ ( x − iγ ) φ 0 ( x − iγ ) P  η ( x − iγ )  def = F ( x ) , (B.2) g ∗ ( x ) T − f ( x ) = φ ∗ 0 ( x ) Q ∗  η ∗ ( x )  p V ∗ ( x ) p V ( x + iγ ) φ 0 ( x + iγ ) P  η ( x + iγ )  def = G ( x ) . (B.3) Then w e hav e ( T + g ) ∗ ( x ) f ( x ) = φ ∗ 0 ( x + iγ ) Q ∗  η ∗ ( x + iγ )  p V ∗ ( x ) p V ( x + iγ ) φ 0 ( x ) P  η ( x )  = F ( x + iγ ) , (B.4) ( T − g ) ∗ ( x ) f ( x ) = φ ∗ 0 ( x − iγ ) Q ∗  η ∗ ( x − iγ )  p V ( x ) p V ∗ ( x − iγ ) φ 0 ( x ) P  η ( x )  = G ( x − iγ ) . (B.5) By using ( B.1 ) and the ‘realit y’ of φ 0 ( x ), η ( x ), P ( η ), Q ( η ), w e obtain g ∗ ( x ) T + f ( x ) = V ( x ) φ 0 ( x ) 2 Q  η ( x )  P  η ( x − iγ )  = F ( x ) , (B.6) g ∗ ( x ) T − f ( x ) = V ∗ ( x ) φ 0 ( x ) 2 Q  η ( x )  P  η ( x + iγ )  = G ( x ) , (B.7) ( T + g ) ∗ ( x ) f ( x ) = V ( x + iγ ) φ 0 ( x + iγ ) 2 Q  η ( x + iγ )  P  η ( x )  = F ( x + iγ ) , (B.8) ( T − g ) ∗ ( x ) f ( x ) = V ∗ ( x − iγ ) φ 0 ( x − iγ ) 2 Q  η ( x − iγ )  P  η ( x )  = G ( x − iγ ) . (B.9) Therefore the necessary and sufficien t condition for the hermiticit y of the Hamiltonian b e- comes Z x 2 x 1  F ( x ) + G ( x )  dx = Z x 2 x 1  F ( x + iγ ) + G ( x − iγ )  dx. (B.10) Of course it is required that there is no singularity on the in tegra t ion con tours. Let C ± b e the rectangular con tours x 1 → x 2 → x 2 ± iγ → x 1 ± iγ → x 1 and D ± b e the regions surrounded b y C ± including the contours. Under the assumption t ha t F ( x ) and G ( x ) do not ha ve singularities on C + and C − resp ectiv ely 2 , the residue theorem implies that (B.10) is rewritten as Z γ 0  F ( x 2 + iy ) − F ( x 1 + iy ) − G ( x 2 − iy ) + G ( x 1 − iy )  dy = 2 π γ | γ |  X x : p ole in D + Res x F ( x ) − X x : pole in D − Res x G ( x )  . ( B.1 1) W e will men tion sev eral sufficien t conditions for (B.11). If F ( x ) and G ( x ) are holo mor phic in D + and D − resp ectiv ely , the r .h.s. of (B.11) v anishes. In the following w e assume this. 2 If there are sing ularities o n the contours x 2 → x 2 ± iγ o r x 1 ± iγ → x 1 , w e deform the cont ours and redefine C ± and D ± in order to avoid singularities o n C ± . F or simplicity we have assumed no singula rity on C ± in the text. 28 case 1: x 1 = −∞ , x 2 = ∞ . If φ 0 ( x ) is rapidly decreasing (e.g. exp onen tia l in η ( x )) at x ∼ ±∞ , then (B.11) is satisfied. case 2 : x 1 = 0, x 2 = ∞ . If φ 0 ( x ) is rapidly decreasing (e.g. exp onential in η ( x )) at x ∼ ∞ , then (B.11 ) b ecomes R γ 0  F ( iy ) − G ( − iy )  dy = 0. This is satisfied if F ( iy ) = G ( − iy ), y ∈ (0 , γ ). As a sufficien t condition for F ( iy ) = G ( − iy ), w e giv e the follow ing three reflection prop erties: φ 0 ( − x ) = φ 0 ( x ) , η ( − x ) = η ( x ) , V ∗ ( x ) = V ( − x ) . (B.12) case 3 : x 2 = x 1 + ω (0 < ω < ∞ ). The condition (B.1 1 ) b ecomes R γ 0  F ( x 1 + ω + iy ) − F ( x 1 + iy ) − G ( x 1 + ω − iy ) + G ( x 1 − iy )  dy = 0. This is satisfied if F ( x 1 + ω + iy ) = G ( x 1 + ω − iy ) and F ( x 1 + iy ) = G ( x 1 − iy ), y ∈ (0 , γ ). As a sufficien t condition for F ( x 1 + ω + iy ) = G ( x 1 + ω − iy ) and F ( x 1 + iy ) = G ( x 1 − iy ), w e giv e the follo wing reflection relations and the p erio dicit y: φ 0 ( − x + x 1 ) = φ 0 ( x + x 1 ) , η ( − x + x 1 ) = η ( x + x 1 ) , V ∗ ( x + x 1 ) = V ( − x + x 1 ) , φ 0 ( x + 2 ω ) = φ 0 ( x ) , η ( x + 2 ω ) = η ( x ) , V ( x + 2 ω ) = V ( x ) . (B.13) C Eigen v alues and eig en v ectors for an upp er triang u - lar matrix Let A = ( a ij ) 1 ≤ i,j ≤ n b e an upp er triangular matrix, namely a ij = 0 for i > j . Its eigenv alue are α i = a ii ( i = 1 , . . . , n ). When α i ’s are m utually distinct, the eigen vec tor corresp o nding to the eigen v alue α i is giv en b y the determinan t            e 1 e 2 e 3 · · · e i α 1 − α i a 12 a 13 · · · a 1 i α 2 − α i a 23 · · · a 2 i . . . . . . . . . 0 α i − 1 − α i a i − 1 i            , (C.1) where { e i } is the natural basis, ( e i ) j = δ ij . References [1] L. E. 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