Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 5 (2009), 104, 16 pages Bethe Ansatz Soluti ons to Quasi Exactly Solv able Dif ference Equations ⋆ Ryu SASAKI † , Wen-Li Y AN G ‡§ and Y ao-Zhong ZHA NG § † Y u kawa Institute for The or etic al Physics, Kyoto U niversity, Kyoto 606-85 02, Jap an E-mail: ryu@yukawa.kyoto-u.ac.jp ‡ Institute of Mo dern Physics, Northwest University, Xian 710069 , P.R. China E-mail: w lyang@nwu.e du.cn § Scho ol of Mathematics and Physics, The University of Que enslan d, Brisb ane, QLD 4072, Austr alia E-mail: yzz@maths.uq.e du.au Received Septem be r 20, 2009, in f inal for m Novem ber 10 , 20 09; P ublis hed online Nov ember 18, 2009 doi:10.38 42/SIGMA.20 09.104 Abstract. Bethe ansatz f ormulation is presented for several explicit examples o f quasi exactly solv a ble dif ference equa tio ns of o ne degree o f freedo m which are introduced rec e ntly by one of the present authors. These e q uations ar e defo r mation of the well-known exactly solv able dif ference equations of the Meixner–Pollaczek, contin uous Ha hn, contin uous dual Hahn, Wilson and Askey–Wilson p olynomials . Up to an overall factor of the s o -called pseudo gr o und state wa v efunction, the eigenfunctions within the exactly so lv a ble subspace are giv en b y polyno mia ls whose ro ots are solutions o f the as so ciated Bethe ansa tz equatio ns . The cor resp onding eigenv a lues are expressed in terms of these ro ots. Key wor ds: Bethe ansatz solution; q ua si-exactly solv a ble mo dels 2000 Mathematics S ubje ct Classific ation: 35Q40 ; 37 N20; 39 A7 0; 82B2 3 1 In tro duction Bethe ansatz metho d is one of the wel l-kno wn solution metho d s for exactly solv able qu an tum systems (see, e.g. [1]) as w ell as for v arious sp in m o dels and statistical lattice m o dels. In recen t y ears the concept of exactly solv a ble quan tum systems was drastically enlarged to in clude v arious examples of th e so-called ‘discrete’ qu an tum mechanica l systems [2, 3, 4, 5, 6, 7], in whic h th e Sc hr¨ odinger equation is a dif f er en ce equ ation in stead of dif ferential. Kno wn examples of exactly solv able ‘discrete’ quantum mec hanics are deformations of exactly solv able quan tum mechanics, in wh ich the momentum op erators a pp ear in exp onen tiated forms in stead of p olynomials in ordinary quantum mec hanics. Th eir eigenfun ctions are ( q -)Ask ey sc heme of hypergeometric orthogonal p olynomials [8, 9], wh ic h are deform ations of the classical orthogonal p olynomials, e.g. the Hermite, L aguerr e and Jacobi p olynomials. It is inte resting to note that these examples are exactly solv able b oth in the S c hr¨ odinger and Heisenberg pictures [4, 5]. That is, for eac h Hamiltonian of these examples, the Heisen b erg op erator solution f or a sp ecial co ordinate called the ‘ sinusoidal ’ co ordinate can b e constructed, as w ell as the complete set of the eigenv alues and the corresp ond ing eigenfunctions. The eigenfunctions consist of the ab o v e-men tio ned ( q -)Ask ey sc heme of h yp ergeometric orthogonal p olynomials in the ‘sin usoidal’ co ordinate. See [6, 7] for a comprehensive introdu ction of the ‘discrete’ quantum m echanics and the recent develo pment s. ⋆ This pap er is a contribution to the Pro ceedings of the 5-th Microconference “A n alytic and Algebraic Me- tho ds V”. The full collection is a v ailable at http://www.emis .de/journals/SIGMA/Prague2009.h tml 2 R. Sasaki, W.-L. Y ang and Y.-Z. Zh ang The domain of the Hamiltonian, its hermiticit y (self-adjoin tness), the exact Heisen b erg op e- rator solutions, the creation-annihilation op er ators and the d ynamical sym metry algebras are explained in some detail. In this p ap er we presen t Bethe ansatz f orm ulations and solutions for a family of Quasi-Exactly Solv able (QES) dif ference equations, whic h were recently introdu ced by one of the pr esent au- thors [10, 11]. One of the main purp oses of the pr esent pap er is to provide a go o d list of exp licit Bethe-ansatz equations for the qu asi-exactl y solv able dif ference equations (2.11), (3.11), (3.15), (4.9), (4.17), and (5.5). The list w ill b e h elpful for future researc h in v arious context s of m ath- ematical physics. A quantum mechanica l system is called quasi-exactly solv able (QES), if only a f inite n umber ( ≥ 2) of eigen v alues and corresp onding eigen v ecto rs can b e obtained exactly [12, 13]. Among v arious charac terisation/iden tif ication of QES systems [12 , 13, 14, 15], we promoted a simple view that a QES system is obtained by a certain deformation of an exactly solv able quantum system. As sh o wn explicitly for ordinary quantum mec hanics by Sasaki– T ak asaki [15], the deformation pro cedu re applies to systems of m any degrees of freedom as we ll as f or single degree of freedom systems. This is in sharp con trast to the sl (2 , R ) c haracterisa- tion [13], whose app licabilit y is limited to essentia lly single degree of freedom systems. Recen tly the deformation pro cedure w as applied to exactly solv able ‘discrete’ qu an tum systems of one and many degrees of fr eedom to ob tain corresp ondin g ‘d iscrete’ QES systems [10, 11, 16]. In this p ap er w e present Bethe ansatz s olutions for these QES dif ference equations. There are only a ve ry limited num b er of examples of (qu asi) exactly solv able d if f erence equations which hav e b een s olved so far by the Bethe ansatz metho d. T o our kn owledge, the Bethe ansatz solutions we re known only for some QES dif ference equations in connection with U q ( sl (2)) [17], and f or exactly solv able dif ference equations [18, 19, 20] of the elliptic Rusijsenaars–Sc hneider system. Here we apply the Bethe ansatz formulat ion to sev eral explicit examples of QES dif ference equations [10, 11] as deformation of exactly solvable ‘discrete’ quan- tum mec hanics [2, 3], wh ic h are dif ference analogues of the well-kno wn qu asi exactly solv able systems, the harmon ic oscillator (with/without the cen trifugal p oten tia l) d eformed by a sextic p oten tial and the 1 / sin 2 x p oten tial deformed by a cos 2 x p oten tial. As will b e sho wn explicitly in the m ain text, these Bethe ansatz equations (2.11), (3.11), (3.15), (4.9), (4.17), and (5.5) can b e considered as deformations of the equations (2.16), (4.14), (5.9) determining th e ro ots of the corresp ondin g ( q -)Ask ey sc heme of h yp ergeometric orth ogonal p olynomials [9] (the Meixner– P ollacz ek, con tin uous Hahn, con tin uous dual Hahn, Wilson and Aske y–Wilson p olynomials and their restrictions) wh ic h constitute the eigenfunctions of the u ndeformed exactly solv able quan- tum systems. The general structure of the quasi exactly solv able d if ference equations to b e discussed in th is pap er and their solutions were exp lained in some detail in [10, 11, 16], inclu- ding the domains of the Hamiltonians, the Hilb ert spaces and hermiticit y and th e r ole p la y ed b y the p seudo ground state wa vefunctions. This pap er is organised as follo ws. In S ection 2, a QES discrete qu antum mec hanics is solv ed in the Bethe ansatz formalism. The Hamiltonian of the system is obtained by ‘crossing’ those of the Meixner–P olla czek and the con tin uous Hahn p olynomials as derive d in [11]. Section 3 pro vides the Bethe ansatz formulat ion of the QES discrete quantum mec hanical systems which are d eformations of the harmonic oscillator w ith a s extic p oten tial as deriv ed in [10]. The cor- resp ond ing eigenfunctions are deformations of the Meixner–P ollacz ek and th e con tin uous Hahn p olynomials. S ection 4 giv es the Bethe ansatz s olutions of th e QES discrete quan tum mec hanical systems w hic h are d eformations of the h arm onic oscillator with a cen trifugal b arrier and a sextic p oten tial as derive d in [10]. T he corresp ond in g eigenfunctions are deformations of the con tin uous dual Hahn and the Wilson p olynomials. Section 5 of fer s a Bethe ansatz solution to a dif ference equation analogue of a QES system with the 1 / sin 2 x p oten tia l deformed by a cos 2 x p otentia l as deriv ed in [10]. Th e corresp ondin g eigenfunctions are d eformations of the Ask ey–Wilson p oly- nomials and th eir v arious restrictions [8, 9]. The f inal section is for a s ummary and comments. Bethe Ans atz Solutions to Qu asi Exactly Solv able Dif ference Equations 3 2 Dif ference equation of the M eixner–P ollaczek t yp e In this s ection we w ill discuss Bethe ansatz solutions for the discrete quantum mechanics ob- tained by deforming that of the Meixner–P olla czek p olynomials in [11]. T o b e more precise, the corresp ond ing discrete quantum mechanics is obtained b y crossing th ose of the Meixner– P ollacz ek and the con tin uous Hahn p olynomials, that is, with the qu ad r atic p otent ial function of the con tin uous Hahn p olynomial m ultiplied b y a constan t ph ase factor e − iβ of the Meixner– P ollacz ek type. It was sho wn in [11] that this system is quasi exactly solv able. T he corresp ond ing Hamiltonian is: H def = p V ( x ) e − i∂ x p V ( x ) ∗ + p V ( x ) ∗ e + i∂ x p V ( x ) − V ( x ) − V ( x ) ∗ + α M x (2.1) = A † A + α M x, α M def = − 2 M sin β , M ∈ Z + , (2.2) A † def = p V ( x ) e − i 2 ∂ x − p V ( x ) ∗ e i 2 ∂ x , A def = e − i 2 ∂ x p V ( x ) ∗ − e i 2 ∂ x p V ( x ) , (2.3) V ( x ) def = ( a 1 + ix )( a 2 + ix ) e − iβ , a 1 , a 2 ∈ C , Re( a 1 ) > 0 , Re( a 2 ) > 0 . It should b e noted that the Hamiltonian is n o longer p ositiv e semi-def inite but the hermiticit y is preserve d. Let us introd u ce the so-called pseudo gr ound state w a v efunction φ 0 ( x ) [10, 11]: φ 0 ( x ) def = e β x q Γ( a 1 + ix )Γ( a 2 + ix )Γ( a ∗ 1 − ix )Γ( a ∗ 2 − ix ) , as the zero mo de of the A op erator (2.3), Aφ 0 = 0. The similarit y transformed Hamiltonian e H in terms of φ 0 , e H def = φ − 1 0 ◦ H ◦ φ 0 = V ( x )  e − i∂ x − 1  + V ( x ) ∗  e i∂ x − 1  + α M x (2.4) = ( a 1 + ix )( a 2 + ix ) e − iβ  e − i∂ x − 1  + ( a ∗ 1 − ix )( a ∗ 2 − ix ) e iβ  e i∂ x − 1  − 2 M sin β x, acts on the p olynomial part of the wa vefunction. In th e exactly solv able Meixner–Po llaczek case with V ( x ) = ( a + ix ) e − iβ and the conti n uous Hahn case with V ( x ) = ( a 1 + ix )( a 2 + ix ), the eigenfunctions are of the form φ 0 ( x ) P ( η ( x )) in w h ic h P ( η ( x )) is a p olynomial in η ( x ) = x. After the deformation, it is obvio us that e H maps a p olynomial in η ( x ) = x in to another and it is easy to verify e H x n = 2( −M + n ) sin β x n +1 + lo w er order terms , n ∈ Z + . This m eans th at the sys tem is not exactly solv able without the comp ensation term, but it is quasi exactly solv able, since e H has an inv arian t p olynomial sub space of degree M : e HV M ⊆ V M , (2.5) V M def = Span  1 , x, x 2 , . . . , x M  , dim V M = M + 1 . (2.6) Let Ψ ( x ) b e one of the eigenfunctions of e H and E b e the corresp onding eigen v alue: e H Ψ( x ) = E Ψ( x ) , 4 R. Sasaki, W.-L. Y ang and Y.-Z. Zh ang namely , ( a ∗ 1 − ix )( a ∗ 2 − ix ) e iβ (Ψ( x + i ) − Ψ( x )) + ( a 1 + ix )( a 2 + ix ) e − iβ (Ψ( x − i ) − Ψ( x )) − 2 M sin β x Ψ( x ) = E Ψ( x ) . (2.7) Equations (2.5) and (2.6) imp ly that the eigenfunctions in the su bspace V M ha v e the follo wing form Ψ( x ) = M Y l =1 ( η ( x ) − η ( x l )) = M Y l =1 ( x − x l ) , (2.8) where { x l | l = 1 , . . . , M} are some parameters w hic h will b e sp ecif ied later b y the asso ciated Bethe ansatz equations (2.11) b elo w. Sub stituting th e ab o v e equation into (2.7 ) and dividing b oth sides b y Ψ( x ), we hav e E = − V ( x ) − V ( x ) ∗ − 2 M sin β x + V ( x ) M Y l =1 x − x l − i x − x l + V ( x ) ∗ M Y l =1 x − x l + i x − x l . (2.9) The r.h.s. of (2.9) is a meromorphic function of x , wh ereas the l.h.s. is a constan t. T o mak e them equal, w e must null the residu es of the r.h.s. It is easy to see that the singularities of the r.h.s. only app ear at x = x j , j = 1 , . . . , M and x = ∞ . Th e r esidues at x = x j v anish if the parameters { x j } s atisfy the follo wing Bethe ansatz equations M Y l 6 = j x j − x l − i x j − x l + i = V ( x j ) ∗ V ( x j ) η ( x j + i ) − η ( x j ) η ( x j ) − η ( x j − i ) (2.10) = V ( x j ) ∗ V ( x j ) e 2 iβ ( a ∗ 1 − ix j )( a ∗ 2 − ix j ) ( a 1 + ix j )( a 2 + ix j ) , j = 1 , . . . , M . (2.11) Throughout this pap er we use the complex conjugate p otent ial f unction V ( x ) ∗ in the ‘analytical’ sense in x , that is, for complex x V ( x ) = ( a 1 + ix )( a 2 + ix ) e − iβ , V ( x ) ∗ = ( a ∗ 1 − ix )( a ∗ 2 − ix ) e iβ , x ∈ C . This con v en tion is necessary for the ab ov e t w o equations (2.10), (2.11) to b e v a lid, since the Bethe ro ots { x j } are in general complex. O ne can c hec k that the r .h.s. of (2.9) is indeed regular at x = ∞ , i.e., th e residu e at x = ∞ v anishes. By the Liouville theorem the r.h.s. of (2.9) is a constan t provided that (2.11 ) is satisf ied. One can get the v alue of the corresp ond ing eigen v alue E by taking the limit of x → ∞ for the r.h.s. of (2.9). Here w e p resen t the resu lt: E = M ( M − 1) cos β + M  ( a 1 + a 2 ) e − iβ + ( a ∗ 1 + a ∗ 2 ) e iβ  + 2 sin β M X l =1 x l , (2.12) where { x l } s atisfy th e Bethe ansatz equation (2.11). Th e f inal term can b e written as 2 sin β M X l =1 η ( x l ) . (2.13) The wa ve function Ψ( x ) (2.8 ) b ecomes th e eigenfun ction of e H in the su b space V M (2.6) provided that the ro ots of the p olynomial Ψ( x ) (2.8) are the solutions of (2.11), and then the corresp onding eigen v alue is giv en by (2.12). S ince all the ro ots { x l } are on the same fo oting, it is n atural that the eigen v alue E d ep ends on the symmetric com bination of them (2.13). Bethe Ans atz Solutions to Qu asi Exactly Solv able Dif ference Equations 5 A few corollaries ensue from th ese results. F or the sp ecial case of β = 0, the Hamilto- nian (2.1) is exactly solv able and the corresp ond ing eigen v ectors are related to the conti n uous Hahn p olynomials. In fact, we obtain fr om (2.12) lim β → 0 E = M ( M + a 1 + a 2 + a ∗ 1 + a ∗ 2 − 1) , (2.14) whic h is the eigen v alue corresp ond ing to the d egree M con tin uous Hahn p olynomial [2 , 3, 6]. The Bethe ansatz equation (2.11) now determines the zeros of the con tin uous Hahn p olynomial. F or a 1 , a 2 ∈ R + , and a 2 → ∞ limit, the ab o v e Hamiltonian H (2.1) divided by a 2 reduces to an exactly solv able one corresp onding to the Meixner–P ollacze k p olynomials [2, 3 , 6]. C orr esp on- dingly the eigenv alue form ula (2.12) gives lim a 2 →∞ E /a 2 = 2 M cos β . (2.15) The latter is th e eigenv alue of the degree M Meixner–Po llaczek p olynomial [6]. Note that th e parameter β is related to the standard parameter φ of the Meixner–P ollacze k p olynomials as β = π 2 − φ . The corresp on d ing Bethe ansatz equation M Y l 6 = j x j − x l − i x j − x l + i = e 2 iβ ( a 1 − ix j ) ( a 1 + ix j ) , j = 1 , . . . , M , (2.16) no w determines the zeros of the d egree M Meixner–P ollacz ek p olynomial. As sh o wn in § 4 of [2], this equation red uces to that determines th e zeros of the Hermite p olynomial in an appropriate limit. 3 Dif ference equation analogue of h armonic oscillator deformed b y sextic p oten tial There are tw o typ es of d if f erence equations wh ic h are dif ference analogues of the sextic p oten tial Hamiltonian [10]. The Hamiltonian is given by H def = p V ( x ) e − i∂ x p V ( x ) ∗ + p V ( x ) ∗ e i∂ x p V ( x ) − ( V ( x ) + V ( x ) ∗ ) + α M ( x ) , (3.1) T yp e I : V ( x ) def = ( a + ix )( b + ix ) V 0 ( x ) , V 0 ( x ) def = c + ix, a, b, c ∈ R + , (3.2) α M ( x ) def = 2 M x 2 , T yp e I I : V ( x ) def = ( a + ix )( b + ix ) V 0 ( x ) , V 0 ( x ) def = ( c + ix )( d + ix ) , a, b, c, d ∈ R + , (3.3) α M ( x ) def = M ( M − 1 + 2( a + b + c + d )) x 2 . (3.4) Here as u sual V ( x ) ∗ is the ‘analytical’ complex conjugate of V ( x ). 3.1 T yp e I theory Here we will consider the dif ference equation of t yp e I, w hile th e d if ference equation of type I I will b e give n in the next su bsection. If V is r eplaced by V 0 in (3.2) and th e last term in (3.1), α M ( x ), is remo v ed, H b ecomes the exactly solv able Hamiltonian of a d if ference analogue of the har- monic oscilla tor, or the deforme d harmonic oscil lator in ‘d iscrete’ quantum mec hanics [2, 3]. Its eigenfunctions consist of the Meixner–P ollac zek p olynomials, with a sp ecial ph ase angle β = 0, 6 R. Sasaki, W.-L. Y ang and Y.-Z. Zh ang whic h is a deform ation of the Hermite p olynomials [2, 3, 21]. Th e qu adratic p olynomial factor ( a + ix )( b + ix ) can b e consider ed as multiplica tiv e d eformation, although the p arameters a , b and c are on the equal fo oting. On th e other hand one can consider it as a multiplicativ e deformation b y a linear p olynomial in x : V ( x ) = ( a + ix ) V 01 ( x ) , V 01 ( x ) def = ( b + ix )( c + ix ) , with V 01 describing another dif ference version of an exactly solv able analogue of the harmonic oscillato r [2, 3]. Its eigenfunctions consist of the contin u ous Hahn p olynomials. Next let us in tro duce the similarit y transformation in terms of the pseudo ground s tate w a v efunction φ 0 ( x ) as the zero mo de of th e A op erator (2.3 ), Aφ 0 = 0: φ 0 ( x ) def = p Γ( a + ix )Γ( a − ix )Γ( b + ix )Γ( b − ix )Γ( c + ix )Γ( c − ix ) , e H def = φ − 1 0 ◦ H ◦ φ 0 = V ( x )  e − i∂ x − 1  + V ( x ) ∗  e i∂ x − 1  + 2 M x 2 . (3.5) Since the p arit y is conserve d, that is H| x →− x = H , it is easy to verify the action of the Hamiltonian e H (3.5) on monomials of x : e H x n =                [ n/ 2+1] X j =0 a n,j x n +2 − 2 j , n ≤ M − 2 , a n,j ∈ R , [ M / 2] X j =0 a ′ n, j x M− 2 j , n = M , a ′ n,j ∈ R . (3.6) Here [ m ] is the standard Gauss’ symb ol denoting th e greatest integ er n ot exceeding or equ al to m . According to the parit y of the p olynomials, there are t w o t yp es of inv arian t subsp aces V M of e H : e H V M ⊆ V M , (3.7) V M def = ( Span  1 , η ( x ) , . . . , η ( x ) k , . . . , η ( x ) M / 2  , M : even , x S pan  1 , η ( x ) , . . . , η ( x ) k , . . . , η ( x ) ( M− 1) / 2  , M : o d d , η ( x ) = x 2 , (3.8) dim V M =  M / 2 + 1 , M : even , ( M + 1) / 2 , M : o dd . 3.1.1 The case of even M Let us introdu ce a p ositiv e in tege r N su c h that M = 2 N . Equations (3.7) an d (3.8 ) imply that the eigenfunctions of e H in the subsp ace V M are of the form Ψ( x ) = N Y l =1 ( η ( x ) − η ( x l )) = N Y l =1 ( x − x l )( x + x l ) . (3.9) Analogous calculation sho ws that the p olynomial Ψ ( x ) b ecomes the eigenfunction of e H if the ro ots of the p olynomial satisfy the Bethe ansatz equations N Y l 6 = j ( x j − x l − i )( x j + x l − i ) ( x j − x l + i )( x j + x l + i ) = V ( x j ) ∗ V ( x j ) η ( x j + i ) − η ( x j ) η ( x j ) − η ( x j − i ) (3.10) Bethe Ans atz Solutions to Qu asi Exactly Solv able Dif ference Equations 7 = ( a − ix j )( b − ix j )( c − ix j )(2 x j + i ) ( a + ix j )( b + ix j )( c + ix j )(2 x j − i ) , j = 1 , . . . , N . (3.11) The corresp onding eigenv alue E is giv en by E = 1 3 M ( M − 1)( M − 2) + ( a + b + c ) M ( M − 1) + 2( ab + ac + bc ) M − 4 N X l =1 x 2 l , (3.12) where { x l } satisfy the Bethe ans atz equations (3.11). Again the f inal term is symmetric in { x j } and can b e written as − 4 P N j =1 η ( x j ). 3.1.2 The case of o dd M Let u s in tro duce a p ositiv e intege r N su c h th at M = 2 N + 1. (3.7) and (3.8) imply th at the eigenfunctions of e H in the su bspace V M are of the form Ψ( x ) = x N Y l =1 ( η ( x ) − η ( x l )) = x N Y l =1 ( x − x l )( x + x l ) . (3.13) Analogous calculation sho ws that the p olynomial Ψ ( x ) b ecomes the eigenfunction of e H if the ro ots of the p olynomial satisfy the Bethe ansatz equations ( x j − i ) ( x j + i ) N Y l 6 = j ( x j − x l − i )( x j + x l − i ) ( x j − x l + i )( x j + x l + i ) = V ( x j ) ∗ V ( x j ) ( η ( x j + i ) − η ( x j )) ( η ( x j ) − η ( x j − i )) (3.14) = ( a − ix j )( b − ix j )( c − ix j )(2 x j + i ) ( a + ix j )( b + ix j )( c + ix j )(2 x j − i ) , j = 1 , . . . , N . (3.15) The corresp onding eigenv alue E is giv en by E = 1 3 M ( M − 1)( M − 2) + ( a + b + c ) M ( M − 1) + 2( ab + ac + bc ) M − 4 N X l =1 x 2 l , (3.16) where { x l } satisfy the Bethe ansatz equations (3.15). The expression for the eigenv alue is exactly the same as the eve n M case (3.12). In th is example, exactly solv able limits are also obtained b y making one or t w o parameters go to inf inity; for example a → ∞ or b oth a → ∞ and b → ∞ . In the former case ( a → ∞ ), the scaled Hamiltonian (3.1) H /a giv es that of the con tin uous Hahn p olynomials with real parameters b and c . The eigen v alue formulas (3.12) and (3.16 ) reduce to that of the con tin uous Hahn p olynomials [2, 3, 6]: lim a →∞ E /a = M ( M + 2( b + c ) − 1) . (3.17) The Bethe ansatz equations (3.11) and (3.15) are equiv alen t to (2.11) with β = 0 and a 1 = b ∈ R + , a 2 = c ∈ R + . This assertion can b e easily verif ied sin ce the solutions of (2.11) are alw a ys paired { x j , − x j } in clud ing a zero x j = 0 for o dd M . I n the latter case ( a → ∞ , b → ∞ ), the scaled Hamiltonian (3.1) H / ( ab ) giv es that of the Meixner–Po llaczek p olynomials with β = 0, a = c . The eigen v alue form ulas (3.12) and (3.16) reduce to that of the Meixner–P ollacze k p olynomials [2, 3, 6]: lim a,b →∞ E / ( ab ) = 2 M . (3.18) Again the Bethe ansatz equ ations (3.11) and (3.15) are equiv alen t to (2.11) with β = 0 and a 1 = c ∈ R + . 8 R. Sasaki, W.-L. Y ang and Y.-Z. Zh ang 3.2 T yp e I I theory Another dif ference analogue of the sextic p oten tia l Hamiltonian has the same form as (3.1), with only the p oten tial fu nction V ( x ) and the comp ensation term α M ( x ) are d if feren t: V ( x ) def = ( a + ix )( b + ix ) V 0 ( x ) , V 0 ( x ) def = ( c + ix )( d + ix ) , a, b, c, d ∈ R + , α M ( x ) def = M ( M − 1 + 2( a + b + c + d )) x 2 . This Hamiltonian can b e considered as a deformation by a qu adratic p olynomial f actor ( a + ix )( b + ix ) of the exactly solv able ‘discrete’ quantum mec hanics h a ving the con tin uous Hahn p olynomials as eigenfunctions [3], another dif ference analogue of the harmonic oscillato r. S ee the commen ts in Section 5 of [16]. The p seudo ground state wa vefunction φ 0 ( x ) is φ 0 ( x ) def = p Γ( a + ix )Γ( a − ix )Γ( b + ix )Γ( b − ix )Γ( c + ix )Γ( c − ix )Γ( d + ix )Γ( d − ix ) . Again it has n o no de and it is square in tegrable. The similarit y transf ormed Hamiltonian acting on the p olynomial s pace is e H def = φ − 1 0 ◦ H ◦ φ 0 = V ( x )  e − i∂ x − 1  + V ( x ) ∗  e i∂ x − 1  + M ( M − 1 + 2( a + b + c + d )) x 2 . (3.19) It is straightforw ard to v erify the relationship (3.6) and to establish the existence of th e in v ari- an t p olynomial sub spaces. In the follo wing su bsections, we p resen t Bethe ansatz solutions to eigenfunctions and eigen v alues of the Hamiltonian (3.19). 3.2.1 The case of even M Let u s in tro duce a p ositiv e integ er N su c h that M = 2 N and a p olynomial function Ψ ( x ) of the form (3.9 ). Then Ψ ( x ) b ecomes th e eigenfunction of e H (3.19) with an even M : M = 2 N pro vided that th e r o ots of the p olynomial satisfy the Bethe ansatz equations N Y l 6 = j ( x j − x l − i )( x j + x l − i ) ( x j − x l + i )( x j + x l + i ) = V ( x j ) ∗ V ( x j ) ( η ( x j + i ) − η ( x j )) ( η ( x j ) − η ( x j − i )) (3.20) = ( a − ix j )( b − ix j )( c − ix j )( d − ix j )(2 x j + i ) ( a + ix j )( b + ix j )( c + ix j )( d + ix j )(2 x j − i ) , j = 1 , . . . , N . The corresp onding eigenv alue E is giv en by E = 2 4 X j =1  M j  ∆ j − (4∆ 3 + (4 M − 6)) N X l =1 x 2 l , (3.21) with th e b inomial co ef f icien ts  M j  = M ! j ! ( M − j )! , j = 0 , . . . , M , and the co ef f icien ts { ∆ j | j = 1 , . . . , 4 } , whic h are the element ary symmetric p olynomials in th e parameters { a, b, c, d } def ined by V ( x ) = ( a + ix )( b + ix )( c + ix )( d + ix ) def = 4 X j =0 ∆ j ( ix ) j , ∆ 4 = 1 . (3.22) Bethe Ans atz Solutions to Qu asi Exactly Solv able Dif ference Equations 9 3.2.2 The case of o dd M Let us in tro duce a p ositiv e in teger N such th at M = 2 N + 1 and a function Ψ( x ) of the form (3.13). The p olynomial Ψ( x ) b ecomes the eigenfunction of e H (3.19) with an o dd M : M = 2 N + 1 pro vided that th e ro ots of the p olynomial satisfy the Bethe ansatz equations ( x j − i ) ( x j + i ) N Y l 6 = j ( x j − x l − i )( x j + x l − i ) ( x j − x l + i )( x j + x l + i ) = V ( x j ) ∗ V ( x j ) ( η ( x j + i ) − η ( x j )) ( η ( x j ) − η ( x j − i )) (3.23) = ( a − ix j )( b − ix j )( c − ix j )( d − ix j )(2 x j + i ) ( a + ix j )( b + ix j )( c + ix j )( d + ix j )(2 x j − i ) , j = 1 , . . . , N . The corresp ondin g eigen v alue E is given b y the same expressions as (3.21)–(3.22) bu t w ith a d if feren t v alue of M : M = 2 N + 1. 4 Dif ference equation analogues of h armonic oscillator with cen trifugal p oten tial deformed b y sextic p oten tia l The Bethe ansatz solutions f or the d if f erence equation analogues of the harmonic oscillator with the cen trifugal p oten tial deformed b y a s extic p oten tial [10] are d iscu ssed h ere. There are tw o t yp es corresp ond ing to the linear and quadratic p olynomial deformations as d iscussed in [10]. The corresp ond ing exactly solv able d if f erence equ ation has the Wilson p olynomials [2, 3, 8 , 9] as th e eigenfun ctions. The Hamiltonians hav e the same form as (3.1), (2.2) and (2.3), with only the p otent ial function V ( x ) and the comp ensation term α M ( x ) are dif ferent: T yp e I : V ( x ) def = ( b + ix ) V 0 ( x ) , α M ( x ) def = M x 2 , T yp e I I : V ( x ) def = ( a + ix )( b + ix ) V 0 ( x ) , (4.1) α M ( x ) def = M ( M − 1 + ( a + b + c + d + e + f )) x 2 , (4.2) with a common V 0 ( x ) V 0 ( x ) def = ( c + ix )( d + ix )( e + ix )( f + ix ) 2 ix (2 ix + 1) , a, b, c, d, e, f ∈ R + − { 1 / 2 } . (4.3) None of the parameters a , b , c , d , e or f should tak e the v alue 1 / 2, since it would cancel the denominator. Because of the cen trifugal barrier, the dynamics is constr ained to a half line; 0 < x < ∞ . Th e t yp e I case can also b e considered as a quadratic p olynomial deformation of the exactly solv able dynamics with V 01 ( x ): T yp e I : V ( x ) def = ( b + ix )( c + ix ) V 01 ( x ) , V 01 ( x ) def = ( d + ix )( e + ix )( f + ix ) 2 ix (2 ix + 1) , (4.4) whic h has the contin u ous dual Hahn p olynomials [2 , 3, 8, 9] as eigenfunctions. This re-in terpre- tation d o es not c hange the dynamics, since the Hamiltonian and A and A † op erators dep end on V ( x ). 4.1 T yp e I theory Here we consider the d if ference equation of type I. Th e p seudo ground state wa ve function φ 0 ( x ) is determined as the zero mo d e of the A op erator (2.3 ), Aφ 0 = 0: T yp e I : φ 0 ( x ) def = s 6 Q j =2 Γ( a j + ix )Γ( a j − ix ) p Γ(2 ix )Γ( − 2 ix ) , 10 R. Sasaki, W.-L. Y ang and Y.-Z. Zh ang in w h ic h the num b ering of the parameters a 1 def = a, a 2 def = b, a 3 def = c, a 4 def = d, a 5 def = e, a 6 def = f , (4.5) is used. It is obvious that φ 0 has n o no d e in the half line 0 < x < ∞ . The similarit y trans formed Hamiltonian acting on th e p olynomial s p ace has the same form as b efore (2.4) e H def = φ − 1 0 ◦ H ◦ φ 0 = V ( x )  e − i∂ x − 1  + V ( x ) ∗  e i∂ x − 1  + M η ( x ) . (4.6) Again the Hamiltonian is parity inv ariant, that is H| x →− x = H . Although the p oten tial V ( x ) has the harmf ul lo oking denomin ator 1 / { 2 i x (2 ix + 1) } , it is straigh t- forw ard to verify that e H maps a p olynomial in η ( x ) = x 2 in to another, as 2 ix + 1 ∝ η ( x − i ) − η ( x ): e H x 2 n =              n +1 X j =0 a n,j x 2 n +2 − 2 j , n ≤ M − 1 , a n, j ∈ R , M X j =0 a ′ n,j x 2 M− 2 j , n = M , a ′ n,j ∈ R . This is b ecause V 0 , whic h h as the ab ov e denominator, ke eps the p olynomial sub space of any ev en d egree inv arian t, ref lecting th e exact solv ability . In other wo rds, the exactly solv able discrete quantum mec hanics corresp onding to the un d eformed p otent ials V 0 ( x ) or V 01 ( x ) has the eigenfunction φ 0 ( x ) P ( η ( x )) , η ( x ) = x 2 , in wh ich P ( η ) is either the conti n uous d ual Hahn p olynomial ( V 01 ( x ), (4.4)), or the Wilson p olynomial ( V 0 ( x ), (4.3) ). This establishes that e H k eeps the p olynomial space V M in v ariant , e HV M ⊆ V M , V M def = Span  1 , x 2 , . . . , x 2 k , . . . , x 2 M  , dim V M = M + 1 . (4.7) The ab o v e equations imp ly that the eigenfunctions of e H in the su bspace V M are of the form Ψ( x ) = M Y l =1 ( η ( x ) − η ( x l )) = M Y l =1 ( x − x l )( x + x l ) . Analogous calculation sho ws that the p olynomial Ψ ( x ) b ecomes the eigenfunction of e H if the ro ots of the p olynomial satisfy the Bethe ansatz equations M Y l 6 = j ( x j − x l − i )( x j + x l − i ) ( x j − x l + i )( x j + x l + i ) = V ( x j ) ∗ V ( x j ) η ( x j + i ) − η ( x j ) η ( x j ) − η ( x j − i ) (4.8) = ( b − ix j )( c − ix j )( d − ix j )( e − ix j )( f − ix j ) ( b + ix j )( c + ix j )( d + ix j )( e + ix j )( f + ix j ) , j = 1 , . . . , M . (4.9) Note that the kinematical factors ± 2 ix + 1 of V ( x ) and V ( x ) ∗ are cancelled b y η ( x ∓ i ) − η ( x ). The corresp onding eigenv alue E is giv en by E = 2 3 M ( M − 1)( M − 2) +  b + c + d + e + f + 1 2  M ( M − 1) Bethe Ans atz Solutions to Qu asi Exactly Solv able Dif ference Equations 11 + { b ( c + d + e + f ) + c ( d + e + f ) + d ( e + f ) + ef } M − M X l =1 x 2 l , (4.10) where { x l } s atisfy th e Bethe ansatz equations (4.9). As exp ected, exactl y s olv able limits are obtained by t w o dif ferent wa ys; either one or tw o parameters go to inf init y . In the f ormer case, the scaled Hamiltonian (3.1) H /f giv es th at of the Wilson p olynomials with four real parameters, b , c , d and e . The eigen v alue f orm ula (4.10) reduces to that of the Wilson p olynomials [2, 3, 6] lim f →∞ E /f = M ( M + b + c + d + e − 1) . (4.11) In the latter case ( e, f → ∞ ), the scaled Hamiltonian (3.1) H / ( ef ) giv es that of the con tin uous dual Hahn p olynomials with thr ee parameters, b , c and d . The eigen v alue f orm ula (4.10) reduces to that of the con tin uous dual Hahn p olynomials [2, 3, 6 ] lim e, f →∞ E / ( ef ) = M . (4.12) The Bethe ansatz equations (4.9) in these limits determine the zeros of the Wilson and the con tin uous d ual Hahn p olynomials, r esp ectiv ely: f → ∞ : M Y l 6 = j ( x j − x l − i )( x j + x l − i ) ( x j − x l + i )( x j + x l + i ) = ( b − ix j )( c − ix j )( d − ix j )( e − ix j ) ( b + ix j )( c + ix j )( d + ix j )( e + ix j ) , (4.13) e, f → ∞ : M Y l 6 = j ( x j − x l − i )( x j + x l − i ) ( x j − x l + i )( x j + x l + i ) = ( b − ix j )( c − ix j )( d − ix j ) ( b + ix j )( c + ix j )( d + ix j ) . (4.14) 4.2 T yp e I I theory Here we consider the dif f erence equation of t yp e I I. The p seudo ground state w a v efunction φ 0 ( x ) is determined again as the zero mo de of the A op erator (2.3), Aφ 0 = 0: T yp e I I : φ 0 ( x ) def = q Q 6 j =1 Γ( a j + ix )Γ( a j − ix ) p Γ(2 ix )Γ( − 2 ix ) , where the same num ber in g of the parameters as those in (4.5) has b een u sed. The similarit y transformed Hamiltonian acting on the p olynomial space has the similar form as (4.6) but with a d if feren t p oten tial fu nction V ( x ) (4.1) and comp ensation term α M ( x ) (4.2) e H def = φ − 1 0 ◦ H ◦ φ 0 = V ( x )  e − i∂ x − 1  + V ( x ) ∗  e i∂ x − 1  + M ( M − 1 + ( a + b + c + d + e + f )) x 2 . (4.15) It can b e verif ied that th e resulting Hamiltonian e H (4.15) ke eps th e p olynomial space V M def ined in (4.7) inv ariant. Lik e the t yp e I case, this allo w s us to searc h the corresp onding eigenfunction Ψ( x ) of form Ψ( x ) = Q M l =1 ( x − x l )( x + x l ). An alogous calculation shows th at the p olynomial Ψ( x ) b ecomes the eigenfunction of e H (4.15) pro vided that th e ro ots of th e p olynomial satisfy the Bethe ans atz equations M Y l 6 = j ( x j − x l − i )( x j + x l − i ) ( x j − x l + i )( x j + x l + i ) = V ( x j ) ∗ V ( x j ) ( η ( x j + i ) − η ( x j )) ( η ( x j ) − η ( x j − i )) (4.16) 12 R. Sasaki, W.-L. Y ang and Y.-Z. Zh ang = ( a − ix j )( b − ix j )( c − ix j )( d − ix j )( e − ix j )( f − ix j ) ( a + ix j )( b + ix j )( c + ix j )( d + ix j )( e + ix j )( f + ix j ) , j = 1 , . . . , M . (4.17) The corresp onding eigenv alue E is giv en by E = ∆ 3  M 1  + (2∆ 4 + ∆ 5 )  M 2  + 4(∆ 5 + 1)  M 3  + 8  M 4  − (∆ 5 + 2( M − 1)) M X l =1 x 2 l . Here the co ef f icien ts { ∆ j | j = 1 , . . . , 6 } are the elemen tary symmetric p olynomials in the para- meters { a, b, c, d, e, f } def ined by ( a + ix )( b + ix )( c + ix )( d + ix )( e + ix )( f + ix ) def = 6 X j =0 ∆ j ( ix ) j , ∆ 6 = 1 . V arious limits to the exactly solv able cases are almost the s ame as in th e t yp e I theory and w ill not b e listed her e. It is interesting to n ote that the typ e I I Hamiltonian of Section 3, (3.1), (3.3), (3.4), is obtained from the t yp e I I Hamiltonian of Section 4, (3.1), (4.1), (4.2) as a f ormal limit e → 0, f → 1 / 2: 4 H Section 4 | e → 0 ,f → 1 / 2 = H Section 3 | M→ 2 M . The Bethe ansatz equ ations together with the eigen v alue form ulas are related in similar wa ys. 5 Dif ference equation analogue of 1 / sin 2 x p oten tial deformed b y cos 2 x p oten tial The last example is the d if ference analogue of the mo del discus sed in Sub section 2.1.2 of [10], 1 / sin 2 x p oten tial deformed b y a cos 2 x p oten tial. In this case th e corresp ond ing exactly solv able dif ference equation has the Askey–W ilson p olynomials [2, 3, 8, 9] as eigenfunctions. Th e basic idea f or sh o wing qu asi exact solv ability is almost the same as sho wn ab o v e. As introdu ced and explored in [10], this system is a quasi exactly solv able deformation of the exactly solv ab le dynamics whic h has the Ask ey–Wilso n p olynomials [2, 3, 8, 9] as eigenfunctions. Here w e s lightly c hange the notation from that of [10 ] for consistency with the rest of this pap er. The r ange of the parameter x is n o w f inite, to b e chosen as 0 < x < π , and we introd u ce a complex v ariable z and the sinusoidal co ordinate η ( x ): z = e ix , η ( x ) def = cos x = ( z + z − 1 ) / 2 . The unit of the shift is changed from 1 to a real constant γ def = log q , 0 < q < 1. Then the shift op erator e γ p can b e written as e γ p = e − iγ d dx = q D , D def = z d dz , whose action on a fun ction of x can b e expr essed as z → q z : e γ p f ( x ) = f ( x − iγ ) = q D ˇ f ( z ) = q z d dz ˇ f ( z ) = ˇ f ( q z ) , with f ( x ) = ˇ f ( z ) . Note that γ < 0. Bethe Ans atz Solutions to Qu asi Exactly Solv able Dif ference Equations 13 The Hamiltonian tak es the 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 ) ∗ + α M ( x ) , = p V ( x ) q D p V ( x ) ∗ + p V ( x ) ∗ q − D p V ( x ) − ( V ( x ) + V ( x ) ∗ ) + α M ( x ) , (5.1) = A † A + α M ( x ) , α M ( x ) def = − 2 abcdeq − 1  1 − q M  η ( x ) , A † def = − i  p V ( x ) q D 2 − p V ( x ) ∗ q − D 2  , A def = i  q D 2 p V ( x ) ∗ − q − D 2 p V ( x )  , (5.2) V ( x ) def = (1 − az ) V 0 ( x ) , V 0 ( x ) def = (1 − bz )(1 − cz )(1 − dz )(1 − ez ) (1 − z 2 )(1 − q z 2 ) , − 1 < a, b, c, d, e < 1 . (5.3) The Hamiltonian is obtained by deforming th e p oten tia l fun ction V 0 ( z ) by a linear p olynomial in z . T he parameter range (5.3) could b e enlarged to one real parameter (sa y , a ) and tw o complex conjugate p airs (for example, b = c ∗ , d = e ∗ ), but the absolute v alues must b e less than 1, | a | < 1, . . . , | e | < 1. The pseudo ground state wa vefunction φ 0 ( x ) is determined as the zero mo de of the A op era- tor (5.2), Aφ 0 = 0: φ 0 ( x ) def = s ( z 2 , z − 2 ; q ) ∞ ( az , az − 1 , bz , bz − 1 , cz , cz − 1 , dz , dz − 1 , ez , ez − 1 ; q ) ∞ , where ( a 1 , . . . , a m ; q ) ∞ def = Q m j =1 Q ∞ n =0 (1 − a j q n ). O bviously φ 0 has no no de or singularit y in 0 < x < π . W e lo ok for exact eigenv alues and eigenfun ctions of the Hamiltonian (5.1) in the form: H φ = E φ, φ ( x ) = φ 0 ( x )Ψ( η ( x )) , η ( x ) = cos x = ( z + z − 1 ) / 2 , in which Ψ( η ( x )) is a p olynomial in η ( x ) or in ( z + 1 /z ) / 2 = cos x . The similarit y transformed Hamiltonian acting on th e p olynomial space has the form e H def = φ − 1 0 ◦ H ◦ φ 0 = V ( z )  q D − 1  + V ( z ) ∗  q − D − 1  − abcdeq − 1  1 − q M   z + 1 z  . Without th e deformation factor 1 − az and the comp ensation term, the ab o v e Hamiltonian e H is exactly solv able, that is, it k eeps the p olynomial subs p ace in η ( x ) = ( z + 1 /z ) / 2 of an y degree in v ariant . Th e deformed Hamiltonian e H is parit y inv arian t H| x →− x = H and it is straigh tforw ard to sh o w the existence of an inv ariant p olynomial sub space: e HV M ⊆ V M , V M def = Span  1 , η ( x ) , . . . , η ( x ) k , . . . , η ( x ) M  , dim V M = M + 1 . The ab o v e equations imp ly that the eigenfunctions of e H in the su bspace V M are of the form Ψ( x ) = M Y l =1 ( η ( x ) − η ( x l )) = M Y l =1 (cos x − cos x l ) ≡ P M (cos x ) . Analogous calculatio n shows that Ψ( x ) b ecomes the eigenfunction of e H if the parameters { x l } satisfy the Bethe ans atz equations M Y l 6 = j cos( x j − iγ ) − cos x l cos( x j + iγ ) − cos x l = V ( x j ) ∗ V ( x j ) η ( x j + iγ ) − η ( x j ) η ( x j ) − η ( x j − iγ ) (5.4) 14 R. Sasaki, W.-L. Y ang and Y.-Z. Zh ang = ( z j − a )( z j − b )( z j − c )( z j − d )( z j − e ) (1 − az j )(1 − bz j )(1 − cz j )(1 − dz j )(1 − ez j ) z j , j = 1 , . . . , M . (5.5) Note that z j = e ix j and η ( x j − iγ ) = cos( x j − iγ ) = ( q z + q − 1 z − 1 ) / 2, etc. Again the kinematical factors (1 − z ± 2 )(1 − q z ± 2 ) of V ( x ) and V ( x ) ∗ are cancelled b y η ( x ∓ iγ ) − η ( x ). The corresp onding eigen v alue E is given by E = ( abcd + abce + abde + acde + bcde ) q − 1  q M − 1  + q −M − 1 − 2 abcde q M− 1  1 − q − 1  M X l =1 cos x l , (5.6) where { x l } s atisfy th e Bethe ansatz equations (5.5). In this example, there are many w a ys to obtain exactly solv able dynamics; b y making either one ( e ), t w o ( d, e ), three ( c, d, e ), four ( b, c, d, e ) or f iv e ( a, b, c, d, e ) parameters v anish. Th e corresp ondin g Hamiltonians are those describing the d ynamics of the Askey–Wi lson, con tin uous dual q -Hahn, Al-Salam–Chihara, big q -Hermite and q -Hermite p olynomials, r esp ectiv ely [8, 9, 2, 3, 6]. T he eigenv alue formula (5.6) reduces to that of the Aske y–Wilson p olynomials for e = 0 [2, 3 , 6]; E =  q −M − 1  1 − abcdq M +1  , (5.7) and to a un iv ersal formula E = q −M − 1 , (5.8) for the r est [2, 3, 6]. The Bethe ansatz equations (5.5) for th e restricted case of e = 0 M Y l 6 = j cos( x j − iγ ) − cos x l cos( x j + iγ ) − cos x l = ( z j − a )( z j − b )( z j − c )( z j − d ) (1 − az j )(1 − bz j )(1 − cz j )(1 − dz j ) , j = 1 , . . . , M , (5.9) then determin e the zeros of the Aske y–Wilson p olynomials. F ur th er restrictions d, e = 0, c, d, e = 0, b, c, d, e = 0 and a, b, c, d, e = 0 determine the zeros of the contin u ous d ual q -Hahn, Al-Salam– Chihara, b ig q -Hermite and q -Hermite p olynomials, resp ectiv ely [9, 6]. 6 Summary and commen ts W e hav e constructed Bethe ansatz solutions for the quasi exactly solv able dif ference equations of one degree of freedom introdu ced in [10] and [11]. These qu asi exactly s olv able d if ference equa- tions are deformations of th e well-kno w n exactly solv able dif ference equations of the Meixner– P ollacz ek, con tin uous Hahn, con tin uous du al Hahn, Wilson and Ask ey–Wilso n p olynomials. The eigenfunctions within the exactly solv a ble subsp ace are exp licitly give n by some p olynomials (mo dule a pseud o ground state wa vefunction φ 0 ) whose ro ots are solutions of the asso ciated Bethe ansatz equations. Th e corresp onding eigen v alues are expressed in terms of the solutions of the Bethe ansatz equations. Th ese are dif ference equation counterparts of the results of Sasaki–T ak asaki [15], wh ic h gav e a Bethe ans atz form ulation of the Q E S s y s tems corresp onding to the harmon ic oscillator (with/without a cen trifugal b arrier) deformed by a sextic p oten tial and th e 1 / sin 2 x p oten tial d eformed by a cos 2 x p oten tial. As in the exactly solv able quantum mec hanics, the sin usoidal co ord inates η ( x ) [4, 5] pla y an essen tial r ole. The exactly solv able sector is spann ed by it Span  1 , η ( x ) , . . . , η ( x ) k , . . . , η ( x ) M  , Bethe Ans atz Solutions to Qu asi Exactly Solv able Dif ference Equations 15 then the Bethe ansatz equations take an almost u niv ersal form, (2.10), (3.10) , (3.14 ), (3.20), (3.23), (4.8), (4.16) and (5.4 ). Th e limits or restrictions to the exactly solv able dynamics are demonstrated in detail in clud ing v arious eigenv alue formulas (2.14), (2.15), (3.17), (3.18), (4.11), (4.12), (5.7 ) and (5.8). Th e Bethe ansatz equations redu ce to those determining the zeros of the corresp ondin g orthogonal p olynomials, for example (2.16), (4.13), (4.14) and (5.9). All the known quasi exactly solv able dynamics ha v e the exactly solv able subsp ace consisting of p olynomials in a certain v ariable (see, e.g. [22]). Our emph asis h ere is that the v ariable is the s inusoidal co ordin ate which plays the cen tral role in the corresp onding exactly solv able limits [4 , 5, 6]. It s hould b e mentio ned that there exist some examples of deriving (quasi) exactly solv able dif ference equations in terms of L ie algebraic deformations of exactly solv able d ynamics in tro- duced b y T urbin er and h is collab orators [23]. Th ese dif ference equations ha v e shifts in th e real direction ψ ( x ± 1) and th e corresp onding eigenfunctions ha v e d iscrete orthogonalit y measures, in con trast to those discussed in th is pap er w hic h h a v e pure imaginary shifts, ψ ( x ± i ) or ψ ( x ± iγ ), γ ∈ R , and the corresp ond ing eigenfunctions hav e con tin uous orthogonalit y measur es. The deformations of exactly solv able dynamics for obtaining the quasi exactly solv able quan- tum s y s tems introdu ced in [15, 10, 11] and d iscussed in this p ap er in detail, are of the simplest t yp e, in whic h the comp ens ation term is linear in the sinusoidal co ord inate. Po ssibilit y of fu rther deformations includ ing quadratic comp ensation terms will b e discussed in [24], in p articular, for those q u an tum s y s tems having discrete orthogonalit y measures [7]. Ac kno wledgemen ts The f inancial su pp ort from Australian R esearch Council is gratefully ac kno wledged. 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