Quasi-exactly solvable models based on special functions

Quasi-exactly solv able mo dels based on sp ecial functions S. N. Doly a ∗ W e suggest a systematic method of extension of quasi-exactl y solv able (QES) systems. W e construct finite-dimensional s ubspaces on t he b asis of sp eci al functions (h yp ergeometric, Airy , Bessel ones) inv arian t with resp ect to th e actio n of differen tial op erators of the second order with p olynomial coefficients. As a example of physical applications, we sh o w that the kn own tw o-photon Rabi Hamiltonian b ecomes qu asi- exactly solv able at certain v alues of paramete rs when it can b e expr essed in t erms of corresp onding QES op erators related to th e hyp ergeo metric f unction. P ACS num b ers: 03 .65.Ge, 02.3 0.Gp, 02.30.Tb I. INTR ODUCTI ON Usually , the mo dels of the quan tum mec hanics are considered to b e exactly-solv able (when the eigen v alues and eigenfunctions are kno wn) or non-solv ab e at all (when the eigen v alues and eidenfunctions are unknown, and t hey can b e found n umerically only). It b ecame a great surprise that an in termediate case is also p ossible, when inside the Hilb ert space there is an in v a rian t subspace for which the eigen v alues and eigenfunctions could b e found from algebraic equations. This t yp e of systems is called quasi-exactly solv able (QES). F or the wide range of the one-dimensional QES the Hamilto nia n p ossesses hidden Lie alg ebra sl (2) and represen ts a bilinear form of first-order differen tial op erators whic h satisfy the same comm utator relations as the spin ones [1], [2], [3], [4]. The QES Sc hr¨ odinger op erators based on the sl (2) represen tation w ere studied in [1], [5], [6 ], [7], [8]. Later it b ecame p ossible to go b ey ond the Lie a lg ebraic con text. In particular, starting from the second-order differen tial op erators T = q ( x ) · d 2 dx 2 + p ( x ) · d dx + r ( x ) (1) ∗ Electronic addre s s: sdo ly a@gmail.c om 2 suc h that they p ossess the finite-dimensional represen tations b oth in the subspace P n = span { 1 , x, ..., x n − 1 , x n } o f all monomials of t he degree ≤ n and in the mo no mial subspace span { 1 , x 2 , ... , x n − 1 , x n } [9], [10], [11]. In [12], [13],.the extension of the the corresp onding subspace w as suggested ν = P n ∪ g ( x ) P n = span { 1 , x 2 , ..., x n − 1 , x n , g ( x ) , g ( x ) · x, ..., g ( x ) · x n } (2) for certain functions g ( x ) [12], [13]. In the presen t pap er we consider the problem of constructing QES op erators whic h pre- serv e subspaces o f a more general form: M n = span { f 1 ( x ) , f 2 ( x ) , ..., f n − 1 ( x ) , f n ( x ) } . (3) The general features of differen tial op erators with in v arian t subspace M n (3) w ere discussed in [15]. The main result o btained is the fo llo wing: an y linear op erator P n ∈ P ( M n ), that preserv e M n (3), ma y b e represen ted as follows (a form ula 5.8 in [15]): P n = X i,j a ij · f i ( x ) · L j + X R m · K m , (4) where a ij are arbitr a ry constants, R ν ∈ D are ar bitrary linear differen tial op erators, D = D ( F ) is a space of the differential op erators with co efficie nts b elonging to the functional space F (hereafter denoted P n ), K m are op erators b elonging to the annihilato r A ( M n ): A ( M n ) = { K ∈ D | K ( f ( x )) = 0, for a ll f ( x ) ∈ M n } , (5) L j − are op erators b elonging to the a ffine annihilator K ( M n ): K ( M n ) = { L ∈ D | L ( f ( x )) = c ∈ R , for all f ( x ) ∈ M n } (6) c is a constan t. All the definitions and sym b ols are in agreemen t with the ones used in [15]. In spite of the fact that the theorem of existence sp ec ifies a general form of the required differen tial op erators (4), their finding is, generally sp eaking, not trivial task. The purp ose of the giv en w ork consists in explicit construction of suc h QES op erators, for whic h the set of functions f n ( x ) do es not reduce t o p olynomials and represen ts sp ecial functions. With this purp ose, we offer a metho d of QES extension. It consists in the following. T o construct the QES Hamiltonians on the basis of the subspace (3), we select in (6) the subspace of differen tial op erators K 2 ( M n ) of the degree t w o or less: K 2 ( M n ) = { L ∈ K ( M n ) | o rder ( L ) ≤ 2 } . (7) 3 The general approach suggested in [15] do es not supply us with informatio n, whether o r not the space K 2 ( M n ) (7) is empty , without the constructing K ( M n ) fo r the c hosen M n (3). T o simplify matter, w e start from the t w o- dime nsional subspace M 2 = span { f 1 ( x ) , f 2 ( x ) } , for which op erators L i ∈ K 2 ( M 2 ) are known to exist and can b e explicitly constructed [15]. Let us try to extend the subspace M 2 b y adding function f 3 ( x ) to it in suc h a w ay that the new subspace M 3 = span { f 1 ( x ) , f 2 ( x ) , f 3 ( x ) } b e the three-dimensional and the space of corresp onding op erators K 2 ( M 3 ) (7) b e non-empt y . Th us, transformation M 2 − → M 3 has to lead to the change o f the co efficien ts at d dx in the op erators L i ∈ K 2 ( M 2 ), with the order of the op erators L i ∈ K 2 ( M 2 ) preserv ed: or der ( L i ) ≤ 2. Rep eating this op eration n times, w e obtain the QES-extension of M 2 . In other w ords, the init ia l subspace M 2 = span { f 1 ( x ) , f 2 ( x ) } p ermits the QES-extension if M 2 ma y b e extended to M n under the condition t ha t the affine annihilat or K 2 ( M n ) (7) is not empt y . (It is w orth not ing another metho d of constructing inv ariant subspaces (3) whic h is based on the conditional symmetries [14], but it will not b e discusse d in the presen t pap er.) I I. CONSTR UCTIO N OF THE INV ARIANT SUBSP ACES Belo w w e suggest the metho d that enables one to construct in v arian t subspaces M n . Let us start from the simplest case n = 2. Let us c ho ose a linear indep enden t ba s is { f + 0 ( x ) , f − 0 ( x ) } for the subspace M 2 = span { f + 0 ( x ) , f − 0 ( x ) } in the follo wing wa y . Let us consider t he function f ( x ), whic h satisfies the second or der homogeneous differen tial equation with p olynomial co efficien ts: q ( x ) d 2 dx 2 f ( x ) + p ( x ) d dx f ( x ) + r ( x ) f ( x ) = 0. The case when f ( x ) = constant · d dx f ( x ) is eliminated. W e select the function f + 0 ( x ) = f ( x ) and its deriv ative f − 0 ( x ) = d dx f ( x ) = f ′ ( x ) as the basis comp onen ts of M 2 : M 2 = span { f + 0 ( x ) , f − 0 ( x ) } . (8) F or simplicit y , w e shall w ork directly with the o perator (4) instead of t he op erators L i represen ting affine annihilator K ( M 2 ) (6). Our strategy can b e describ ed as follows: I W e find a general form o f t he op erator of the second order P 2 for which subspace M 2 (9) is preserv ed. The existence o f suc h an op erator is supp orted of Lemma 4.1 0 [15]. 4 I I W e mak e extension of the subspace M 2 → M 4 = span { f + 0 , f + 1 , f − 0 , f − 1 } . I II W e find a general for m of the o p erator o f t he second order P 4 for which subspace M 4 is preserv ed. If a non-t r ivial solution P 4 6 = const exists for the chos en w ay of the extension, we pass to item IV, otherwise we c hange a w ay of extension. IV W e mak e comparison of t w o op erators P 2 and P 4 and try to guess a general form of co efficie nts whic h enable us to r ep eat extension in the c hain ( M 2 → M 4 → ... → M 2( N +1) ) ( N = 0 , 1 , 2 , ... ) ) that leav es these subspaces in v arian ts. A s a result, w e obtain the explicit for m of op erator P 2( N +1) that acts on the elemen ts of the subspace M 2( N +1) = { f + 0 , f + 1 , f + 2 ... f + N , f − 0 , f − 1 , f − 2 , ..., f − N } , N = 0 , 1 , 2 , ... . This is not the end of story since the general form do es not fix b y itself the concrete expression for the co efficie nts. V With the guessed general form of the op erator P 2( N +1) at hand, w e require that it lea v e the corresp onding subspace in v arian t, whence w e find the concrete v alues of its co efficie nts. F or explicit demonstration of the a foremen tioned algorithm, we shall consider a concrete c hoice o f the function f + 0 ( x ) : f + 0 ( x ) = 0 F 1  − s ; x  [16] where 0 F 1  − s ; x  is a hy p ergeometric function, f − 0 ( x ) = 0 F 1  − s +1 ; x  is its deriv ativ e up to a constan t f actor. The differen tial equation which the h yp ergeometric function ob eys  x · d 2 dx 2 + s · d dx − 1  f + 0 ( x ) = 0 as we ll as its other prop erties can b e found, e.g., in [16].Then, we hav e: M 2 = span n 0 F 1  − s ; x  , 0 F 1  − s +1 ; x  o , (9) Let us c ho ose the op erator of the second or der P 2 = a 3 ( x ) d 2 dx 2 + a 2 ( x ) d dx + a 0 ( x ) and write do wn the condition of inv ariance of the subspace M 2 (9) for the given op erator: P 2  f + 0  = c 1 · f + 0 + c 2 · f − 0 (10) P 2  f − 0  = c 3 · f + 0 + c 4 · f − 0 (11) Using r ules of differen tiation d dx f + 0 = f + 0 x − f − 0 x , d dx f − 0 = − s ( s + 1) f + 0 x 2 + ( s + s 2 + x ) f − 0 x 2 and 5 equating co efficien ts at functions f + 0 , f − 0 (10, 11) w e obtain the solution: a 3 ( x ) = − c 3 · x 2 s + c 2 · s · x, a 2 ( x ) = − c 3 · x + c 2 · s ( s + 1) , (12) a 1 ( x ) = c 3 · x s − c 2 · s + c 4 + c 3 c 1 = c 4 + c 3 . Substituting obtained co efficien ts a k ( x ), ( k = 1 , 2 , 3) ( 1 2) into the expression for the op erator P 2 w e ha ve : P 2 = c 2 · s · P 1 2 − c 3 s · P 2 2 + c 4 + c 3 + s · c 2 (13) where P 1 2 = x d 2 dx 2 + ( s + 1) · d dx , P 2 2 = x 2 d 2 dx 2 + s · x d dx − x. (14) One can c hec k that op erators P 1 2 , P 2 2 act on the basis functions according to the formulas P 1 2  f + 0  = f + 0 + 1 s · f − 0 , P 1 2  f − 0  = f − 0 , P 2 2  f + 0  = 0, P 2 2  f − 0  = s · f − 0 − s · f + 0 and, th us, preserv e the subspace M 2 (9). This is agreemen t with the fact that the a ffine annihilator K 2 ( M ) has the prop ert y of the ve ctor space and p ossesses the op erator basis (14) (see in [15]). The extension of the subspace M 2 (9) can b e realized in man y w a ys. The basic require- men t here consists in that after adding new elemen ts of the basis M 2 ∪ span  f + 1 , f − 1  = span  f + 0 , f + 1 , f − 0 , f − 1  = M 4 the op erator basis of P 2 (13, 14) should remain tw o -dimens ional for t he new subspace M 4 . W e consider the simplest w a y of extension - m ultiplication of the basis functions by the p o w er function x n : M 2 → M 4 → ... → M 2( N +1) ( N = 0 , 1 , 2 , .... ), { f + 0 , f − 0 } → { f + 0 , f + 1 , f − 0 , f − 1 } → ... → { f + 0 , f + 1 , ...f + N , f − 0 , f − 1 , ..., f − N } (15) where f + n = x · f + n − 1 = x n · f + 0 , f − n = x · f − n − 1 = x n · f − 0 , (16) n = 0 , 1 , .., N − 1 , N . T o demonstrate that suggested extension is suitable, w e shall consider subspace M 4 =  f + 0 , f + 1 , f − 0 , f − 1  and shall write dow n the condition of its inv ariance with resp ec t to the op erator P 4 = b 3 ( x ) d 2 dx 2 + b 2 ( x ) d dx + b 0 ( x ): P 4  − → f  = b d · − → f (17) where − → f =  f + 0 , f + 1 , f − 0 , f − 1  T - is v ector, b d = [ d i,j ], ( i = 1 , ..., 4; j = 1 , ..., 4) - is the matrix of co efficien ts. The condition (1 7) r ep resen ts the system of the linear equations on functions 6 b k ( x ), k = 1 , 2 , 3 and quan tities d i,j , ( i = 1 , ..., 4; j = 1 , ..., 4 ) . Solving this linear system of the equations we hav e: b 3 ( x ) = d 13 · s · x − d 14 · s x 2 2 , (18) b 2 ( x ) = d 13 · s ( s + 1 ) + d 14 · x · s  1 − s 2  , (19) b 1 ( x ) = d 13 · ( − s ) + d 14 ·  sx 2 + s 2 − s + 1  (20) d 11 = d 14 ·  s 2 − s  + d 44 , d 21 = d 13 ·  s 2 + s  , d 22 = d 14 · s 2 2 + d 44 , d 24 = 3 d 13 , (21) d 31 = d 14 · 3 s 2 2 , d 33 = − d 14 ·  s 2 2 + s  + d 44 , d 41 = 2 d 13 · s 2 , (22) d 42 = d 14 · s 2 2 , d 43 = d 13 ·  s − s 2  , d 12 = d 23 = d 32 = d 34 = 0 . (23) The existence of the non-trivial solution (1 8 -20) means that the wa y by whic h extension (15) w as made is suitable. Let us analyze the g ene ral form of the op erator P 4 for the obtained solutio n: P 4 = d 13 · s · P 1 4 − d 14 · s 2 P 2 4 + d 44 − d 13 · s + d 14 ·  s 2 − s  (24) where P 1 4 = x d 2 dx 2 + ( s + 1) · d dx , P 2 4 = x 2 d 2 dx 2 + ( s − 2) · x d dx − x. (25) The op erator P 4 (24) as we ll as op erator P 2 (13), up to an additiv e constant , dep ends on t w o free parameters d 13 , d 14 (24). F rom the general solution (24) w e select tw o op erators P 1 4 , P 2 4 (25) whic h do not dep end on parameters d 13 , d 14 , d 44 . W e see that the co efficien ts of the op erator P 1 4 coincide with those of the op erator P 1 2 (14). This suggests the idea to implemen t the expression (2 5) fo r the op erator P 1 2( N +1) for any N and subspace (15). The situation with the op erator P 2 m is somewhat more complicated. The natural general guessed form for this op erator reads P 2 2( N +1) = x 2 d 2 dx 2 + ( s − α N ) · x d dx − x (26) where the quantit y α N dep ends on N and, b esides, satisfies the conditio ns α 0 = 0, α 1 = 2 as it follows from (14, 25). T o find dep enden ce α N from N , w e shall consider the result of the action of the op erator P 2 2( N +1) on the elemen t f + n (15): P 2 2( N +1)  f + n  = n ( n + s − 1 − α N ) · f + n + (2 n − α N ) s · f − n +1 . (27) 7 If we put n = N in (27) and require that the subspace b e finite-dimensional, the condition of cut off at n = N giv es us a N = 2 · N , whence the op erator P 2 2( N +1) lo oks lik e P 2 2( N +1) = x 2 d 2 dx 2 + ( s − 2 · N ) · x d dx − x . As a result, w e obtained a pair of op erators that indeed lea v e the subspace M 2( N +1) in v arian t. Corresp ondingly , any linear com bination P 2( N +1) = κ 1 · P 1 2( N +1) + κ 2 · P 2 2( N +1) where κ 1 , κ 2 arbitrary constants has the same prop ert y . F ollo wing the w ay w e describ ed ab o v e, w e f ound a series of concrete examples 1, 2, 4, 5, 6. The list of o btained finite-dimensional subspaces together with the basic op erators for whic h corresp onding subspaces are in v arian t is giv en b elo w. Mean while, the suggested w a y of the QES - extension ( 16 ) o f the op erators (4) is not unique. As the alternativ e approac h, in an example 3 we used ano t he r ”natural” pro cedure of QES - extension in that a n in teger n lab eling the basis was added to the parameter o f the function: α → α + n . 1 ) F inite dimensional function subspace R 1 N = span { f + 0 , ..., f + N , f − 0 , ..., f − N } ( N = 0 , 1 , 2 , .... ) , dim ( R 1 N ) = 2 ( N + 1), formed b y functions f + n = x n · 0 F 1  − s ; x  , f − n = x n · 0 F 1  − s +1 ; x  ( n = 0 , 1 , .., N − 1 , N ) [16], is in v arian t for the op erators J + 1 ≡ P 1 2( N +1) , J − 1 ≡ P 2 2( N +1) : J − 1 = x d 2 dx 2 + ( s + 1) d dx J + 1 = x 2 d 2 dx 2 + ( s − 2 N ) · x d dx − x (28) The op erators J + 1 , J − 1 act on the functions f + n ( x ), f − n ( x ) ∈ R 1 N as follo ws: J + 1  f + n f − n  =  n ( A n − 1 + s ) · f + n + 2 B n s · f − n +1 ( n − s ) ( A n − 1) · f − n + s (2 B n − 1) · f + n  J − 1  f + n f − n  =  f + n + n ( n + s ) · f + n − 1 + 1+2 n s · f − n f − n + n ( n − s ) · f − n − 1 + 2 ns · f + n − 1  (29) , where A n = n − 2 N , B n = n − N . The function f + 0 ( x ) satisfies the differen tial equation  x d 2 dx 2 + s d dx − 1  f + 0 ( x ) = 0. 2 ) F inite dimensional function subspace R 2 N = span { f + 0 , ..., f + N , f − 0 , ..., f − N } ( N = 0 , 1 , 2 , .... ) , dim ( R 2 N ) = 2 ( N + 1), f o rmed b y functions f + n = x n · 1 F 1  α s ; x  , f − n = x n · 1 F 1  α +1 s +1 ; x  ( n = 0 , 1 , .., N − 1 , N ) [16], is in v arian t fo r t he op erators J + 2 , J − 2 : J − 2 = x d 2 dx 2 + (1 + s − x ) d dx J + 2 = x 2 d 2 dx 2 + ( s − 2 N − x ) · x d dx + x ( N − α ) (30) 8 The op erators J + 2 , J − 2 act on the functions f + n ( x ) , f − n ( x ) ∈ R 2 N as follo ws: J + 2  f + n f − n  =  n ( A n − 1 + s ) · f + n − B n · f + n +1 + 2 αB n /s · f − n +1 ( s − n ) (1 − A n ) · f − n + B n · f − n +1 + s (2 B n − 1) · f + n  (31) J − 2  f + n f − n  =  ( α − n ) f + n + n ( n + s ) · f + n − 1 + α (1 + 2 n ) /s · f − n ( α + n + 1 ) f − n + n ( n − s ) · f − n − 1 + 2 ns · f + n − 1  (32) , where A n = n − 2 N , B n = n − N . The function f + 0 ( x ) satisfies the differen tial equation  x d 2 dx 2 + ( s − x ) d dx − α  f + 0 ( x ) = 0. 3 ) Finite dimensional function subspace R 3 N = span { f 0 , f 1 , ..., f N } ( N = 0 , 1 , 2 , .... ) , dim ( R 3 N ) = N + 1 , formed b y functions f n = 1 F 1  α + n s ; x  ( n = 0 , 1 , .., N − 1 , N ) [1 6], is in v arian t for the o perators J + 3 , J − 3 : J − 3 = x d 2 dx 2 + ( s − x ) d dx J + 3 = x 2 d 2 dx 2 + ( s − N − x ) · x d dx − αx (33) The op erators J + 3 , J − 3 act on the functions f n ( x ) ∈ R 3 N as follows : J − 3 ( f n ) = ( n + α ) · f n (34) J + 3 ( f n ) = ( sn + ( α + n ) C n ) · f n + ( α + n ) B n · f n +1 + n ( α + n − s ) · f n − 1 , whe re C n = N − 2 n , B n = n − N . It is w orth while to note that replacemen t  s − → s − 1, α − → α − 1 2 + N 2 , N − → N 2 − 1 2  transforms the o p erators J + 2 , J − 2 in to J + 3 , J − 3 , if N is o dd. 4 ) Finite-dimensional function subspace R 4 N = span { f + 0 , ..., f + N , f − 0 , ..., f − N } ( N = 0 , 1 , 2 , .... ) , dim ( R 4 N ) = 2 ( N + 1 ) , formed by functions f + n = x n · Air y ( x ) , f − n = x n · d dx Air y ( x ) ( n = 0 , 1 , .., N − 1 , N ) [16], is in v arian t for t he op erators J + 4 , J − 4 : J − 4 = x d 2 dx 2 − (1 + 2 N ) d dx − x 2 J + 4 = d 2 dx 2 − x (35) The op erators J + 4 , J − 4 act on the functions f + n ( x ) , f − n ( x ) ∈ R 4 N as follo ws: J + 4  f + n f − n  =  n ( n − 1) · f + n − 2 + 2 n · f − n − 1 (1 + 2 n ) · f + n + n ( n − 1) · f − n − 2  (36) J − 4  f + n f − n  =  ( A n − 2) n · f + n − 1 + (2 B n − 1) · f − n 2 B n · f + n +1 + ( A n − 2) n · f − n − 1  (37) where A n = n − 2 N , B n = n − N . The function f + 0 ( x ) to satisfy the differen tial equation  d 2 dx 2 − x  f + 0 ( x ) = 0. 9 5 ) F inite dimensional function subspace R 5 N = span { f + 0 , ..., f + N , f − 0 , ..., f − N } ( N = 0 , 1 , 2 , .... ) , dim ( R 5 N ) = 2 ( N + 1), formed by functions f + n = x n · B esselK ( ν , x ) , f − n = x n · B esselK ( ν + 1 , x ) ( n = 0 , 1 , .., N − 1 , N ) [16], is in v arian t for the o perators J + 5 , J − 5 : J − 5 = x d 2 dx 2 + 2 d dx − ν 2 + ν + x 2 x J + 5 = x 2 d 2 dx 2 + x ( 1 − 2 N ) d dx − x 2 (38) The op erators J + 5 , J − 5 act on the functions f + n ( x ) , f − n ( x ) ∈ R 5 N as follo ws: J + 5  f + n f − n  =  ( ν + n ) ( ν + A n ) · f + n − 2 B n · f − n +1 ( n − 1 − ν ) ( A n − 1 − ν ) · f − n − 2 B n · f + n +1  (39) J − 5  f + n f − n  =  n (1 + n + 2 ν ) · f + n − 1 − (1 + 2 n ) · f − n − (1 + 2 n ) · f + n + n ( n − 2 ν − 1 ) · f − n − 1  (40) , where A n = n − 2 N , B n = n − N . The function f + 0 ( x ) satisfies the differen tial equation  x 2 d 2 dx 2 + x d dx − ( x 2 + ν 2 )  f + 0 ( x ) = 0. 6 ) F inite dimensional function subspace R 6 N = span { f + 0 , ..., f + N , f − 0 , ..., f − N } ( N = 0 , 1 , 2 , .... ) , dim ( R 6 N ) = 2 ( N + 1 ), formed by functions f + n = x n · 1 F 1 h α 1 / 2 ; x 2 i , f − n = x n · 1 F 1 h α +1 3 / 2 ; x 2 i ( n = 0 , 1 , .., N − 1 , N ) [16], is in v arian t for t he op erators J + 6 , J − 6 : J − 6 = d 2 dx 2 − 2 x d dx J + 6 = x d 2 dx 2 − (2 x 2 + 1 + 2 N ) d dx + 2 x ( N − 2 α ) (41) The op erators J + 6 , J − 6 act on the functions f + n ( x ), f − n ( x ) ∈ R 6 N as follo ws: J + 6  f + n f − n  =  − 2 B n · f + n +1 + n ( A n − 2) · f + n − 1 + 4 α (2 B n − 1) · f − n n ( A n − 2) · f − n − 1 + (2 B n − 1) · f + n + 2 B n · f − n +1  (42) J − 6  f + n f − n  =  2 (2 α − n ) · f + n + ( n 2 − n ) · f + n − 2 + 8 αn · f − n − 1 2 n · f + n − 1 + 2 (2 α + 1 + n ) · f − n + ( n 2 − n ) · f − n − 2  (43) where A n = n − 2 N , B n = n − N . F or the pro of of equalities (2 9 , 31, 32, 34, 36, 37, 39, 40, 4 2, 43) it is enough to use rules of differen tiation of special functions or their represe ntations b y the p o w er series [16]. The comm utation r ules fo r the o p erators J + k , J − k ( k = 1 . . . 6 ) ar e giv en in app endix A. Let us no w consider an explicit example of application of the constructed subspaces to a ph ysical system - t w o- pho t o n Ra bi Hamiltonian. 10 I I I. QES TWO-PHOTON RABI H AMIL TONIAN The tw o -photon Rabi Hamiltonian (TPRH) is a n obvious extension of the original Rabi Hamiltonian [18], which t ak es into account the a t o mic transitions induced b y the absorption and emission of tw o photons rather than one [17]. The corresponding system as a whole is not in tegrable. Let us pro v e that (33, 34), a t some c hoice o f parameters, is in v arian t subspace for TPRH: H = ω 0 2 σ z + ω · b + b + g  b 2 +  b +  2  · ( σ + + σ − ) (44) where σ z , σ ± = σ x ± i · σ y are Pauli matrices, b and b + are the annihilation and creation op erators resp ectiv ely ([ b, b + ] = 1). F ollo wing the approac h of the article [17], a pply the Bogoliub o v transformatio n to the op erators b and b + ( b ( t ) = cos ( t ) · b + sin ( t ) · b + , b + ( t ) = cos ( t ) · b + − sin ( t ) · b ), where t ∈ [0 , 2 π ] is a para meter. W e will use the coheren t state represen tation (F o c k-Bargma n represen tation): b = d dz , b + = z . The eignev alue problem for the op erator (44) take s the form :   ω 0 2   0 1 1 0   +   ∧ c 0 0 ∧ c   +   ∧ a 0 0 − ∧ a       ψ 1 ψ 2   = E   ψ 1 ψ 2   , (45) where ∧ a = g  b ( t ) 2 + b + ( t ) 2  = 2 g  d 2 dz 2 + z 2  , ∧ c = ω · b + ( t ) b ( t ) = ω 2 · h − sin (2 t ) d 2 dz 2 + 2 cos (2 t ) z d dz + sin (2 t ) z 2 + cos (2 t ) − 1 i . R em oving the function ψ 1 ( z ) from the system (45) we come to the four t h- order differential equation fo r the f unction ψ 2 ( z ): L ( t ) ψ 2 ( z ) ≡  1 ω ·  ∧ a + ∧ c   ∧ a − ∧ c  + 2 E ω 2 · ∧ c + ω 2 0 4 ω 2 − E 2 ω 2  ψ 2 ( z ) = 0 (46) Let us try to reduce eq. (46) to the algebraic problem using the finite-dimensional pre- sen tation of the op erators J + 3 , J − 3 (33). F or this purp ose, we shall presen t the op erator L ( t ) (46) as p olynomial on the op erators J + 3 , J − 3 . T o this end, we use a gauge transformation φ ( z ) − 1 L ( t ) φ ( z ) = P  J + 3 , J − 3  | x = µ ( z ) , where P ( x 1 , x 2 ) is a p olynomial of the second degree in non- comm utativ e v ariables x 1 and x 2 . W e restrict ourselv es b y a sp ecial case of φ ( z ) and µ ( z ): φ ( z ) = φ 1 ( z ) = exp ( η · z 2 ) or φ ( z ) = φ 2 ( z ) = z · exp ( η · z 2 ), µ ( z ) = ξ · z 2 . Th us, a searc h of a p olynomial represen tation for L ( t ) is reduced to a search of co efficien ts of p olynomial P ( x 1 , x 2 ) and constan ts η , ξ , t . The obtained solutions read: 11 T yp e I, s = 1 2 , α = − 1 4 − N 2 : L ( t 0 ) e η · z 2 = e η · z 2 3  2  J − 3  2 +  J + 3 , J − 3  − 7 J + 3 + 4 J − 3 + C 1  | x = − ξ z 2 (47) T yp e I I, s = 3 2 , α = 5 4 − N 2 : L ( − t 0 ) e − η · z 2 = z e − η · z 2 3  2  J − 3  2 +  J + 3 , J − 3  − 7 J + 3 − 8 J − 3 + C 2  | x = ξ z 2 (48) where t 0 = 1 4 arctan  10 √ 2 23  , E ω = N +1 √ 3 − 1 2 , η = √ 2 8 , C 1 = 3 ω 2 0 4 ω 2 − N 2 − N 4 + 1 8 , C 2 = 3 ω 2 0 4 ω 2 − N 2 + 13 N 4 + 49 8 , g ω = 1 2 √ 6 , ξ = 3 √ 2 8 . Tw o t yp es of the solutions (47,48) ha v e differen t symmetries with resp ect to the par ity op erator Π = exp ( iπ b + b ) [17] ([Π , H ] = 0). F urther, w e will consider the solution (47) (t yp e–I) in more details, the second solu- tion (48) can b e considered in a similar w ay . T a king in to accoun t the equality (4 7 ), the problem (46) can b e reduced to a searc h of v ectors φ k satisfying the matrix equation  2  j − 3  2 +  j + 3 , j − 3  − 7 j + 3 + 4 j − 3 + C 1  φ k = 0, where j + 3 , j − 3 are finite- dimensional repre- sen tations of the o perators J + 3 , J − 3 (33) in the subspace R 3 N . F or eac h of φ k , taking into accoun t ( 4 7 ), we hav e L ( t 0 ) e η · z 2 φ k = 0 that is equiv alen t to (46) under t he condition ψ 2 ≡ ψ k 2 = e η · z 2 φ k . The functions ψ k 2 obtained are entire ones and b elong to the F o c k space:   ψ k 2 ( z )   2 = O  | z | p exp  − √ 2 4 ( z 2 + z 2 )  (t yp e I),   ψ k 2 ( z )   2 = O  | z | p exp  √ 2 4 ( z 2 + z 2 )  (t yp e I I). W e list the corresp onding functions explicitly for the sp ec ial case N = 2 . t yp e I: (49) ψ 2 = e η · z 2   1 F 1 h − 5 / 4 1 / 2 ; − ξ z 2 i ·  57 20 + 7 √ 42 15  + 1 F 1 h − 1 / 4 1 / 2 ; − ξ z 2 i ·  5 3 + √ 42 30  + 1 F 1 h 3 / 4 1 / 2 ; − ξ z 2 i   (50) ω 0 2 ω = s 11 12 + √ 42 3 (51) t yp e I I: (52) ψ 2 = z e − η · z 2   1 F 1 h 1 / 4 31 / 2 ; ξ z 2 i ·  5 4 + √ 10 7  + 1 F 1 h 5 / 4 3 / 2 ; ξ z 2 i ·  31 21 + √ 10 42  + 1 F 1 h 9 / 4 3 / 2 ; ξ z 2 i   (53) ω 0 2 ω = s √ 10 3 − 5 12 (54) 12 where ξ = 3 √ 2 8 , η = √ 2 8 , E ω = √ 3 − 1 2 . The second comp onen t ψ 1 of the sp ectral problem (45) can b e obtained from the equation ψ 1 = 2 ω 0 ·  E + ∧ a − ∧ c  ψ 2 . In ta ble 1 the v alues of relativ e frequencies 2 ω ω 0 for b oth types I and I I are giv en for different dimensions of the subspace R 3 N . The obtained solutions (47, 48 and t a ble 1 ) concern a non-resonan t case 2 ω ω 0 6 = 1 and ar e related to the known isolated solutions (Juddian solution), found earlier in [17]. It is w orth stressing t hat in [17] the eigenfunctions w ere constructed on the basis of elemen tary function, whereas the solutions (47, 48) are constructed on the basis of functions 1 F 1  α + n s ; x  . F or the v alues of the parameters s = 1 / 2, α = − 1 / 4 − N / 2 (type I), s = 3 / 2, α = 5 / 4 − N / 2 (type I I) the h yp ergeometric function do es not degenerate into p olynomial ( − α / ∈ N ). T a ble1. V alues of para meters Hamiltonian (44,45), for differen t dim( R 3 N ), g ω = 1 2 √ 6 , E ω = N +1 √ 3 − 1 2 . dim( R 3 N ) 3 5 6 7 8 I 2 ω ω 0 0.44315 1.6 8 889 3.03496 2.72766 3.60267 2.10305 3.74421 3.90266 I I 2 ω ω 0 0.79838 0.7 9 838 2.23006 2.75234 3.43545 2.66128 4.08801 E ω 1.23205 2.3 8 675 2.96410 3.54145 4.11880 IV. CONCLUSIO N T o the b est of my know ledge, the only previously kno wn example of QES related to sp ec ial functions was found in [19].where these function app eared ”by chance ” for a partic- ular pro blem connected with quartic Bose Hamiltonians. No w, w e dev elop ed a systematic approac h of QES-extension that enables us to generate new QES op erators based on sp e- cial functions. This extends considerably the family of QES systems and can find ph ysical applications, one of whic h (tw o-photon Rabi Hamilatonian) w as discussed in the presen t ar- ticle. The main features of our approach include 1) the construction of the affine annihilator K ( M 2 ) [15]; 2) multiplication at the p o we r function x n . One can think that the approach suggested in the g iv en work, will g iv e rise to further essen tial expansion of classe s of quasi exactly solv able mo dels. 13 V. A CKNOWLEDGMENTS The author thanks O. B Zasla vskii f o r useful discussions. VI. APPENDIX A. The constructed o perators J + k , J − k ( k = 1 . . . 6), satisfy to the following comm utatio n relations:  J − k , J + k  = S k (55)  J + k , S k  = c + 1 ·  J − k  2 + c + 3 · J + k J − k + c + 4 · S k + c + 5 · J + k + c + 6 · J − k + c + 7 (56)  J − k , S k  = c − 1 ·  J − k  2 + c − 2 ·  J + k  2 + c − 5 · J + k + c − 6 · J − k + c − 7 (57) where k = 1 . . . 6, c ± i ( i = 1 , . . . , 7) are constants, theirs v alue are g iven in the table 2 . The op erators S k (55) ha v e t he g eneral structure S k = β 3 ( x ) d 3 dx 3 + β 2 ( x ) d 2 dx 2 + β 1 ( x ) d dx + β 0 ( x ), β m ( x ) ∈ P n ( m = 0 , 1 , 2 , 3). How ev er, w e do no t list them here explicitly since the corresp onding expressions are rather cum b ersome. T a ble2. V alues of constan ts c ± i ( i = 1 , . . . , 7 ) included in the comm utation r elat io ns (55- 57), C N = 2 + 2 α + s , α N = 2 + N + 2 α , β N = − 4 · (1 + ν 2 + 2 N + ν ), γ N = ( α − N ) · ( s + 1) · C N , δ N = − 8 · (2 α − N ) · α N , S N = s · α · ( s − N − 2), A N = s + N + 2 α , B N = (2 N − s ) · ( s − 2 − 2 N ), D N = ( s + 1) · ( s − 2 − 2 N ), G N = ( α − N ) · ( s + 1). 14 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 c + 1 0 0 0 0 0 − 6 c − 1 2 2 2 0 2 0 c − 2 0 0 0 6 0 0 c + 3 − 4 − 4 − 4 0 − 4 0 c + 4 − 2 − 2 − 2 0 − 2 0 c + 5 2 C N A N − 2 0 0 c − 5 0 1 1 0 4 4 c + 6 B N B N s − N 0 1 − 4 N 2 12 + 32 α c − 6 − 2 − C N − A N 2 0 0 c + 7 D N γ N S N 0 0 δ N c − 7 0 G N s · α 0 β N 0 [1] O. B. Zasla vskii, and V. V. Ulya nov, So v. Ph ys. JE T P 60, 991 (1984). [2] O. B. Zasla vskii, and V. V. Ulya nov, Theor. Math. Phys. 71, 520 (1987). [3] M. A. S hifman, Int. J. Mo d. Phys. A 4, 3305 (1989). [4] O. B. Zasla vskii, Sov. P h ys. J. 33, 12 (1990). [5] M. A. S hifman, and A. V. T urbiner, Commun. Math. Ph ys. 126, 347 (1989). [6] A. V. T u rbiner, Commun. Math. Ph ys. 118, 467 (1988). [7] A. Ushv eridze, Quasi-Exactly Solvable Mo dels in Quantum Me chanics (Bristol: Intitute of Ph ysics Publishing, 1994). [8] V. V. Ulyano v, and O . B. Zasla vskii, Phys. Rep. 216, 179 (1992). [9] D. Gomez-Ullate , N. Kamr an , and R. Milson, Pr eprint arXiv nlin.SI/0601053, (2006 ). [10] D. Gomez-Ullat e, N. Kamran an d R. Milson, Pr eprint arXiv nlin.SI/0610065, (2006). [11] G. P ost, and A. T urb iner, Pr eprint arXiv fu nct-a n.S I/9 307001, (1993) . [12] Y. Briha ye , Pr eprint arXiv math-ph/0401005, (2004). [13] Y. Briha ye , J. Ndimubandi, and B. P . Mandal Pr eprint arXiv math-ph /0 601004, (2006). [14] R. Z. Zh dano v, J. Math. Phys. 37, 3198 (1996) . [15] N. Kamran, R. Milson, and P . J. Olv er, Adv.in Math. 156, 286 (2000). [16] M. Abr amo witz, and I. A. Stegun (eds) H a ndb o ok of Mathematic al F unctions 15 (New Y ork:Do v er, 1970) or http:// fun cti ons.wolfram.co m/Hyp ergeometricF un ct ions/ , h ttp://functions. w olfram.com/BesselAiryStruv eF unctions/ [17] C. Emary and R. F. Bishop, J . Ph ys. A: Math. Gen. 35, 8231 (2002 ). [18] I. I. Rabi, Ph ys. Rev. 51, 652 (1937). [19] S. N. Doly a, and O. B. Zasla vskii, J. of Phys. A: Math. Gen. 34, 5995 (2001).