Quantum tops as examples of commuting differential operators

Quan tum tops as examples of comm uting differen tial op erators V.E. Adler, V.G. Marikhin, A.B. Shabat 26 Septem b er 201 1 Abstract. W e s tu dy the quan tu m analogs of tops on Lie algebras so (4) and e (3) represen ted b y differen tial op erators. 1 In tro duction The comm utativ e rings of d ifferen tial op erators p la y an imp ortan t role in mathematical ph ys ics. The problem o f their description, in the case of o n e indep endent v ariable, wa s set u p an d solv ed in the w orks by Sc h u r [ 1 ] and Burc hn all–Chaundy [ 2 ]. Later on, these results h a v e foun d n umerous ap- plications in the theory of in tegrable equ atio n s of the Kortew eg–de V ries t yp e; we mention only the description of finite-gap op erators obtained by No vik ov, Kric hever and others [ 3 , 4 ], and the algorithm of c hec kin g necessary in tegrabilit y cond itio ns dev elop ed b y Shabat and others [ 5 ]. In the case of tw o ind ep en den t v ariable, the generalizations in different directions are p ossible. In general, the problem on a pair of comm utin g op erators [ ˆ H , ˆ K ] = 0 is n ot very w ell p osed and the qu estion ab out addi- tional assumptions arises. Apart from the conditions suggested by the logic of the p roblem itself (for instance, comm utativit y of a triple of op erators), it is also natur al to s elec t the classes of op erators wh ic h are imp ortan t in applications. Here w e consider one suc h class r elated to the quant u m tops on so (4) and e (3). The corresp onding op erator ˆ H is of a v ery sp ecial form ˆ H = a ( x ) D 2 x + 2 f ( x, y ) D x D y + b ( y ) D 2 y + . . . where a, b are f ou r th degree p olynomials, f ( x, y ) is a biquadratic p olynomial and dots den ote the low er ord er terms, also with sp ecial co efficien ts. This form app ears if we consider a Hamiltonian which is quadratic with resp ect to the generators of Lie algebra, for in stance ˆ H = ( ˆ U , A ˆ U ) + ( ˆ V , B ˆ V ) + 2( ˆ U , F ˆ V ) + ( ~ c, ˆ U ) + ( ~ d, ˆ V ) 1 in the case of so (4), and use a differentia l representa tion suc h that generators U, V are replaced with differen tial op erators of fir st order with quadr atic co efficien ts (see equations ( 6 ), ( 8 ) b elo w). The order of op erator ˆ K d oes not exceed 4 in the known examples and, resp ectiv ely , the degree of its co efficien ts do es n ot exceed 8. The original classical v ersion of th e p roblem deals w ith Hamiltonians p olynomial in m omen ta w h ic h are in inv olution with resp ect to P oisson–Darb oux brac ke t, instead of commuting d ifferen tial op erators. Th e conditions of “c ommutativit y of the principal part” [ 6 ], that is, cancellation of th e leading terms in the comm u tato r , coincide for th e classical an d q u an tum cases. In more details, this relation is illustrated by the d iagram { H, K } = 0 quan tization on the Lie algebra − − − − − − − − − − − − − − − − − − − − − − − → [ ˆ H , ˆ K ] = 0 R ∗   y   y ˆ R ∗ { H D , K D } = 0 quan tization in the Darb oux co ordinates − − − − − − − − − − − − − − − − − − − − − − − − − − − − → [ ˆ H D , ˆ K D ] = 0 Here R den ote s a representa tion of the L ie–P oisson brac ket in the Darb oux co ordinates, and H D is the image of the Hamiltonian H un der the pullbac k map, that is H D = R ∗ ( H ) = H ◦ R . Analogously , ˆ R denotes a repr esen tation of the Lie algebra of differenti al op erators and ˆ H D = ˆ R ∗ ( ˆ H ) = ˆ H ◦ ˆ R . The horizon tal arro ws corresp ond to the passage fr om the Lie–Po isson brac ket to the comm u tato r on the Lie algebra and fr om the Po iss on –Darb oux brac ket to the Heisen b erg algebra { p i , x i } = δ ij → [ D x i , x j ] = δ ij . This passage, on b oth lev els, is p erf orm ed by c ho osing a su itable ord er in g of the monomials of Hamiltonians H , K , which is equiv alen t to add ing the lo w er ord er terms (“quantum corrections”). Let us remind that the description of th e classical tops admitting an additional fi r st in tegral is still an op en problem. In the s o (4) case, the necessary condition w ere d er ived by V eselo v [ 7 ] (see [ 8 ] for a m ore precise result) which define a certain v arieties in the parameter space. Ho wev er, the num b er of parameters in the known examp les is less at least by 1 than the dimension of these v arietie s. Some classification results are presented in pap er [ 9 ]. Is it p ossible to mak e progress in this problem, or, m ore precisely , in its q u an tum ve r sion, b y us e of the language of differential op erators? As a first step, in this pap er we constru ct the differentia l op erators whic h cor- resp ond to seve r al most imp ortant examples. In particular, these examples suggest that pr obably the simplest necessary condition of integ r ab ility is that the discriminant of the leading p art of ˆ H is factorable: f ( x, y ) 2 − a ( x ) b ( y ) = w ( x, y ) ˜ w ( x, y ) (1) 2 where w , ˜ w are, as w ell as f , b iqu adratic p olynomials. Notice that this relation is imp ortan t in the problem of separation of v ariables [ 10 ]. It is not clear no w, ho w to pr o v e this p rop ert y in the general setting, usin g only the assumption that there exists a comm u tin g op erator ˆ K of arbitrary order. F rom the compu tatio n al p oin t of view, the adv an tages of the d ifferen tial represent ation is that it tak es care of suc h issues as ord ering in the op erator algebra and Casimir op erators. Notice that represent ation ˆ R is defin ed on a fi xed lev el set of Casimir op erators and their v alues o ccur in ˆ H D , ˆ K D as indep endent parameters (their num b er is equ al to the rank of Lie algebra, that is 2 in s o (4) and e (3) cases). Splitting of the comm u tation relatio n with resp ect to these parameters mak es its analysis more easy . Con tents of the p ap er are the follo wing. The so (4) case considered in sec- tion 2 includes the tops of: Sc hottky–Manak o v [ 11 , 12 ], Stekloff [ 13 ], Adler– v an Mo er b ek e [ 14 , 8 ] and Sokol ov [ 15 ]. The q u an tization of the S c hottky– Manak o v top w as studied in p ap er [ 16 ]. Th e cases of Adler–v an Mo erb ek e and Sok olo v are more complicated (the op erator ˆ K is of fourth order) and, up to our kno w ledge, their quanti zation is obtained h ere for the fi rst time. In section 3 , w e consider tops on e (3): the Clebsc h top [ 17 ], the Ko walevsk a y a top [ 18 ] (quan tization is due to Lap orte [ 19 ]), the Kow alevsk a y a gyrostat and the Gory ac hev–Chaplygin case. Th e quan tization of mo dels on e (3) is r ather w ell stu d ied by Komaro v and others [ 20 , 21 , 22 , 23 , 24 ]. Certainly , the ab o ve examples do n ot exhaust all kno wn cases. An ac- coun t of the kn o wn resu lts an d a detailed b ibliograph y can b e foun d in b o ok [ 25 ]. Other directions in the problem on comm uting differen tial op erators w ere dev elop ed by many authors, we mentio n only p ap er s [ 26 , 27 ]. Notations. W e denote x = x 1 , y = x 2 for short. The definitions of Poisso n brac ket s and comm utators do n o include v a n ishing b r ac k ets. The s u bscripts i, j, k alw a ys d enote an arbitrary p ermutat ion of 1 , 2 , 3. The imaginary un it is denoted as i . The notations ~ a = ( a 1 , a 2 , a 3 ) , ( ~ a, ~ b ) = a 1 b 1 + a 2 b 2 + a 3 b 3 are used. The follo wing p olynomials are often used in writing a compact form of Hamiltonians in Darb oux co ordinates: W ( a, b, c ; x, y ) = a ( x 2 − 1)( y 2 − 1) − b ( x 2 + 1)( y 2 + 1) + 4 cxy , R ( a, b, c ; x ) = W ( a, b, c, x, x ) = ( a − b )( x 4 + 1) + 2(2 c − a − b ) x 2 . (2) Notice that op erator W ( a, b, c ; x, y ) D x D y is form in v ariant un d er the in v er- sion x → 1 /x , y → 1 / y . 3 2 T ops on so (4) ≃ so (3) ⊕ so (3 ) 2.1 Darb oux co ordinates and op erator represen tation The Lie–Po iss on brac k et on so (3) and its C asimir function are defin ed by equations { U i , U j } = i ε ij k U k , ( U, U ) = s 2 1 (3) where U denotes th e v ector ( U 1 , U 2 , U 3 ). A repr esen tation in Darb ou x co- ordinates is giv en b y relations U 1 = − 1 2 ( x 2 − 1) p 1 + s 1 x, U 2 = − i 2 ( x 2 + 1) p 1 + i s 1 x, U 3 = − xp 1 + s 1 . (4) In order to quantize , w e pass to the op erator algebra [ ˆ U i , ˆ U j ] = i ε ij k ˆ U k , ( ˆ U , ˆ U ) = j 1 ( j 1 + 1) (5) whic h is r epresen ted by differential op erators as follo w s: ˆ U 1 = − 1 2 ( x 2 − 1) D x + j 1 x, ˆ U 2 = − i 2 ( x 2 + 1) D x + i j 1 x, ˆ U 3 = − xD x + j 1 . (6) The Lie algebra so (4) ≃ s o (3) ⊕ so (3) is represent ed as the direct sum w ith a double und er the follo wing renaming: U → V , x → y , p 1 → p 2 , s 1 → s 2 , ˆ U → ˆ V , D x → D y , j 1 → j 2 . The tops on so (4) are defined by Hamiltonians of the general form ˆ H = ( U, AU ) + ( V , B V ) + ( U, F V ) + ( ~ c, U ) + ( ~ d, V ) . (7) The matrices A and B can b e c hosen diagonal without loss of generalit y . The use of representa tion ( 6 ) yields a differenti al op erator of th e sp ecial form ˆ H D = a ( x ) D 2 x + f ( x, y ) D x D y + b ( y ) D 2 y −  2 j 1 − 1 2 a ′ ( x ) + 2 c ( x ) + j 2 f y ( x, y )  D x −  2 j 2 − 1 2 b ′ ( y ) + 2 d ( y ) + j 1 f x ( x, y )  D y + j 1 (2 j 1 − 1) 6 a ′′ ( x ) + j 2 (2 j 2 − 1) 6 b ′′ ( y ) + 2 j 1 c ′ ( x ) + 2 j 2 d ′ ( y ) + j 1 j 2 f xy ( x, y ) + κ (8) 4 where κ is an arbitrary constan t and a ( x ) = a 0 ( x 4 + 1) + a 2 x 2 , b ( y ) = b 0 ( y 4 + 1) + b 2 y 2 , c ( x ) = c 2 x 2 + c 1 x + c 0 , d ( y ) = d 2 y 2 + d 1 y + d 0 , f ( x, y ) = f 22 x 2 y 2 + · · · + f 00 . The corresp ondence b et ween ( 8 ) and ( 7 ) is giv en by relations A = diag(2 a 0 , − 2 a 0 , a 2 ) + αI , B = d iag(2 b 0 , − 2 b 0 , b 2 ) + β I , ~ c = 2( c 2 − c 0 , − i ( c 2 + c 0 ) , c 1 ) , ~ d = 2( d 2 − d 0 , − i ( d 2 + d 0 ) , d 1 ) , F =   f 00 − f 02 − f 20 + f 22 i ( f 00 + f 02 − f 20 − f 22 f 21 − f 01 i ( f 00 − f 02 + f 20 − f 22 ) − f 00 − f 02 − f 20 − f 22 − i ( f 21 + f 01 ) f 12 − f 10 − i ( f 12 + f 10 ) f 11   , 3 κ = (3 α + a 2 ) j 1 ( j 1 + 1) + (3 β + b 2 ) j 2 ( j 2 + 1) with arbitrary constan ts α, β . The follo wing remark is usefu l for computing sp ectra. It is ab ou t the re- lation b et ween differential representati on ( 6 ) and the matrix representa tion [ 28 , p. 88] ˆ U 1 ± i ˆ U 2 = ˆ U ± , ˆ U 3 | m, j i = m | m, j i , ˆ U + | m, j i = p ( j − m )( j + m + 1) | m + 1 , j i , ˆ U − | m, j i = p ( j + m )( j − m + 1) | m − 1 , j i . (9) In the p hysical case we consider an orbit j = const such that j an d m are sim ultaneously in teger or half-inte ger and m tak es the v alues from − j to j , whic h corresp onds to the matrices of the finite size (2 j + 1) × (2 j + 1). F or- mally , one can w aive these restrictions and consider the infin ite trid iagonal matrices whic h are in terpr eted as the difference op erators a ( m ) T m + b ( m ) + c ( m ) T − 1 m where T m : m 7→ m + 1 is the sh ift op erator (the matrix with units on the sub diagonal). A conjugation ˆ A → f − 1 ˆ Af allo ws to get rid of square r oots, and the representa tion of so (3) by sh ift op erators app ears: ˆ U 1 = − 1 4 ( m − j )( m + j + 1) T m + T − 1 m , ˆ U 2 = − i  1 4 ( m − j )( m + j + 1) T m + T − 1 m  , ˆ U 3 = m. (10) Another difference representat ion can b e obtained directly from ( 6 ) via the c hange (“F ourier transformation”) D x → mT m , x → T − 1 m 5 whic h p reserv es the Heisen b erg algebra: [ D x , x ] = 1 , [ mT m , T − 1 m ] = 1 , and yields the op erators ˆ U 1 = m 2 T m +  j + 1 − m 2  T − 1 m , ˆ U 2 = − i m 2 T m + i  j + 1 − m 2  T − 1 m , ˆ U 3 = j − m + 1 . (11) One can c hec k th at rep resen tations ( 10 ) and ( 11 ) are equiv alen t up to a certain linear transformation from S O (3) and the conjugation by op erator g ( m ) T − j − 1 where fun ction g satisfies relation g ( m + 2) /g ( m + 1) = 2 j − m . 2.2 Sc hottky–Manak ov top The comm uting op erators are of the form ˆ H = − α 2 1 ˆ U 2 1 − α 2 2 ˆ U 2 2 − α 2 3 ˆ U 2 3 − α 2 1 ˆ V 2 1 − α 2 2 ˆ V 2 2 − α 2 3 ˆ V 2 3 + 2 α 2 α 3 ˆ U 1 ˆ V 1 + 2 α 3 α 1 ˆ U 2 ˆ V 2 + 2 α 1 α 2 ˆ U 3 ˆ V 3 , (12) ˆ K = α 1 ˆ U 1 ˆ V 1 + α 2 ˆ U 2 ˆ V 2 + α 3 ˆ U 3 ˆ V 3 . (13) These can b e obtained from the classical Hamiltonians in inv olution by the simple exc h ange U → ˆ U , V → ˆ V , b ecause the problem of norm al order- ing do es n ot app ear in this example (eac h m onomial con tains comm u ting v ariables only). In con trast to some other examples (Stekloff, Ko walevsk ay a tops), no linear terms can b e added to the Hamiltonians, b oth in classical and quantum cases, as one can pro v e by a dir ect computation with indeter- minate coefficients. Applying map ( 6 ) yields the follo w ing pair of comm u ting differen tial op erators: 4 ˆ H D = r ( x ) D 2 x + 2 z D x D y + r ( y ) D 2 y −  2 j 1 − 1 2 r ′ ( x ) + 2 j 2 z y  D x −  2 j 2 − 1 2 r ′ ( y ) + 2 j 1 z x  D y + 2 j 1 j 2 z xy + j 1 (2 j 1 − 1) 6 r ′′ ( x ) + j 2 (2 j 2 − 1) 6 r ′′ ( y ) − 4 3 ( j 1 ( j 1 + 1) + j 2 ( j 2 + 1))( α 2 1 + α 2 2 + α 2 3 ) , (14) 4 ˆ K D = w D x D y − j 2 w y D x − j 1 w x D y + j 1 j 2 w xy (15) where, using the notations ( 2 ), r ( x ) = − R ( α 2 1 , α 2 2 , α 2 3 ; x ) , z = z ( x, y ) = W ( α 2 α 3 , α 3 α 1 , α 1 α 2 ; x, y ) , w = w ( x, y ) = W ( α 1 , α 2 , α 3 ; x, y ) . 6 The constan t in the last line of ( 14 ) ( κ in equation ( 8 )) has n o effe ct on the comm utativit y and determines the energy shift of the ground state of the system. Alternativ ely , op erators ( 14 ), ( 15 ) can b e obtained by another route along the diagram f rom Introd uction. First, one s hould pass to the Darb oux co ordinates in the classical Hamiltonians H, K , ac cordin gly to f ormulae ( 4 ). This y ields 4 H D = r ( x ) p 2 1 + 2 z p 1 p 2 + r ( y ) p 2 2 − ( s 1 r ′ ( x ) + 2 s 2 z y ) p 1 − ( s 2 r ′ ( y ) + 2 s 1 z x ) p 2 + 2 s 1 s 2 z xy + s 2 1 3 r ′′ ( x ) + s 2 2 3 r ′′ ( y ) − 4 3 ( s 2 1 + s 2 2 )( α 2 1 + α 2 2 + α 2 3 ) , (16) 4 K D = w p 1 p 2 − s 2 w y p 1 − s 1 w x p 2 + s 1 s 2 w xy (17) where r, z , w are the same as ab o v e. Next, p i are r eplace d b y D x i in qu ad r atic terms and the lo wer ord er terms are tak en with indeterminate co efficien ts (preserving the degree with resp ect to x, y ). S o, one searc hes for th e com- m utin g differen tial op erators of the form ˆ H D = r ( x ) D 2 x + 2 z ( x, y ) D x D y + r ( y ) D 2 y + a ( 3 x, 1 y ) D x + b ( 1 x, 3 y ) D y + c ( 2 x, 2 y ) , ˆ K D = w ( x, y ) D x D y + f ( 2 x, 1 y ) D x + g ( 1 x, 2 y ) D y + h ( 1 x, 1 y ) where r , z , w are giv en, a, b, c, f , g , h are p olynomials with indeterminate co- efficien ts, and n umb ers abov e arguments sho w the corresp onding degrees. The resulting system of equ ations is rather bulky , bu t its solution ( 14 ), ( 15 ) can b e easily reco vered by an y system of computer algebra. Th u s, b oth metho ds of quan tization turn out to b e equiv alent , as one shou ld exp ect. In the situation, w hen ˆ H is given and ˆ K is to b e foun d, eac h metho d h as its o wn strong and w eak p oin ts from the p oin t of view of computation com- plexit y . The use of the Lie algebra generators giv es a simpler equations for the co efficien ts, bu t its implement ation is more difficult b ecause it d eals with n on commutativ e v ariables. The language of d ifferen tial representa tion is more fl exible and un iversal, b ut the resulting sy s tem for the co efficien ts of th e op erators is slightly more cum b ers ome. 2.3 Stekloff top The classical Hamiltonians are of the form ( α = α 1 α 2 α 3 ) H = − α 2  1 α 2 1 U 2 1 + 1 α 2 2 U 2 2 + 1 α 2 3 U 2 3  + 2 α ( α 1 U 1 V 1 + α 2 U 2 V 2 + α 3 U 3 V 3 ) , K = 2 α  1 α 1 U 1 V 1 + 1 α 2 U 2 V 2 + 1 α 3 U 3 V 3  − α 2 1 V 2 1 − α 2 2 V 2 2 − α 2 3 V 2 3 . 7 Lik e in th e previous example, the pr oblem of ordering do es not app ear and the comm uting quant u m Hamiltonians are ob tained just by placing h ats o v er U, V . It is easy to c h ec k that the follo win g linear terms can b e added to the Hamiltonians, b oth in the classical and qu an tum cases: H → H − β 1 α 2 α 3 U 1 − β 2 α 1 α 3 U 2 − β 3 α 1 α 2 U 3 , K → K + β 1 V 1 + β 2 V 2 + β 3 V 3 , with arbitrary β i . How ever, we assum e β i = 0 for the sake of simp licit y . The passage to the differentia l op erators yields 4 ˆ H D = r 1 D 2 x + 2 z D x D y −  2 j 1 − 1 2 r ′ 1 + 2 j 2 z y  D x − 2 j 1 z x D y + 2 j 1 j 2 z xy + j 1 (2 j 1 − 1) 6 r ′′ 1 − 4 3 j 1 ( j 1 + 1)( α 2 1 α 2 2 + α 2 2 α 2 3 + α 2 3 α 2 1 ) , 4 ˆ K D = 2 w D x D y + r 2 D 2 y − 2 j 2 w y D x −  2 j 2 − 1 2 r ′ 2 + 2 j 1 w x  D y + 2 j 1 j 2 w xy + j 2 (2 j 2 − 1) 6 r ′′ 2 − 4 3 j 2 ( j 2 + 1)( α 2 1 + α 2 2 + α 2 3 ) where r 1 = r 1 ( x ) = − R ( α 2 2 α 2 3 , α 2 3 α 2 1 , α 2 1 α 2 2 ; x ) , r 2 = r 2 ( y ) = − R ( α 2 1 , α 2 2 , α 2 3 ; y ) , z = z ( x, y ) = α 1 α 2 α 3 W ( α 1 , α 2 , α 3 ; x, y ) , w = w ( x, y ) = W ( α 2 α 3 , α 3 α 1 , α 1 α 2 ; x, y ) . The same op erators are obtained via the quant ization in the Darb oux v ari- ables, starting from the Hamiltonians 4 H D = r 1 p 2 1 + 2 z p 1 p 2 − ( s 1 r ′ 1 + 2 s 2 z y ) p 1 − 2 s 1 z x p 2 + 2 s 1 s 2 z xy + s 2 1 3 r ′′ 1 − 4 3 s 2 1 ( α 2 1 α 2 2 + α 2 2 α 2 3 + α 2 3 α 2 1 ) , 4 K D = 2 w p 1 p 2 + r 2 p 2 2 − 2 s 2 w y p 1 − ( s 2 r ′ 2 + 2 s 1 w x ) p 2 + 2 s 1 s 2 w xy + s 2 2 3 r ′′ 2 − 4 3 s 2 2 ( α 2 1 + α 2 2 + α 2 3 ) . 8 2.4 M. A dler–v an Mo erb ek e top The classical Hamiltonians are of th e follo wing form (the parameters are related b y th e constrain t λ 1 + λ 2 + λ 3 = 0): H = 3 X i =1  − 9 λ 2 j λ 2 k U 2 i + 6 λ j λ k ( λ j − λ i )( λ k − λ i ) U i V i + λ j λ k (4 λ 2 i − λ j λ k ) V 2 i  , (18) K = 3 X i,j λ j ( λ i − λ j ) U i V i V 2 j + X i ( λ i − λ j )( λ i − λ k ) U i V 3 i − 9 X l U 2 l X i λ j λ k U i V i + 3 2 X l U 2 l X i (3 λ 2 i − λ 2 j − λ 2 k ) V 2 i . (19) The quantum Hamiltonian ˆ H is obtained fr om H by fixing hats. How ever, ˆ K con tains monomials w ith noncomm utativ e generators and the problem of ordering arises. The resu lt of direct computation with indeterminate co efficien ts is the follo w ing expression for ˆ K : ˆ K = X i,j λ j ( λ i − λ j )  ˆ U i ˆ V i ˆ V 2 j + ˆ U i ˆ V j ˆ V i ˆ V j + ˆ U i ˆ V 2 j ˆ V i  + X i ( λ i − λ j )( λ i − λ k ) ˆ U i ˆ V 3 i − 9  X l ˆ U 2 l + 1 3  X i λ j λ k ˆ U i ˆ V i + 3 2 X l ˆ U 2 l X i (3 λ 2 i − λ 2 j − λ 2 k ) ˆ V 2 i . (20) Th us, the quant u m Hamiltonian differs from the classical one b y ordering in the first sum and a quan tum correction in the th ir d su m (rec all, that P l ˆ U 2 l is the Casimir fun ctio n ). The follo w ing expression for ˆ H D is obtained b y passage to differen tial op erators according to general form ula ( 8 ): 4 ˆ H D = r 1 ( x ) D 2 x + 2 z D x D y + r 2 ( y ) D 2 y −  2 j 1 − 1 2 r ′ 1 ( x ) + 2 j 2 z y  D x −  2 j 2 − 1 2 r ′ 2 ( y ) + 2 j 1 z x  D y + j 1 (2 j 1 − 1) 6 r ′′ 1 ( x ) + j 2 (2 j 2 − 1) 6 r ′′ 2 ( y ) + 2 j 1 j 2 z xy − 4 3 (9 j 1 ( j 1 + 1) + j 2 ( j 2 + 1))( λ 2 1 + λ 1 λ 2 + λ 2 2 ) 2 (21) where r 1 ( x ) = − 9 R ( λ 2 2 λ 2 3 , λ 2 3 λ 2 1 , λ 2 1 λ 2 2 , x ) , r 2 ( x ) = 1 9 r 1 ( x ) + 4 λ 1 λ 2 λ 3 R ( λ 1 , λ 2 , λ 3 , x ) , z = z ( x, y ) = 3 W ( µ 1 , µ 2 , µ 3 , x, y ) , µ i = ( λ 2 i − λ 2 j )( λ 2 i − λ 2 k ) . 9 The structure of op erator ˆ K D is rather simple: 4 ˆ K D = g 2 D x D 3 y + c 1 g g y D x D 2 y + c 2 g g x D 3 y + ( c 3 g g y y + c 4 g 2 y ) D x D y + ( c 5 g y g x + c 6 g g xy ) D 2 y + ( c 7 g g y y y + c 8 g y g y y ) D x + ( c 9 g y y g x + c 10 g y g xy + c 11 g g xy y ) D y + c 12 g y y y g x + c 13 g y y g xy + c 14 g y g xy y + c 15 g g xy yy , but the co efficien ts are cum b ers ome. Here g denotes the p olynomial g = ( λ 1 − λ 2 )( xy 3 + 1) + 3 λ 3 ( x + y ) y related to the co efficien ts of ˆ H D b y relation z 2 − r 1 ( x ) r 2 ( y ) = 36 λ 1 λ 2 λ 3 ( λ 1 − λ 2 )( λ 2 − λ 3 )( λ 3 − λ 1 ) g ˜ g , where ˜ g = ( λ 1 − λ 2 ) x ( x + y ) − λ 3 ( x 3 y + 1). The coefficients c i are, in tur n, p olynomials in the parameters j 1 , j 2 : c 1 = − 2( j 2 − 1) , c 2 = − 2 j 1 , c 3 = 1 2 (3 j 1 ( j 1 + 1) + j 2 ( j 2 − 3) + 2) , c 4 = − j 1 ( j 1 + 1) + j 2 ( j 2 − 1) , c 5 = 2 j 1 ( j 1 + j 2 ) , c 6 = − 2 j 1 ( j 1 − j 2 + 2) , c 7 = − j 2 6 (9 j 1 ( j 1 + 1) − j 2 ( j 2 + 3) + 4) , c 8 = j 2 2 ( j 1 ( j 1 + 1) − j 2 ( j 2 − 1)) , c 9 = − j 1 2 (3 j 2 1 + 4 j 1 j 2 + j 2 2 + j 1 + j 2 ) , c 10 = 2 j 1 ( j 1 ( j 1 + 1) − j 2 ( j 2 − 1)) , c 11 = − j 1 2 (3 j 2 1 − 4 j 1 j 2 + j 2 2 + 5 j 1 − 7 j 2 + 4) , c 12 = j 1 j 2 6 (9 j 2 1 + 3 j 1 j 2 − j 2 2 + 6 j 1 + 1) , c 13 = j 1 j 2 6 ( − 3 j 2 1 + 5 j 1 j 2 + 3 j 2 2 − 4 j 1 + 2 j 2 − 1) , c 14 = j 1 j 2 6 ( − 3 j 2 1 − 5 j 1 j 2 + 3 j 2 2 − 2 j 1 − 8 j 2 + 1) , c 15 = j 1 j 2 6 (9 j 2 1 − 3 j 1 j 2 − j 2 2 + 12 j 1 − 6 j 2 + 7) . 2.5 Sok olo v t op In this example, it is con venien t to use the v ariables m i = U i + V i , n i = U i − V i . The Sok olo v top on s o (4) is defined b y the Hamiltonian H = 1 2 m 2 1 + 1 2 m 2 2 + m 2 3 + m 3 ( αn 1 + β n 2 ) − 1 2 ( α 2 + β 2 ) n 2 3 (22) 10 and additional fourth order inte gral is of the f orm K = m 2 3  2 H − m 2 3 + ( β m 1 − αm 2 ) 2 + ( αn 1 + β n 2 ) 2  +2 m 3  αm 1 + β m 2 − ( α 2 + β 2 ) n 3  ( m 1 n 1 + m 2 n 2 ) . (23) The quantum Hamiltonian is obtained b y symmetrization of noncommuta- tiv e monomials: ˆ H = 1 2 ˆ m 2 1 + 1 2 ˆ m 2 2 + ˆ m 2 3 + [ ˆ m 3 , α ˆ n 1 + β ˆ n 2 ] + − 1 2 ( α 2 + β 2 ) ˆ n 2 3 (24) where [ a, b ] + = 1 2 ( ab + ba ) . The op erator ˆ K is obtained by symmetrization as w ell, with some sp ecial w eight co efficien ts. It should b e noted that there exist homogeneous p oly- nomials in th e algebra so (4) w h ic h v anish identica lly in virtue of the com- m utation relations, s o that the form of the oper ator ˆ K is not unique and one can try to simplify it by addin g suc h p olynomials. W e write do wn one of p ossible versions. Let u s d en ote A a,b,c ( m, f ) = amf 2 m + b [ m 2 , f 2 ] + + c [ f , mf m ] + + (1 − a − b − c ) f m 2 f , B ( m, f , g ) = mf g + g f m, then the op erator ˆ K = A 1 2 , 0 , 3 4 ( ˆ m 3 , ˆ m 1 ) + A 1 2 , 0 , 3 4 ( ˆ m 3 , ˆ m 2 ) − ( α 2 + β 2 ) ˆ m 2 3 ˆ n 2 3 + A 1 2 , 1 4 , 1 ( ˆ m 3 , β ˆ m 1 − α ˆ m 2 ) + A − 3 4 , − 1 , 7 2 ( ˆ m 3 , ˆ m 3 + α ˆ n 1 + β ˆ n 2 ) + B ( ˆ m 3 , α ˆ m 1 + β ˆ m 2 − ( α 2 + β 2 ) ˆ n 3 , ˆ m 1 ˆ n 1 + ˆ m 2 ˆ n 2 ) , − 1 2 [ ˆ m 3 ˆ n 3 , [ ˆ m 3 , α ˆ m 1 + β ˆ m 2 ]] (25) comm utes w ith ˆ H and coincides with K if all v ariables are comm utativ e. In this example, it is also conv enient to mo dify sligh tly our differen tial represent ation of so (4), by changing signs y → − y , D y → − D y . T his brings to the op erators whic h are symmetric with r esp ect to x, y . Let u s denote (in general, the parameters α, β are complex) ξ 1 = α + i β , ξ 2 = − α + i β , r ( x ) = x ( ξ 1 x + 1)( x + ξ 2 ) , z = z ( x, y ) = ξ 1 xy ( x + y ) + ( x + y ) 2 + 2(1 − ξ 1 ξ 2 ) xy + ξ 2 ( x + y ) 11 then ˆ H is written in the form ( 8 ) as f ollo ws: 2 ˆ H D = r ( x ) D 2 x + z D x D y + r ( y ) D 2 y −  2 j 1 − 1 2 r ′ ( x ) + j 2 z y  D x −  2 j 2 − 1 2 r ′ ( y ) + j 1 z x  D y + j 1 (2 j 1 − 1) 6 r ′′ ( x ) + j 2 (2 j 2 − 1) 6 r ′′ ( y ) + j 1 j 2 z xy + 1 3 ( j 1 ( j 1 + 1) + j 2 ( j 2 + 1))( ξ 1 ξ 2 + 4) . Notice that the p olynomial r is of degree 3, b ecause Hamiltonian ( 22 ) tak es the f orm ( 7 ) with nondiagonal matrices A, B if one r etur ns to the v ariables U, V . The second op erator is to o bulky and we present explicitly only the leading terms: ˆ K D = w 2 ( xD x + y D y ) 2 D x D y − 2 j 2 x 2 ww y D 3 x − 2 j 1 y 2 ww x D 3 y − 2 xw (( j 1 − 1) xw x + ( j 1 + j 2 − 1) w + (2 j 2 − 1) y w y ) D 2 x D y − 2 y w ((2 j 1 − 1) xw x + ( j 1 + j 2 − 1) w + ( j 2 − 1) y w y ) D 2 x D y + . . . where w = ξ 1 xy + x + y + ξ 2 . 2.6 The classical limit The Planck constant is introd uced by simple scaling of the generators, so that comm utation relations ( 5 ) are rep lace d with [ ˆ U i , ˆ U j ] = i ~ ε ij k ˆ U k , ( ˆ U , ˆ U ) = ~ 2 j 1 ( j 1 + 1) and representa tion ( 6 ) is replaced with ˆ U 1 = ~  − 1 2 ( x 2 − 1) D x + j 1 x  , ˆ U 2 = ~  − i 2 ( x 2 + 1) D x + i j 1 x  , ˆ U 3 = ~ ( − xD x + j 1 ) . The equations for the v ariables V are c hanged analogously . The passage to the classical limit for any qu an tum op erator ˆ A is defined according to the form u la A = lim ~ → 0 e − i ~ ( p 1 x + p 2 y )  ˆ A e i ~ ( p 1 x + p 2 y )    j i = s i ~ , in particular the comm u tato r brac ket and the Casimir f u nctions for eac h cop y of so (3) are mapp ed int o the Lie–P oisson brac ket ( 3 ), and th e f ormulae for the generators themselve s are mapp ed int o the Darb oux co ordinates represent ation ( 4 ). Applying of th is pro cedure to ˆ H D , ˆ K D giv es the same expressions for H D , K D as the intermediate passage to th e Darb oux co ord inates in the classical Hamiltonians H , K . Th is is gu aranteed by the “corresp ondence p rinciple” 12 whic h is inv arian t, that is it does n ot dep end explicitly on the c h oice of rep- resen tation of the algebra so (4). The c h ec k of th e corresp on d ence prin ciple is trivial for the Sc hottky–Manak ov and Stekloff tops, b ecause there are no quan tu m corrections in these systems. In the Adler–v an Moerb ek e case, the correct p assage to the classical limit is ac hieved b y changing one term in th e expression for ˆ K ( 20 ):  X l ˆ U 2 l + 1 3  X i λ j λ k ˆ U i ˆ V i →  X l ˆ U 2 l + ~ 2 3  X i λ j λ k ˆ U i ˆ V i . After this, all terms b ecome homogeneous w ith resp ect to ~ , and the limit ~ → 0 gives rise to the classical Hamiltonian ( 19 ). It should b e noted th at the Casimir functions ~ 2 j i ( j i + 1) are of the quan tu m nature, b ecause j i tak e in teger/half-in teger v alues. The passage to the limit ~ → 0 b rings to th e classical (finite) qu an tities s i = ~ j i . On the other hand, if we consider spin s th en the v alues j i are finite and therefore s i → 0, in accordance w ith a statemen t that the spin is a pure quan tu m concept. 3 T ops on e (3) 3.1 Darb oux co ordinates and op erator represen tations The Lie–P oisson brac ket on e (3) is of the form { M i , M j } = − ε ij k M k , { M i , γ j } = − ε ij k γ k , { γ i , γ j } = 0 and the Casimir functions are ( M , γ ) = l , ( γ , γ ) = a 2 where M = ( M 1 , M 2 , M 3 ), γ = ( γ 1 , γ 2 , γ 3 ). W e use the follo wing repr esen- tation in the Darb oux co ordinates: M 1 = − i 2 ( x 2 − 1) p 1 − i 2 ( y 2 − 1) p 2 + l 2 a ( x − y ) , M 2 = − 1 2 ( x 2 + 1) p 1 − 1 2 ( y 2 + 1) p 2 − i l 2 a ( x − y ) , M 3 = i ( xp 1 + y p 2 ) , γ 1 = a 1 − xy x − y , γ 2 = i a 1 + xy x − y , γ 3 = a x + y x − y . (26) 13 A represen tation with real Darb oux co ordinates [ 29 ] sh ould b e mentioned as w ell: M 1 = − p 1 q 1 q 2 + 1 2 p 2 ( q 2 1 − q 2 2 − 1) + lq 1 ( q 2 1 + q 2 2 + 1) 2 a ( q 2 1 + q 2 2 ) , M 2 = p 2 q 1 q 2 + 1 2 p 1 ( q 2 1 − q 2 2 + 1) + lq 2 ( q 2 1 + q 2 2 + 1) 2 a ( q 2 1 + q 2 2 ) , M 3 = p 1 q 2 − p 2 q 1 , γ 1 = 2 aq 1 ( q 2 1 + q 2 2 + 1) , γ 2 = 2 aq 2 ( q 2 1 + q 2 2 + 1) , γ 3 = a ( q 2 1 + q 2 2 − 1) ( q 2 1 + q 2 2 + 1) . (27) Quant ization replaces the Lie–P oisson brac k et with the e (3) comm utator [ ˆ M i , ˆ M j ] = i ε ij k M k , [ ˆ M i , ˆ γ j ] = i ε ij k γ k , [ ˆ γ i , ˆ γ j ] = 0 (28) and the Casimir op erators are ( ˆ γ , ˆ M ) = l , (ˆ γ , ˆ γ ) = a 2 . (29) This op erator algebra admits the follo win g rep resen tation: ˆ M 1 = 1 2 (1 − x 2 ) D x + 1 2 (1 − y 2 ) D y + l 2 a ( x − y ) , ˆ M 2 = i 2 (1 + x 2 ) D x + i 2 (1 + y 2 ) D y − i l 2 a ( x − y ) , ˆ M 3 = xD x + y D y , ˆ γ 1 = a 1 − xy x − y , ˆ γ 2 = i a 1 + xy x − y , ˆ γ 3 = a x + y x − y . (30) Notice that all op erators are inv ariant with r esp ect to the c h ange x ↔ y , a → − a. A matrix repr esen tation can b e obtained b y in tro du cing the basis func- tion | ψ i = | m, n i = ( x + y ) m ( x − y ) n . Th is c hoice is motiv ated b y th e denominator of generators γ i in r epresen tation ( 30 ) and the symmetry ar- gumen ts. Easy compu tation yields ( M ± = M 1 ± i M 2 , γ ± = γ 1 ± i γ 2 ) ˆ M 3 | m, n i = ( m + n ) | m, n i , ˆ M − | m, n i = 2 m | m + 1 , n − 1 i , ˆ M + | m, n i = −  n + m 2  | m + 1 , n i − m 2 | m − 1 , n + 2 i + l a | m, n + 1 i , ˆ γ 3 | m, n i = a | m + 1 , n − 1 i , ˆ γ − | m, n i = 2 a | m, n − 1 i , ˆ γ + | m, n i = a 2 | m, n + 1 i − a 2 | m + 2 , n − 1 i . 14 In ord er to p ass to the classical limit, the Planc k constan t is introdu ced as follo ws: [ ˆ M i , ˆ M j ] = i ~ ε ij k M k , [ ˆ M i , ˆ γ j ] = i ~ ε ij k γ k , [ ˆ γ i , ˆ γ j ] = 0 , and the op erator representa tion is replaced by equations ˆ M 1 = ~  1 2 (1 − x 2 ) D x + 1 2 (1 − y 2 ) D y  + l 2 a ( x − y ) , ˆ M 2 = ~  i 2 (1 + x 2 ) D x + i 2 (1 + y 2 ) D y  − i l 2 a ( x − y ) , ˆ M 3 = ~ ( xD x + y D y ) , ˆ γ 1 = a 1 − xy x − y , ˆ γ 2 = i a 1 + xy x − y , ˆ γ 3 = a x + y x − y . The form ula f or the classical limit is analogous to the s o (4) case: A = lim ~ → 0 e − i ~ ( p 1 x + p 2 y )  ˆ A e i ~ ( p 1 x + p 2 y )  , ho wev er, notice that here the Casimir oper ators ( 29 ) are pure cla ssical. Ap- plying th is pro cedure to the generators M i yields the b rac ket ( 26 ). In the Ko wal evsk a ya case this p ro cedure r esu lts in c hanging of a co efficien t in op- erator ˆ K ( 35 ) (cf [ 19 ]): ˆ K = 1 2 ( ˆ k + ˆ k − + ˆ k − ˆ k + ) + 4 ~ 2 ( ˆ M 2 1 + ˆ M 2 2 ) . 3.2 The Clebsc h top There is no problem of ordering in this case and the quan tum top is defined b y the Hamiltonians ˆ H = 1 2 3 X i =1  ˆ M 2 i + λ i ˆ γ 2 i  , (31) ˆ K = 1 2 3 X i =1  λ i ˆ M 2 i − λ ˆ γ 2 i λ i  , λ = λ 1 λ 2 λ 3 . (32) 15 The use of repr esentati on ( 30 ) yields the follo win g comm uting differenti al op erators: 2 ˆ H D = − ( x − y ) 2 D x D y + l a ( x − y )( D x + D y ) + a 2 z ( x − y ) 2 + a 2 ( λ 1 + λ 2 + λ 3 ) , 8 ˆ K D = r ( x ) D 2 x + 2 z D x D y + r ( y ) D 2 y +  a − l 2 a r ′ ( x ) + l a z y  D x +  a + l 2 a r ′ ( y ) − l a z x  D y + ( λ 1 − λ 2 ) l 2 a 2 ( x − y ) 2 − ( λ 1 − λ 2 ) l a ( x 2 − y 2 ) + a 2 r ( x ) r ( y ) − z 2 ( x − y ) 4 where r ( x ) = R ( λ 1 , λ 2 , λ 3 ; x ) , z = z ( x, y ) = W ( λ 1 , λ 2 , λ 3 ; x, y ) . Notice, that in this case identit y ( 1 ) tak es the form z 2 − r ( x ) r ( y ) = 4( x − y ) 4 ( λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 ) + 4( x − y ) 2 W ( λ 2 λ 3 , λ 3 λ 1 , λ 1 λ 2 ; x, y ) , therefore the last term in ˆ K D partially ca n cels. The form of classical Hamil- tonians in the Darb oux co ord inates is analogous, with slightly different co- efficien ts: 2 H D = − ( x − y ) 2 p 1 p 2 + 2 l a ( x − y )( p 1 + p 2 ) + a 2 z ( x − y ) 2 + a 2 ( λ 1 + λ 2 + λ 3 ) , 8 K D = r ( x ) p 2 1 + 2 z p 1 p 2 + r ( y ) p 2 2 − l a ( r ′ ( x ) − 2 z y ) p 1 + l a ( r ′ ( y ) − 2 z x ) p 2 + 4( λ 1 − λ 2 ) l 2 a 2 ( x − y ) 2 + a 2 r ( x ) r ( y ) − z 2 ( x − y ) 4 . 3.3 Ko walevs k a y a top The classical top is defin ed by Hamiltonians H = 1 2 ( M 2 1 + M 2 2 + 2 M 2 3 ) − 1 2 γ 1 , K = k + k − (33) where k ± = ( M 1 ± i M 2 ) 2 + γ 1 ± i γ 2 . 16 The expressions in the Darb oux co ordinates ( 26 ) are: 2 H = x 2 p 2 1 + (4 xy − x 2 − y 2 ) p 1 p 2 + y 2 p 2 2 + 2 l a ( x − y )( p 1 + p 2 ) + a xy − 1 x − y , K =  x 2 p 1 + y 2 p 2 − 2 l a ( x − y )  2 − 2 axy x − y !  ( p 1 + p 2 ) 2 + 2 a x − y  . The quant u m version of the Ko wal evsk a ya top is of the form [ 19 , 22 ] ˆ H = 1 2 ( ˆ M 2 1 + ˆ M 2 2 + 2 ˆ M 2 3 ) − 1 2 ˆ γ 1 , (34) ˆ K = 1 2 ( ˆ k + ˆ k − + ˆ k − ˆ k + ) + 4( ˆ M 2 1 + ˆ M 2 2 ) ( 35) where ˆ k ± = ( ˆ M 1 ± i ˆ M 2 ) 2 + ˆ γ 1 ± i ˆ γ 2 . The use of repr esentati on ( 30 ) yields the follo wing comm uting differenti al op erators: 2 ˆ H D = x 2 D 2 x + (4 xy − x 2 − y 2 ) D x D y + y 2 D 2 y , + 1 a (( a + l ) x − l y ) D x + 1 a ( lx + ( a − l ) y ) D y + a xy − 1 x − y , ˆ K D =  f 2 − 2 axy x − y , g 2 + 2 a x − y  + + 4[ f , g f g ] + − 2[ f g , g f ] + − 2[ f 2 , g 2 ] + where [ a, b ] + = 1 2 ( ab + ba ) and f = x 2 D x + y 2 D y − l a ( x − y ) , g = D x + D y . It is worth noticing that Hamiltonians ( 34 ), ( 35 ) admit the follo wing gener- alizati on (Ko wale vs k a ya gyrostat) [ 23 ]: 2 ˆ H = ˆ M 2 1 + ˆ M 2 2 + 2 ˆ M 2 3 − ˆ γ 1 + c ˆ M 3 , ˆ K = 1 2 ( ˆ k + ˆ k − + ˆ k − ˆ k + ) + 4( ˆ M 2 1 + ˆ M 2 2 ) − 2 c ( ˆ M 2 1 + ˆ M 2 2 ) ˆ M 3 + 2 c 2 ˆ M 2 3 + c ( c 2 + 1) ˆ M 3 − 2 c ˆ M 1 ˆ γ 3 − c 2 ˆ γ 1 − i c ˆ γ 2 . 3.4 Gory ach ev–Chaplygin case The quant ization w as considered in [ 21 ]. The Hamiltonians ˆ H = ˆ M 2 1 + ˆ M 2 2 + 4 ˆ M 2 3 − ˆ γ 1 + c ˆ M 3 , ˆ K = 4( ˆ M 2 1 + ˆ M 2 2 ) ˆ M 3 + 2 ˆ M 1 ˆ γ 3 − 4 c ˆ M 2 3 + (1 − c 2 ) ˆ M 3 + c ˆ γ 1 + i ˆ γ 2 satisfy the relation [ ˆ H , ˆ K ] = 4 i l ˆ M 2 17 where l = ( ˆ γ , ˆ M ) is one of the Casimir op erators on e (3). Th erefore, an in tegrable case o ccurs at l = 0. The passage to operator repr esen tation ( 30 ) (at l = 0) yields the commuting p air ˆ H D = 3 x 2 D 2 x − ( x 2 − 8 xy + y 2 ) D x D y + 3 y 2 D 2 y +( c + 3)( xD x + y D y ) + a xy − 1 x − y , − ˆ K D − 4( c 3 + 1) ˆ H D = 4 x 3 D 3 x + 4( x 2 + xy + y 2 )( xD x + y D y ) D x D y + 4 y 3 D 3 y +( c 3 + 3)(4( x − y ) 2 D x D y − ( c + 3)( xD x + y D y )) + a x − y  ( x + y )(( x 2 − 1) D x + ( y 2 − 1) D y ) − ( c 3 + 5) xy + c 3 + 3  . 4 Sp ectra The quant ization in terms of the generators of Lie algebra is u niv ersal, bu t the s etting of a b oundary v alue problem and compu tati on of the sp ectra dep end on the c hoice of a concrete repr esen tation. As an application, we consider here the eigen v alue pr oblem for the Eu ler top on s o (3) u sing r ep- resen tation ( 6 ). Recall, that quantiza tion of this mo del w as obtained by Kramers–Ittmann [ 30 ]. In the case of representa tion ( 6 ), it is n atural to de- fine the sp ectrum by condition that eigenfunctions are p olynomial. This can b e compared w ith Komarov– K uznetso v pap er [ 22 ] where the sp ectrum wa s found for the m atrix rep resen tation, and the recent Grosset–V eselo v pap er [ 31 ] where the sp ectrum was studied for the r ep resen tation in elliptic coordi- nates and it w as s ho wn th at co efficien ts of the c haracteristic p olynomial at a giv en level set j = s are expressed through the so-called elliptic Bernoulli p olynomials. The sp ectral problem for the tops on so (4) is rather compli- cated and w e restrict ourselv es by d eriv ation of equations for eig enf unctions in the Schot tky–Manak o v case. 4.1 Matrix represen tation of so (3) Let us introdu ce the wa ve fu nction | m, j i = x j − m , then r epresen tation ( 6 ) has the follo wing matrix elemen ts in this basis: ˆ U 1 ± i ˆ U 2 = ˆ U ± , ˆ U 3 | m, j i = m | m, j i , ˆ U + | m, j i = ( j − m ) | m + 1 , j i , ˆ U − | m, j i = ( j + m ) | m − 1 , j i . (36) In this case the condition that j − m is inte ger follo w s from the condition that the basis functions x j − m m us t b e single-v alued. If the p roblem admits the time-reversal symmetry m → − m then j + m sh ould b e integer as w ell, and this implies that j and m are simultaneously integ er or half-in teger. The form of the m atrix element s imp lies that m = − j, . . . , j , and the cond ition j > 0 follo w s fr om the condition that the b asis fu nction m u st b e analytic. 18 It is clear from its form that wa ve fun ctio n on the orb it j = const is a p olynomial of d egree 2 j . The w av e f unction and th e sp ectral pr oblem on s o (3) can b e written as follo ws: | ψ i = j X m = − j C ( m, j ) | m, j i , H | ψ i = λ | ψ i Analogously , the wa ve function and the sp ectral prob lem on so (4) r ead: | ψ i = j 1 X m 1 = − j 1 j 2 X m 1 = − j 2 C ( m 1 , m 2 ; j 1 , j 2 ) | m 1 , m 2 , j 1 , j 2 i , H | ψ i = λ | ψ i . In the differen tial representa tion, the basis wa ve function on so (4) is c h osen as | m 1 , m 2 , j 1 , j 2 i = x j 1 − m 1 y j 2 − m 2 . Th en the general wa ve function is a p olynomial of degree 2 j 1 with resp ect to x and degree 2 j 2 with resp ect to y . 4.2 Sp ectrum of the E uler top on so (3) The eigen v alue pr oblem is of the form ˆ H ψ λ j ( x ) = λψ λ j ( x ) , ˆ C ψ λ j ( x ) = j ( j + 1) ψ λ j ( x ) where ˆ H = α 1 ˆ U 2 1 + α 2 ˆ U 2 2 + α 3 ˆ U 2 3 , ˆ C = ˆ U 2 1 + ˆ U 2 2 + ˆ U 2 3 . W e will assume that α 1 + α 2 + α 3 = 0, up to an unessential shift of the sp ectrum. Let us use the differentia l op erators represen tation for so (3) algebra ( 36 ) and represent w av e fun ctio n in the form ψ λ j ( x ) = 2 j X k =0 ˜ C λ j ( k ) x k . Then the eigen v alue p roblem is rewritten in the form of recurr ent relation 1 4 (2 j + 1 − k )(2 j + 2 − k ) C Λ j ( k − 2) +  1 2 ( j ( j + 1) − 3( j − k ) 2 ) ξ − Λ  C Λ j ( k ) + 1 4 ( k + 1)( k + 2) C Λ j ( k + 2) = 0 where ξ = α 1 + α 2 α 1 − α 2 , Λ = λ α 1 − α 2 , C Λ j = ˜ C λ j , with th e b oun dary conditions on the left end C Λ j ( − 2) = C Λ j ( − 1) = 0 , C Λ j (0) = C Λ j (1) = 1 . 19 The problem is splitting for o dd and ev en p olynomials. If j is integ er th en the b oun dary conditions on th e righ t end is C Λ j (2 j + 2) = 0 for the ev en p olynomials and C Λ j (2 j + 1) = 0 for the o dd ones. If j is half-in teger then , vice-v ersa, the b oundary conditions on the righ t end is C Λ j (2 j + 1) = 0 for the eve n p olynomials and C Λ j (2 j + 2) = 0 for the o dd ones. As a result, the eigen v alues of the Hamiltonian ˆ H are zero es of the p olynomial P j (Λ) = C Λ j (2 j + 1) C Λ j (2 j + 2) , deg P j (Λ) = 2 j + 1 . Let u s explicitly write d o wn sev eral p olynomials P j (Λ) n ormalize d by the condition that the co efficien t of the leading term Λ 2 j +1 is unit: P 0 (Λ) = Λ , P 1 / 2 (Λ) = Λ 2 , P 1 (Λ) = 1 4 (Λ − ξ )(2Λ + ξ + 1)(2Λ + ξ − 1) , P 3 / 2 (Λ) = 1 16 (4Λ − 9 ξ 2 − 3) 2 , P 2 (Λ) = 1 4 (Λ + 3 ξ )(2Λ − 3 ξ + 3)(2Λ − 3 ξ − 3)(Λ − 9 ξ 2 − 3) 2 , P 5 / 2 (Λ) = (Λ 3 − 7Λ(3 ξ 2 + 1) + 20 ξ ( ξ 2 − 1)) 2 . All p olynomials P j ( λ ) with half-in teger j are full squares in virtue of the Kramers theorem [ 28 , p. 225] ab out the double degenerati on of the systems with half-in teger v alue of the spin. 4.3 Matrix represen tation for the Sc hottky–Manak ov top In the Sc hottky–Manak o v case w e ha ve t wo consisten t eigen v alue problems ˆ H ψ = λψ , ˆ K ψ = µ ψ . T he wa ve function is of the form ψ λ,µ j 1 ,j 2 ( x, y ) = 2 j 1 X k =0 2 j 2 X l =0 C λ,µ j 1 ,j 2 ( k , l ) x k y l . In th e lattice representati on , the 5-p oin t equation app ears for the eigen v alues µ : ( α 1 − α 2 )(2 j 1 + 1 − k )(2 j 2 + 1 − l ) C λ,µ j 1 ,j 2 ( k − 1 , l − 1) + ( α 1 − α 2 )( k + 1)( l + 1) C λ,µ j 1 ,j 2 ( k + 1 , l + 1) + 2(2( j 1 − k )( j 2 − l ) α 3 − µ ) C λ,µ j 1 ,j 2 ( k , l ) + ( α 1 + α 2 )(2 j 1 + 1 − k )( l + 1) C λ,µ j 1 ,j 2 ( k − 1 , l + 1) + ( α 1 + α 2 )( k + 1)(2 j 2 + 1 − l ) C λ,µ j 1 ,j 2 ( k + 1 , l − 1) = 0 , 20 and 9-p oin t one for the eigenv alues λ : ( α 2 2 − α 2 1 )( k + 1)( k + 2) C λ,µ j 1 ,j 2 ( k + 2 , l ) + ( α 2 2 − α 2 1 )( l + 1)( l + 2) C λ,µ j 1 ,j 2 ( k , l + 2) + 2 α 3 ( α 2 − α 1 )( k + 1)( l + 1) C λ,µ j 1 ,j 2 ( k + 1 , l + 1) + 2 α 3 ( α 2 + α 1 )( k + 1)(2 j 2 + 1 − l ) C λ,µ j 1 ,j 2 ( k + 1 , l − 1) +  ( α 1 − α 2 ) 2  (2 j 1 + 1 − k ) 2 + (2 j 2 + 1 − l ) 2 − 2( j 1 + j 2 + 1 − k )( j 1 + j 2 + 1 − l ) − ( j 1 − j 2 ) 2 − j 1 ( j 1 + 1) − j 2 ( j 2 + 1)  − 4 α 2 3 (( j 1 − k ) 2 + ( j 2 − l ) 2 ) + ( α 2 + α 1 ) 2 (( j 1 + j 2 − k − l ) 2 − j 1 ( j 1 + 1) − j 2 ( j 2 + 1))  C λ,µ j 1 ,j 2 ( k , l ) + ( α 2 2 − α 2 1 )(2 j 1 + 1 − k )(2 j 1 + 2 − k ) C λ,µ j 1 ,j 2 ( k − 2 , l ) + ( α 2 2 − α 2 1 )(2 j 2 + 1 − l )(2 j 2 + 2 − l ) C λ,µ j 1 ,j 2 ( k , l − 2) + 2 α 3 ( α 2 + α 1 )(2 j 1 + 1 − k )( l + 1) C λ,µ j 1 ,j 2 ( k − 1 , l + 1) + 2 α 3 ( α 2 − α 1 )(2 j 1 + 1 − k )(2 j 2 + 1 − l ) C λ,µ j 1 ,j 2 ( k − 1 , l − 1) = 0 . The b ound ary conditions are: C λ,µ j 1 ,j 2 ( k , l ) = 0 if the pair ( k , l ) lies outside the r ecta n gle with the ve r tices (0 , 0), ( j 1 , 0), ( j 1 , j 2 ), (0 , j 2 ). Solutions C λ,µ j 1 ,j 2 ( k , l ) split into solutions on t wo su blattice s: a solution is called “ev en” if it v anish es at o dd k + l , and it is called “o dd” if it v anishes at ev en k + l . Actually , one can a v oid solving 9-point equation: it is sufficien t to deter- mine the wa ve fun ctions from the 5-p oint equation and then the substitution in to the 9-p oint one allo ws to determine the r elat ion b etw een λ i and µ i . A plausible answ er is that the pairs ( λ i , µ i ) lie on a certain algebraic curv e. Consider the case j 1 = j , j 2 = 1 2 as an example. T he wa ve fun ction is a sup erp osition of o dd and ev en ones. The ev en wa v e fun ction is of the f orm ψ = C 0 + C 1 xy + C 2 x 2 + C 3 x 3 y + . . . and coefficients satisfy the b oun dary conditions C − 2 = C − 1 = C 2 j +1 = 0 and recurrent relations 1 2 ( α 1 + ( − 1) k α 2 )(2 j + 1 − k ) C k − 1 + ( α 3 ( j − k )( − 1) k − µ ) C k + 1 2 ( α 1 − ( − 1) k α 2 )( k + 1) C k +1 = 0 . Up to the constan t factors, one find s C 0 = 0 , C 1 = µ − j α 3 , C 2 = µ 2 − α 3 µ − α 2 3 j ( j − 1) − 1 2 j ( α 1 − α 2 ) 2 , . . . 21 The o dd wa ve function is of the form ψ = B 0 y + B 1 x + B 2 x 2 y + B 3 x 3 + . . . , the co efficients satisfy the b oundary conditions B − 2 = B − 1 = B 2 j +1 = 0 and recurrent relations ( α 1 − ( − 1) k α 2 )(2 j + 1 − k ) B k − 1 + 2( α 3 ( k − j )( − 1) k − µ ) B k + ( α 1 + ( − 1) k α 2 )( k + 1) B k +1 = 0 . Up to the constan t factors, one find s B 0 = 0 , B 1 = µ + j α 3 , B 2 = µ 2 + α 3 µ − α 2 3 j ( j − 1) − 1 2 j ( α 1 − α 2 ) 2 , . . . 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