Exact polynomial solutions of second order differential equations and their applications
We find all polynomials $Z(z)$ such that the differential equation $${X(z)\frac{d^2}{dz^2}+Y(z)\frac{d}{dz}+Z(z)}S(z)=0,$$ where $X(z), Y(z), Z(z)$ are polynomials of degree at most 4, 3, 2 respectively, has polynomial solutions $S(z)=\prod_{i=1}^n(z…
Authors: Yao-Zhong Zhang
Exact p olynomial solut ions of second order differen tial e quations and the ir applicati ons Y ao-Zho ng Zhang Scho ol of Mathematics and Physics, The Univ e rsity of Que ensland, Brisb ane , Qld 4072, A ustr alia Abstract W e fi nd all p olynomials Z ( z ) such that the differenti al equation X ( z ) d 2 dz 2 + Y ( z ) d dz + Z ( z ) S ( z ) = 0 , where X ( z ) , Y ( z ) , Z ( z ) are p olynomials of degree at most 4, 3, 2 resp ectiv ely , has p olynomial solutions S ( z ) = Q n i =1 ( z − z i ) of d egree n with distinct r o ots z i . W e deriv e a set of n alg ebraic equations whic h determine th ese ro ots. W e also find all p olynomials Z ( z ) whic h giv e p olynomial solutions to the different ial equation w hen the co efficien ts of X ( z ) and Y ( z ) are algebraically dep end en t. As applications to our g eneral resu lts, w e obtai n the exact (c losed-form) sol utions o f th e Schr¨ odinger t yp e differen tia l equations describing: 1) Tw o Coulom bically repelling electrons on a sphere; 2) S c hr¨ odinger equation from kink stabilit y analysis of φ 6 -t yp e field theory; 3) Static p erturbations f or the non-extremal Reissner-Nordstr¨ om solution; 4) P lanar Dirac electron in Coulomb and magnetic fi elds; and 5) O ( N ) in v arian t decatic anh arm onic oscillator. 2000 Mathematics Subje ct Classific ation . 34A05, 30 C15, 81U15, 81Q05, 8 2 B23. P A C S numb ers : 0 2.30.Hq, 03.65 .-w, 03.65.F d, 03.65.G e, 02.30.Ik. Keywor ds : P olynomial solutions, Bethe ansatz, Quasi-exactly solv able syste ms. 1 In t ro ductio n and main results Consider the general 2nd order linear ordinary differen tial equation (ODE) X ( z ) d 2 dz 2 + Y ( z ) d dz + Z ( z ) S ( z ) = 0 , (1.1) where X ( z ) , Y ( z ) , Z ( z ) are p o lynomials of degree at most 4, 3, 2 resp ectiv ely , X ( z ) = 4 X k =0 a k z k , Y ( z ) = 3 X k =0 b k z k , Z ( z ) = 2 X k =0 c k z k . 1 The ODE (1.1) has as many as 12 pa r ameters a k , b k , c k , and contains, as particular cases, the Heun and generalized Heun equations as we ll as v arious confluen t equations. As examples, here we list fiv e ODEs for each of whic h w e pro vide a ph ysical application in section 3 . (i) a 4 = b 3 = c 2 = 0, i.e. deg X ( z ) = 3, deg Y ( z ) ≤ 2 and deg Z ( z ) ≤ 1. If X ( z ) has no m ultiple ro ots, w e can write X ( z ) = Q 3 s =1 ( z − d s ) , Y ( z ) X ( z ) = P 3 s =1 α s z − d s for suitable complex num b ers d s , α s . W e then o btain t he Heun equation, ( d 2 dz 2 + 3 X s =1 α s z − d s d dz + Z ( z ) Q 3 s =1 ( z − d s ) ) S ( z ) = 0 . (1.2) (ii) deg X ( z ) = 4, deg Y ( z ) ≤ 3 and deg Z ( z ) ≤ 2. If X ( z ) has no m ultiple ro ots, w e can write X ( z ) = Q 4 s =1 ( z − e s ) , Y ( z ) X ( z ) = P 4 s =1 µ s z − e s for suitable c-n umbers e s , µ s . Then (1.1) takes the f orm, ( d 2 dz 2 + 4 X s =1 µ s z − e s d dz + Z ( z ) Q 4 s =1 ( z − e s ) ) S ( z ) = 0 . (1.3) This is the generalized He un equation. (iii) deg X ( z ) = 3 (i.e. a 4 = 0), deg Y ( z ) ≤ 3 and deg Z ( z ) ≤ 2. If X ( z ) has no m ultiple ro ots, we can write X ( z ) = Q 3 s =1 ( z − f s ) , Y ( z ) X ( z ) = P 3 s =1 ν s z − f s + ν for suitable c-n um b ers f s , ν s , ν . Then (1.1) has the form, ( d 2 dz 2 + 3 X s =1 ν s z − f s + ν ! d dz + Z ( z ) Q 3 s =1 ( z − f s ) ) S ( z ) = 0 . (1.4) This is the equation considered b y Sc h¨ afk e and Schmid t [1]. (iv) deg X ( z ) = 2 (i.e. a 4 = a 3 = 0), deg Y ( z ) ≤ 3 a nd deg Z ( z ) ≤ 2. If X ( z ) has no m ultiple ro ots, w e can write X ( z ) = ( z − g 1 )( z − g 2 ) , Y ( z ) X ( z ) = σ 1 z − g 1 + σ 2 z − g 2 + σ z + κ for suitable c-n um b ers g 1 , g 2 , σ 1 , σ 2 , σ, κ . Then (1.1) is g iv en by d 2 dz 2 + σ 1 z − g 1 + σ 2 z − g 2 + σ z + κ d dz + Z ( z ) ( z − g 1 )( z − g 2 ) S ( z ) = 0 . (1.5) (v) deg X ( z ) = 1 (i.e. a 4 = a 3 = a 2 = 0), d eg Y ( z ) ≤ 3 a nd deg Z ( z ) ≤ 2. W rite Y ( z ) X ( z ) = η z − h + λz 2 + γ z + δ for suitable c-n um b ers h, η , λ, γ , δ . Then (1.1) reads d 2 dz 2 + η z − h + λz 2 + γ z + δ d dz + Z ( z ) ( z − h ) S ( z ) = 0 . (1.6) Recen tly there is a lot of researc h inte rest in finding p olynomial solutions to second order differen tial equations of the fo rm (1 .1) [2]- [11]. ODEs with p olynomial solutio ns are often called quasi-exactly solv able and ha v e wide-spread applications in phys ics, che mistry 2 and engineering (see e.g. [12] a nd r eferences therein). One of the classic al problems ab out the OD E (1.1), suggested b y E. He ine [13], T. Stieltjes [14] and G. Szego [15], is ∗ Problem Given a p air o f p olynomials X ( z ) , Y ( z ) and a p ositive inte ger n , (a) find al l p olynomials Z ( z ) such that the ODE (1.1) has a p olynomial solution S ( z ) of de gr e e n . (b) find S ( z ) . In this pap er w e solve this problem by means o f the so- called F unctional (o r Analytic) Bethe Ansatz metho d [16 – 19]. Precisely , w e find the explicit v alues of the co efficien ts c 2 , c 1 , c 0 of Z ( z ) whic h give r ise to degree n p olynomial s olutions S ( z ) of (1.1) with r o ots z 1 , z 2 , · · · , z n of multiplic it y one, and we obtain a set of n algebraic equations (the so-called Bethe a nsatz equations) whic h determine these ro ots.. F or cases (i) and (ii) ab o v e with fixed real d s , e s and p ositiv e α s , µ s n um b ers in (1.2) and (1.3), res p ectiv ely , there is a classical r esult, kno wn as Heine -Stieltjes theorem [13]. This theorem sa ys [5] that if the co efficien ts of X ( z ) and Y ( z ) are algebraically indep enden t, i.e. do not satisfy an y algebraic relatio ns with inte ger co efficien ts, then for an a rbitrary p ositiv e integer n there are exactly n + deg X ( z ) − 2 n ! p olynomials Z ( z ) o f degree exactly (deg X ( z ) − 2) suc h that the ODE has a degree n p olynomial solution S ( z ). Ho w ev er, ev en for these tw o cases, no results about the v alues of the coefficien ts c 2 , c 1 , c 0 of Z ( z ) seem t o be previously kno wn. F urthermore, one may ask Question If the c o efficients of X ( z ) and Y ( z ) do satisfy some al g ebr aic r el a tion s with inte ger c o efficients, i.e. ar e algebr aic al ly dep e ndent, then how man y p olynomials Z ( z ) ar e ther e which le ad to de gr e e n p olynomial so lutions of the ODE (1.1)? T o my kno wledge, this question w as not answ ered b y Heine a nd Stieltjes in their theorem. In this pap er we will also provid e an answ er to this question as a b y-pro duct of our g eneral result (theorem 1.1 below ). W e no w state one of our main results o f this paper. Theorem 1.1 Given a p air of p olynomia l s X ( z ) and Y ( z ) , then the values of the c o ef- ficients c 2 , c 1 , c 0 of p olynomial Z ( z ) such that the differ ential e quation (1.1) has de gr e e n p olynomial solution S ( z ) = n Y i =1 ( z − z i ) (1.7) with distinct r o ots z 1 , z 2 , · · · , z n ar e given by c 2 = − n ( n − 1) a 4 − nb 3 , (1.8) ∗ Authors in these references o nly co nsidered the cases c orresp o nding to (1.2) a nd (1.3), while the general ODE (1.1) here a lso con tains many other cases, e.g . differential equa tions (1.4)-(1.6). 3 c 1 = − [2( n − 1) a 4 + b 3 ] n X i =1 z i − n ( n − 1) a 3 − nb 2 , (1.9) c 0 = − [2( n − 1) a 4 + b 3 ] n X i =1 z 2 i − 2 a 4 n X i 32 π 2 g 2 m , the function r 2 − M 4 π r + g 2 m 2 has tw o ro ots r ± = M 8 π ± 1 8 π p M 2 − 32 π 2 g 2 m , (3.35) 12 whic h corresp ond t o the tw o horizons of the blac k hole. Noting tha t r ± ob ey the relations, r + r − = g 2 m 2 , r + + r − = M 4 π , (3.36) then the O D E for φ can be brough t in to the form, φ ′′ + p ( r ) φ ′ + q ( r ) φ = 0 . (3.37) Here p ( r ) = 1 r − r + + 1 r − r − , q ( r ) = − m 2 s + 2 a 2 r 2 + 2 a 2 ( 1 r + + 1 r − ) 1 r + q + r − r + + q − r − r − (3.38) with q + = 1 r + − r − m 2 s r 2 + − 4 a 2 r 2 − g 2 m + m 2 s g 2 m 2 (1 − r + + r − ) , q − = 1 r + − r − m 2 s r 2 − + 4 a 2 r 2 + g 2 m + m 2 s g 2 m 2 (1 − r + + r − ) . (3.39) The Lagrangian has a rigid rescaling symmetry . This scaling freedom can b e us ed to set the horizon lo cation r + = 1. Then black hole mass M can b e determined b y requiring that the horizon is at r + = 1. It follo ws fro m (3.36) that M = 2 π ( g 2 m + 2 ) and r − = g 2 m 2 . The corr esp o nding q ( r ) and p ( r ) reduce to † p ( r ) = 1 r − 1 + 1 r − r − , q ( r ) = − m 2 s + 2 a 2 r 2 + 2 a 2 (1 + 1 r − ) 1 r + a 2 g 2 m − m 2 s (1 + r 2 − ) 1 − r − 1 r − 1 + m 2 s g 2 m r + + 2 a 2 r − 1 − r − 1 r − r − . ( 3 .40) W e now find exact solutions to (3 .37) with q ( r ) , p ( r ) give n b y (3.40). By means of transformation φ ( r ) = r µ e − m s r f ( r ) , µ = 1 2 1 ± √ 1 − 8 a 2 , (3.41) then it can be sho wn that f ( r ) satisfie s the f o llo wing ODE f ′′ + 2 µ r + 1 r − 1 + 1 r − r − − 2 m s f ′ + c 2 r 2 + c 1 r + c 0 r ( r − 1)( r − r − ) f = 0 , (3.42) † Authors in [24] als o co ns idered the differential equation (3.37) but arrived at the different v alues of M , r − as well as the different p ( r ) , q ( r ) functions. 13 where c 2 = 2 a 2 (1 + 1 r − ) − 2 m s ( µ + 1) + 1 1 − r − g 2 m ( a 2 + m 2 s r − ) − m 2 s (1 + r 2 − ) + 2 a 2 r − , c 1 = [ m s (2 µ + 1) + µ ] ( r − + 1 ) − 2 a 2 ( r − + 1 ) 2 r − − 1 1 − r − g 2 m ( a 2 + m 2 s ) r − − m 2 s (1 + r 2 − ) r − + 2 a 2 r − , c 0 = 2 a 2 + 2 a 2 − µ ( m s + 1 ) r − . (3.43) This OD E is of the form (1 .4). Applying our general results in prev ious s ections, w e can sho w that (3.42) has p olynomial s olutions of degree n = 0 , 1 , 2 , · · · , f ( r ) = n Y i =1 ( r − r i ) , f ≡ 1 for n = 0 , (3.44) where r i are t he root s of the ab ov e p olynomial to b e determined, pro vided tha t a, m s , g m ob ey the follo wing relations: 2 a 2 (1 + 1 r − ) − 2 m s ( µ + 1) + 1 1 − r − g 2 m ( a 2 + m 2 s r − ) − m 2 s (1 + r 2 − ) + 2 a 2 r − = 2 m s n, (3.45) [ m s (2 µ + 1) + µ ] ( r − + 1 ) − 2 a 2 ( r − + 1 ) 2 r − − 1 1 − r − g 2 m ( a 2 + m 2 s ) r − − m 2 s (1 + r 2 − ) r − + 2 a 2 r − = 2 m s n X i =1 r i − n [ n + 2 µ + 1 + 2 m s ( r − + 1 ) ] , (3.46 ) 2 a 2 + 2 a 2 − µ ( m s + 1 ) r − = 2 m s n X i =1 r 2 i − 2 [ n + µ + m s ( r − + 1 ) ] n X i =1 r i + n [( n − 2 µ )( r − + 1 ) − 2 m s r − ] , (3.47) where the ro ots r i are determined b y the set of Bethe ansatz eq uations, n X j 6 = i 2 r i − r j + 2 µ r i + 1 r i − 1 + 1 r i − r − − 2 m s = 0 , i = 1 , 2 , · · · , n. (3.48) Note tha t f ( r ) = 1 is a solution of the OD E (3.42) pro vided tha t a, m s , g m satisfy the follo wing relations 2 a 2 (1 + 1 r − ) − 2 m s ( µ + 1) + 1 1 − r − g 2 m ( a 2 + m 2 s r − ) − m 2 s (1 + r 2 − ) + 2 a 2 r − = 0 , 14 [ m s (2 µ + 1) + µ ] ( r − + 1 ) − 2 a 2 ( r − + 1) 2 r − − 1 1 − r − g 2 m ( a 2 + m 2 s ) r − − m 2 s (1 + r 2 − ) r − + 2 a 2 r − = 0 , 2 a 2 + 2 a 2 − µ ( m s + 1 ) r − = 0 . (3.49) These r elations can be obtained from (3.45)-(3.47) by letting n = 0. It follows that φ ( r ) = r 1 2 (1 ± √ 1 − 8 a 2 ) e − m s r , (3.50) where a and m s are determined by equations (3.49). This giv es the first exact solution o f the differen tial equation (3.37) suc h that the horizon is at r + = 1. 3.4 Planar Dirac electron in Coulom b and magnetic fields Consider the (2+1 ) -dimensional relativistic system of a D irac electron (with mass m e ) in the presence of an external electromagnetic field A µ . This syste m w as also examined in [25] via a similar Bethe ansatz approac h. The co v ar ian t Dirac equation (in the unit ~ = c = 1 ) has the f o rm iγ µ ( ∂ µ + ieA µ )Ψ( t, r ) = m e Ψ( t, r ) , (3.51) where m e is the rest mass of the electron, − e ( e > 0) is its electric c harge and the (2 + 1) Dirac ga mma matrices γ µ satisfy the an ti-commutation relations { γ µ , γ ν } = 2 η µν with η µν = η µν = diag(1 , − 1 , − 1). In an external Coulom b and a constan t homogeneous magnetic field B , the ve ctor p oten tial can b e written as A 0 = − Z e r , A 1 = − B y 2 , A 2 = B x 2 . (3.52) Then the Hamiltonian H ( r ) of the syste m can b e e xpressed as i∂ 0 Ψ( t, r ) = H ( r )Ψ( t, r ) , H ( r ) = γ 0 γ k P k + eA 0 + γ 0 m e , (3.53) where P k = − i∂ k + eA k , k = 1 , 2 , is t he op erator of generalized momen tum of the electron. The wa ve function Ψ( t, r ) is a ssumed to hav e the form Ψ( t, r ) = 1 √ r exp( − iE t ) ψ l ( r , θ ) , (3.54) where E is the energy eigen v alue of Hamiltonian, and ψ l ( r , θ ) = F ( r ) e ilθ G ( r ) e i ( l +1) θ ! , (3.55) 15 where l is an in teger. Substituting (3.54) and (3.5 5) in to (3.53) and w orking in the polar co ordinates ( t, r, θ ) reduce the pr o blem to a system of coupled differen tial equations fo r F ( r ) and G ( r ), dF dr − l + 1 2 r + eB r 2 F + E + m e + Z α r G = 0 , (3.56) dG dr + l + 1 2 r + eB r 2 G + E − m e + Z α r F = 0 , (3.57) where α = e 2 = 1 / 1 37 is t he fine structure constan t. Solving G ( r ) from the first equation in terms of F ( r ) and substituting into the second equation, we obtain the second-order ODE for F ( r ), F ′′ + 1 r − 1 r + r 0 F ′ + E 2 − m 2 e − eB ( l + 1) + 2 E Z α + ( l + 1 2 ) /r 0 r − eB 2 r 0 + ( l + 1 2 ) /r 0 r + r 0 − ( l + 1 2 ) 2 − ( Z α ) 2 r 2 − ( eB ) 2 4 r 2 ) F = 0 , (3.58) where r 0 = Z α E + m e . Applying the transforma t io n, F ( r ) = r ξ e − eB r 2 4 f ( r ) , ξ = r ( l + 1 2 ) 2 − ( Z α ) 2 , (3.59) w e obtain f ′′ + 2 ξ + 1 r − 1 r + r 0 − eB r f ′ + c 2 r 2 + c 1 r + c 0 r ( r + r 0 ) f = 0 , (3.60) where c 2 = E 2 − m 2 e − eB ( ξ + l + 3 2 ) , c 1 = 2 E Z α + E 2 − m 2 e − eB ( ξ + l + 5 2 ) r 0 , c 0 = 2 E Z α r 0 + l + 1 2 − ξ . (3.61) This ODE is of the form (1.5) and has, fro m our general results in previous sections, p olynomial solutions of degree n = 0 , 1 , 2 , · · · , f ( r ) = n Y i =1 ( r − r i ) , f ≡ 1 for n = 0 , (3.62) where r i are the ro ots of the ab o v e p olynomial to b e determined, pro vided that E , Z, B are g iven b y E 2 = m 2 e + eB n + l + ξ + 3 2 , 2 E Z α = eB r 0 + n X i =1 r i ! , 2 E Z αr 0 = − n ( n + 2 ξ − 1) + ξ − ( l + 1 2 ) + eB n X i =1 r 2 i + r 0 n X i =1 r i ! , (3 .6 3) 16 and the Bethe ansatz equations n X j 6 = i 2 r i − r j + 2 ξ + 1 r i − 1 r i + r 0 − eB r i = 0 , i = 1 , 2 , · · · , n. (3.64) Let us remark that our (3.63)-(3.64) differ from the corresp onding equations (22 )-(25) of ref. [25]. It w ould b e in teresting to establish the relation b etw een the tw o sets of expressions . Note that f ( r ) = 1 is a solution of the ODE (3.60) pro vided that E , Z , B ob ey the follo wing relations: E 2 = m 2 e + eB l + ξ + 3 2 , (3.65) 2 E Z α = eB r 0 , 2 E Z αr 0 = ξ − ( l + 1 2 ) . (3.66) These r elations can b e obtained from (3.63) by setting n = 0. Solving these relations w e obtain eB = − m 2 e ( l + 1 2 + ξ ) ( l + 1 + ξ ) 2 , E = − m e 2 + 1 2 p m 2 e + 2 eB = − m e 2( l + 1 + ξ ) , ( 3 .67) where ξ is related to the pa rameter Z via the ex pression giv en in (3.59). It follo ws that F ( r ) = r ξ e − eB r 2 4 , G ( r ) = ξ − l − 1 2 + eB r 2 ( E + m e ) r + Z α r ξ e − eB r 2 4 (3.68) with eB and E giv en by (3.67) in terms of ξ (i.e. Z ). As far as w e kno w, (3.68) giv es the first exact solution to the planar Dirac electron sy stem. How ev er, this solution do es not seem to b e squarely in tegra ble. 3.5 Sc hr¨ odinger equation of O ( N ) in v arian t decatic anharmonic oscillator Consider the O ( N ) in v arian t decatic anharmonic o scillator in N dimensions [26]. The Sc hr¨ odinger e quation is − 1 2 ∇ 2 + V ( x ) ψ ( x ) = E ψ ( x ) (3.69) with the p oten tial V ( x ) defined b y V ( x ) = λ 1 x 2 + λ 2 x 2 2 + λ 3 x 2 3 + λ 4 x 2 4 + x 2 5 , x 2 = N X i =1 x 2 i . (3.70 ) 17 Here without loss of generality w e hav e normalized the p oten tial suc h that the co efficien t of ( x 2 ) 5 equals to 1. In N -dimensional spherical co o rdinates, the radial w a ve function R ( r ) satisfies 1 2 − d 2 dr 2 − N − 1 r d dr + l ( l + N − 2) r 2 + 2 ( V ( r ) − E ) R ( r ) = 0 . (3.71) Applying the tr a nsformation R ( r ) = r 1 − N 2 ψ ( r ) yields − ψ ′′ + µ ( µ − 1) r 2 + 2 ( V ( r ) − E ) ψ = 0 , (3.72) where µ = l + 1 2 ( N − 1). Applying the tr ansformation ψ = r µ e − α r 2 2 − β r 4 4 − γ r 6 6 φ, z = r 2 , (3.73) where α , β , γ > 0 are parameters y et to be determine d, yields the ODE fo r φ ( z ) φ ′′ + l + N / 2 z − γ z 2 − β z − α φ ′ + 1 4 z (2 αβ − γ ( N + 2 l + 4 ) − 2 λ 2 ) z 2 + α 2 − β ( N + 2 l + 2 ) − 2 λ 1 z + 2 E − α ( N + 2 l ) φ = 0 , (3.74) where α , β , γ are giv en b y γ = √ 2 , β = λ 4 √ 2 , α = 1 √ 2 λ 3 − λ 2 4 4 (3.75) The ODE (3.74) is of the fo r m (1 .6). Applying our general results in previous section, we obtain that this equation has polynomial solutions of de gree n = 0 , 1 , 2 , · · · , φ ( z ) = n Y i =1 ( z − z i ) , φ ≡ 1 for n = 0 , (3.76) where z i are the ro ots of the a b ov e p olynomial t o b e determined, pro vided that 2 αβ − γ ( N + 2 l + 4) − 2 λ 2 = 4 γ n, (3.77) α 2 − β ( N + 2 l + 2) − 2 λ 1 = 4 γ n X i =1 z i + 4 β n, (3.78) E = α 2 (4 n + N + 2 l ) + 2 β n X i =1 z i + 2 γ n X i =1 z 2 i , (3.79) and the Bethe ansatz equations, n X j 6 = i 2 z i − z j = γ z 2 i + β z i + α − l + N/ 2 z i , i = 1 , 2 , · · · , n. (3.80) 18 Note that φ = 1 is a solution to (3.74) with energy E = 1 2 √ 2 λ 3 − λ 2 4 4 ( N + 2 l ) , (3.81) where λ 3 and λ 4 ob ey the constrain ts, λ 4 λ 3 − λ 2 4 4 − √ 2( N + 2 l + 4) − 2 λ 2 = 0 , (3.82) 1 2 λ 3 − λ 2 4 4 2 − λ 4 √ 2 ( N + 2 l + 2) − 2 λ 1 = 0 , (3.83) whic h can b e obtained fr o m (3.77)-(3.79) by setting n = 0. The real solutions of these constrain t equations (so that the energy E is real) are give n b y λ 4 = − 2 √ 2 λ 1 3( N + 2 l + 2) + − q 2 + r ( q 2 ) 2 + ( p 3 ) 3 1 / 3 + − q 2 − r ( q 2 ) 2 + ( p 3 ) 3 1 / 3 , λ 3 = λ 2 4 4 + √ 2 N + 2 l + 4 + √ 2 λ 2 λ 4 , (3.84) where p = − 24 λ 2 1 9( N + 2 l + 2) 2 , q = 32 √ 2 λ 3 1 27( N + 2 l + 2) 3 − ( N + 2 l + 4 + √ 2 λ 2 ) 2 N + 2 l + 2 . (3.8 5) W e th us obtain the first exact g round state of the r a dial Sc hr¨ odinger equation (3.71), R ( r ) = r l exp − 1 2 √ 2 λ 3 − λ 2 4 4 r 2 − λ 4 4 √ 2 r 4 − 1 3 √ 2 r 6 (3.86) with λ 3 and λ 4 giv en by (3 .84). F or n = 1, w e find that the real ro ot to the Bethe ansatz equation (3.80) is z 1 = λ 4 6 + − v 2 + r ( v 2 ) 2 + ( u 3 ) 3 1 / 3 + − v 2 − r ( v 2 ) 2 + ( u 3 ) 3 1 / 3 , (3.87) where u = λ 2 4 24 − λ 3 2 , v = 5 λ 3 4 432 − 1 2 λ 3 + 1 2 √ 2 ( N + 2 l ) . (3.88) 19 The par ameters λ 3 and λ 4 are determined from the equations (see (3 .7 7) and ( 3 .78)), λ 4 λ 3 − λ 2 4 4 − √ 2( N + 2 l + 8) − 2 λ 2 = 0 , (3.89) 1 2 λ 3 − λ 2 4 4 2 − λ 4 √ 2 ( N + 2 l + 6) − 2 λ 1 = 4 √ 2 z 1 . (3.90) The energy eigen v alue E is giv en b y E = 1 2 √ 2 λ 3 − λ 2 4 4 ( N + 2 l + 4) + √ 2 λ 4 z 1 + 2 √ 2 z 2 1 , (3.91) and the radial w a v e function is R ( r ) = r l r 2 − z 1 exp − 1 2 √ 2 λ 3 − λ 2 4 4 r 2 − λ 4 4 √ 2 r 4 − 1 3 √ 2 r 6 (3.92) with z 1 giv en by (3 .87). This g iv es the firs t excited state of t he system. 4 Conclud ing re marks The main results of this pap er are theorem 1.1, corollary 1.3 and their applications to the exact solutio ns of the physic al systems (examined in section 3 ). As spelled out in theorem 1.1, we ha ve found all p olynomials Z ( z ) suc h that the ODE (1.1) has p olynomial solutions S ( z ) of degree n with distinct ro ots z 1 , z 2 , · · · , z n . W e ha v e also found a set of n a lg ebraic equations whic h determine the ro ots z i and thus the corresp onding p olynomial solutions S ( z ). If t he co efficien ts of p olynomials X ( z ) and Y ( z ) in (1.1) are algebraically dep enden t, i.e satisfy the relations in corolla ry 1.3, then ODE (1.1) allow s an sl (2) algebraization. W e ha ve also found all the p olynomials Z ( z ) and degree n p olynomial s olutions S ( z ) under these conditions. Ha ving as man y a s 12 parameters, the ODE (1.1) is v ery g eneral and con tains, as sp ecial cases, most kno wn second order differen tial equations whic h o ccur in ph ysical, c hemical and engineering applications in the literature (e.g. the relative ly simple Heun equation and its v arious confluent equations). Th us our gene ral results (theorem 1.1 and its cor o llaries) pro vide an unified deriv ation of exact, closed for m solutions fo r all suc h systems . As a pplications to theorem 1.1, we hav e examined five phys ical systems in section 3, which are describ ed by the ODEs corresp onding to (1.2)-(1 .6), resp ectiv ely . W e ha v e sho wn that these systems are quasi-exactly solv able, i.e. the corresp onding differen t ia l equations ha v e p olynomial solutions, if their para meters satisfy certain constraints (sp ecial cases of (1.8)-(1.10)). The quasi-exact solv abilit y has enabled us to use theorem 1.1 t o obtain the closed form expressions for the eigen v alues and eigenfunctions of these systems. Our r esults (theorem 1.1 and corollary 1.3) can be extende d to sec ond order ODE of the form (1 .1) with deg X ( z ) ≤ l , deg Y ( z ) ≤ l − 1 and deg Z ( z ) ≤ l − 2 for l ≥ 5 as 20 w ell as to hig her order ODEs. Researc h on this as w ell as o n applications of theorem 1.1 and corolla r y 1.3 in v arious a r eas of scien ce is in progr ess, and results will b e rep orted elsewhere . Ac kno wledgmen ts: I w o uld lik e to thank Ryu Sa saki for a v ery careful reading of the man uscript and ma ny critical commen ts, and Clare Dunning for a careful reading of the man uscript and helpful suggestions. I also thank T on y Brack en, G ¨ un ter v on Gehlen, Mar k Gould, and Jon Links for commen ts and suggestions. This w ork w as supp or ted by the Australian Rese arc h C ouncil. 5 App endix F or completeness and conv enience of a pplications, in this app endix w e write do wn the explicit fo rm ulas obtained from applying the theorem 1.1 to the sp ecial cases (1 .2)-(1.6). Corollary 5.1 The c o e fficients c 1 , c 0 of Z ( z ) such that the Heun e quation (1.2) has p oly- nomial solution (1.7) ar e given by c 1 = − n " n − 1 + 3 X s =1 α s # , (5.1) c 0 = − " 2( n − 1) + 3 X s =1 α s # n X i =1 z i + n ( n − 1) 3 X s =1 d s + n [ α 1 ( d 2 + d 3 ) + α 2 ( d 1 + d 3 ) + α 3 ( d 1 + d 2 )] , (5.2) wher e the r o ots z 1 , z 2 , · · · , z n ar e determine d by the Bethe ansatz e quations, n X j 6 = i 2 z i − z j + 3 X s =1 α s z i − d s = 0 , i = 1 , 2 , · · · , n. (5.3) W rite c 1 = α β . Then (5.1) is nothing but the so-called F uc hsian relation, α + β + 1 = P 3 s =1 α s , where α = − n and β = P 3 s =1 α s + n − 1. Corollary 5.2 The c o efficients c 2 , c 1 , c 0 of Z ( z ) such t hat the gene r alize d Heun e quation (1.3) has p olynomial solution (1.7) ar e c 2 = − n 4 X s =1 µ s + n − 1 ! , (5.4) c 1 = − 4 X s =1 µ s + 2 ( n − 1) ! n X s =1 z i + n " ( n − 1) 4 X s =1 e s + P # , (5.5) c 0 = − 4 X s =1 µ s + 2 ( n − 1) ! n X s =1 z 2 i + 2 n X i
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