aa r X i v : . [ qu a n t - ph ] M a y The generalized quantum isotonic oscillator
J Sesma ∗ Departamento de F´ısica Te´orica,Facultad de Ciencias,50009 Zaragoza, Spain.November 8, 2018
Abstract
Recently, it has been proved that a nonlinear quantum oscillator, gen-eralization of the isotonic one, is exactly solvable for certain values ofits parameters. Here we show that the Schr¨odinger equation for such anoscillator can be transformed into a confluent Heun equation. We give avery simple and efficient algorithm to solve it numerically, no matter whatthe values of the parameters are. Algebraic quasi-polynomial solutions,for particular values of the parameters, are found.
Two years ago, Cari˜nena, Perelomov, Ra˜nada and Santander [1] considered aquantum oscillator, intermediate between the harmonic and the isotonic ones,whose Schr¨odinger equation reads d dx Ψ − (cid:20) ω x + 2 g x − a ( x + a ) (cid:21) Ψ + 2 E Ψ = 0 . (1)The interest of those authors on that problem lay on the fact that, as theyproved, it is exactly solvable for certain values of the parameters, namely, g = 2,and ω and a such that ωa = 1 /
2. Very recently, Fellows and Smith [2] haveshown that this particular case of generalized isotonic oscillator is a super-symmetric partner of the harmonic oscillator. This fact has allowed them toreproduce the results of Ref. [1] in a very concise and elegant manner, and evento construct an infinite set of oscillators, with potentials approaching that ofan isotonic oscillator as x → ∞ , all of them being partners of the harmonic ∗ [email protected] et al. , another class of exactly solvable problems hasbeen obtained by Kraenkel and Senthilvelan [3]. They have used point canonicaltransformations to convert Eq. (1) into a Schr¨odinger equation with a positiondependent effective mass, which, with adequate mass distributions, may rep-resent different problems encountered in semiconductor physics. On the otherhand, in view of the considerable progress in the synthesis of artificial quan-tized structures, a great variety of shapes for the potential wells and barriersare easily feasible and it seem plausible to see generalized isotonic oscillators aspossible representations of realistic quantum dots.The rising interest on generalizad isotonic oscillators has lead us to try tocontribute to a better understanding of their main features. Dealing with Eq.(1) without any restriction, apart from the trivial ones ω > , g > , a > , about the values of the parameters, we have found that a very natural change ofvariable transforms it into a confluent Heun equation (CHE). Then, the energies E can be easily obtained as the zeros of a function defined by a continuedfraction. Of course, the results of Ref. [1] are reproduced by our procedure.This allows us to relate, by a continuous variation of the parameter g , theenergy levels of the harmonic oscillator ( g = 0) with those of the generalizedisotonic oscillator. Besides this, quasi-polynomial (i. e. product of a rationalfunction times an exponential times a polynomial) wavefunctions appear forspecific energy levels and particular values of the parameters ω , g and a , differentfrom those considered in Ref. [1].In Section 2 we transform the Schr¨odinger equation (1) into a CHE. Tosolve it, we propose an extremely simple algorithm that allows us to obtainthe eigenvalues and the eigenfunctions with a great accuracy. Results of thatalgorithm are shown in Section 3. quasi-polynomial solutions of the Schr¨odingerequation are obtained in Section 4. The relation of the polynomials found byCari˜nena et al. [1] with confluent Heun polynomials is discussed in Section 5.Some final comments are added in Section 6. Equation (1) presents two regular singular points, at x = ± ia , and an irregularone of s -rank 3 at infinity. (Along this paper we adopt the definition of s -rank ofan irregular singular point given in Sec. 1.2 of the book by Slavyanov and Lay[4].) Taking advantage of the symmetry of the potential, the number and/orrank of the singularities can be reduced by a very natural mapping, namely, z = x /a . (2)In this way, the upper half of the x -plane goes into the whole z -plane and thereal axis x ∈ ( −∞ , + ∞ ), relevant from the physical point of view, is mapped2nto the positive real semiaxis in the z -plane, covered from ∞ to 0 along the rayarg z = 2 π and from 0 to ∞ along the ray arg z = 0. The Schr¨odinger equationturns into d Ψ dz + 12 z d Ψ dz + (cid:20) − ω a − g z − z ( z + 1) + Ea z (cid:21) Ψ = 0 , (3)where the two free parameters ωa and g are assumed to be given, whereas Ea represents the eigenvalues to be determined. That equation has two regularsingularities, at − s -rank 2 at infinity. Thissuggests to compare it with some one of the various forms of the CHE, whichpresents the same pattern of singularities. Extensive discussions of the CHEcan be found in Refs. [4] and [5]. Equation (3), when written in the form d Ψ dz + 12 z d Ψ dz + (cid:20) − ω a g/ z − g/ z + 1 − g ( z + 1) + Ea z (cid:21) Ψ = 0 , (4)is an example of CHE in its natural form [5, Eq. (1.1.4)]. The singularity at z = −
1, coming from the singularity at x = ia , has indices ρ = 12 (cid:16) p g (cid:17) , ρ = 12 (cid:16) − p g (cid:17) . (5)The mapping (2) has introduced the singularity (branch point) at z = 0, withindices ν = 0 , ν = 1 / , (6)which correspond, respectively, to even and odd solutions Ψ( x ). The singularityfor z → ∞ corresponds to that for x → ∞ . It is immediate to check that thetwo Thom´e formal solutions of (3) for z → ∞ behave asexp (cid:18) ± ωa z (cid:19) z µ ± (cid:0) O ( z − ) (cid:1) , with µ ± = ∓ E ω − . (7)The problem of finding the energy levels of the generalized isotonic oscillatorreduces, thus, to determine the values Ea such that the Floquet solutions (seriesof increasing powers of z ) of (3), corresponding to the indices ν or ν , vanish atinfinity according to (7) with the minus sign in the argument of the exponential.This is the well known connection problem of the singular points at 0 and at ∞ .The presence of the singular point at z = − z → ∞ . Instead of this, Slavyanovand Lay [4, Sec. 3.6] suggest to carry out a Jaff´e transformation consisting in anadequate linear transformation of the dependent variable followed by a M¨obiustransformation of the independent variable to convert the interval z ∈ [0 , ∞ ) intothe interval [0 ,
1] for the new variable. We have found convenient to substituteΨ( z ) = ( z + 1) µ exp (cid:18) − ωa z (cid:19) w ( z ) , with µ = µ − = E ω − , (8)3n (3) to get d wdz + (cid:18) − ωa + 12 z + 2 µz +1 (cid:19) dwdz + (cid:20) µ ( ωa +1 / z ( z + 1) + µ ( µ − z + 1) − g z − z ( z +1) (cid:21) w = 0 , (9)and then to apply the M¨obius transformation prescribed by Slavyanov and Lay t = zz + 1 , (10)which transforms Eq. (9) into (keeping the same symbol to represent the de-pendent variable in terms of the new independent one) d wdt + (cid:18) − ωa (1 − t ) + 12 t (1 − t ) + 2 µ − − t (cid:19) dwdt + (cid:18) µ ( ωa + 1 / t (1 − t ) + µ ( µ − − t ) − g t − t (1 − t ) (cid:19) w = 0 , (11)to be solved in the interval t ∈ [0 , E suchthat the “even” ( ν = ν = 0) or “odd” ( ν = ν = 1 /
2) series solution w ( t ) = t ν ∞ X n =0 c n t n , c = 0 (12)becomes finite at t = 1. Substitution of (12) in (11) gives for the coefficients c n the recurrence relation A n +1 c n +1 + B n c n + C n − c n − = 0 , (13)where we have abbreviated A m = ( m + ν )( m + ν − / ,B m = ( m + ν )( − m − ν − ωa + 2 µ − /
2) + µ ( ωa + 1 /
2) + g/ ,C m = ( m + ν )( m + 1 + ν − µ ) + µ ( µ − − g . The recurrence relation (13) is an irregular difference equation of the Poincar´e-Perron type. The procedure suggested by Slavyanov and Lay to solve the con-nection problem consists in considering the physically acceptable solution of(13) as a linear combination of its two Birkhoff solutions and adjust the value of Ea so as to reach the cancelation of the coefficient of the exponentially diver-gent one. Instead of this, we have preferred to base ourselves on the algorithmsproposed by Gautschi [6] to find minimal solutions of three-term recurrence re-lations. The crucial consideration is that the solution minimal for n → ∞ turnsout to be dominant when the recurrence is used “from tail to head”. Bearing4his in mind, we write the above recurrence relation in the form c n − c n = − B n C n − − A n +1 C n − c n c n +1 (14)and use it to compute c − /c , starting with the approximate value c n c n +1 ≃ √ ωa n / , for n sufficiently large , (15)obtained from the characteristic equation [4, Sec.1.6.3] of the recurrence relation.The energy levels are obtained from the values of Ea such that one gets c − /c = 0 , (16)which implies c − = 0 and, in view of (13), c − = c − = . . . = 0. We have applied the above described algorithm to the determination of thefour lowest states of a generalized harmonic oscillator with intensity g varyingfrom g = 0 (harmonic oscillator) to g = 20, for two typical values of ωa ,namely ωa = 1 / ωa = 2. The results are shown in Figures 1 and 2. Thebehaviour of the eigenenergies, as g increases from zero, can be easily understoodin view of the probability density of the harmonic oscillator eigenstates and thefact that the additional potential g x − a ( x + a ) is negative for | x | < a and positive for | x | > a [1, Fig. 1]. For low values of g , theeffects on the energy of the positive and negative parts of that potential almostcancel to each other, except for the fundamental state whose density probilityconcentrates near the origin, where the additional potential is negative. Thisexplains the gap between the energies of the fundamental and the first excitedstates encountered by Cari˜nena et al. [1] in the case of ωa = 1 / g = 2.As g increases further, the deeper and deeper potential well at the origin makesto decrease the energies of more and more excited states.For possible numerical comparison with results obtained by other methods,we report ours, for some arbitrarily chosen values of g , in Tables 1 and 2. Theenergies of the four lowest states, given with ten decimal digits, have beenobtained by using a double precision FORTRAN code. The presence, in Table1, of exact values of the energy in the case of g = 2 is not surprising: this is theexactly solvable case discussed by Cari˜nena et al . [1]. More intriguing are theexact values of E a and E a found in Table 1 for g = 12 and those of E a for g = 20 and of E a for g = 42 in Table 2. As we are going to show in thenext Section, these are also cases of quasi-polynomial wave functions. For thosevalues of the parameters, one could speak of quasi-exactly solvable potentials.5 gEa -15-10-505 0 5 10 15 20 gEa -15-10-505 0 5 10 15 20 gEa -15-10-505 0 5 10 15 20 gEa Figure 1: Variation of the four lowest energy levels of the generalized isotonicoscillator with the intensity g . The parameters ω and a of the oscillator areassumed to be such that ωa = 1 / ω and a such that ωa = 1 / g . g E a E a E a E a − − − − − − − − − − − − − gEa -15-10-505 0 5 10 15 20 gEa -15-10-505 0 5 10 15 20 gEa -15-10-505 0 5 10 15 20 gEa Figure 2: Variation of the four lowest energy levels of the generalized isotonicoscillator with the intensity g , for ωa = 2.Table 2: Energies of the four lowest states of the generalized isotonic oscillatorof parameters ω and a such that ωa = 2 and for some chosen values of g . g E a E a E a E a − − − − − − − − − − − − − − − − Quasi-polynomial solutions
The study of the generalized isotonic oscillator done by Cari˜nena et al. [1]revealed the existence of quasi-polynomial solutions for certain values of theparameters. Specifically, they found that, if the parameters are related in theform g = 2 ωa (2 ωa + 1) , (17)equation (1) admits a quasi-polynomial solutionΨ = N ( a + x ) ωa exp (cid:18) − ωx (cid:19) , E = 12 ω − (2 ωa ) , (18) N being a normalization constant. It corresponds to the ground state of thatgeneralized isotonic oscillator. More interestingly, they found that if, besidesEq. (17), one has ωa = 1 / , (19)all eigenstates, of energies E m = ( m − / ω, are represented by quasi-polynomial wave functionsΨ m ( x ) = N m ω ( x + a ) exp (cid:16) − ω x (cid:17) P m ( ω / x ) , m = 0 , , , . . . , (20)the polynomials P m being linear combinations of three consecutive Hermitepolynomials of the same parity.The fact that simple analytic solutions of quantum problems, for particularvalues of the parameters, can serve as a check on numerical calculations wasalready pointed out by Demkov [7] in his study of the motion of a particle inthe field of two Coulomb centers. He found that for certain set of values of thecharges Z and Z of the centers and the distance R between them, the wavefunction becomes quasipolynomial when written in ellipsoidal coordinates.In order to analyze the possible existence of quasi-polynomial solutions ofEq. (3), it is convenient to make explicit the behaviour of such solutions at thesingular points. With this purpose we writeΨ( z ) = ( z + 1) ρ z ν exp (cid:18) − ωa z (cid:19) Φ( z ) , (21)where ρ represents either ρ or ρ , given by (5) and such that ρ ( ρ −
1) = g , and, according to (6), ν = ν = 0 in the case of even wavefunctions and ν = ν = 1 / z ), d Φ dz + (cid:18) − ωa + 2 ν + 1 / z + 2 ρz + 1 (cid:19) d Φ dz + ωa ( µ − ν − ρ ) z + ωa ( µ − ν ) + ρ (2 ν + ρ/ z ( z + 1) Φ = 0 , (22)8ith µ given in Eq. (8), is again an example of CHE. In fact, the change ofvariable z −→ − z permits to compare it with the non-symmetrical canonicalform of the CHE [5, Eq. (1.2.27)]. Polynomial solutions of the CHE havebeen studied by Slavyanov [5, Sec. 3.4]. His analysis, however, is not directlyapplicable to our problem due to the different role played by the parameters inhis equation and in ours.It is possible to write a formal solution of (22) as a power seriesΦ( z ) = ∞ X n =0 a n z n , (23)with coefficients obeying the recurrence relation α n +2 a n +2 + β n +1 a n +1 + γ n a n = 0 , (24)with a − = 0 , a = 0 , arbitrary , where we have abbreviated α m = m ( m + 2 ν − / ,β m = m ( m − ωa + 2 ν + 2 ρ − /
2) + ωa ( µ − ν ) + ρ (2 ν + ρ/ ,γ m = − ωa ( m − ( µ − ν − ρ )) . Obviously, the series in the right hand side of (23) reduces to a polynomialΦ( z ) = Q k = k X n =0 a n,k z n , (25)the coefficients a n,k being solution of (24), if the two conditions γ k = 0 and a k +1 ,k = 0 (26)are satisfied. Assuming that the parameters ω and a are given, the first one ofthose conditions, gives the eigenenergy E = (2 k + 2 ρ + 2 ν + 1 / ω (27)in terms of ρ , whereas the second one determines the values of ρ , and conse-quently of g , for which the quasi-polynomial (21), with Φ( z ) replaced by Q k ( z ),is a solution. Notice that the second condition (26) can be equivalently ex-pressed as the cancelation of the determinant of a tridiagonal ( k + 1) × ( k + 1)matrix, det β α γ β α γ β α . . . . . . . . . γ k − β k − α k γ k − β k = 0 , (28)9ith µ replaced by k + ν + ρ in the expressions of α m , β m and γ m given above.The left hand side of (28) is a polynomial of degree 2( k + 1) in ρ . One of itsroots, for every value of k , is ρ = 0: the pure harmonic oscillator possesses quasi-polynomial solutions of every degree. Obviously, conjugate pairs of complexroots may appear, but these are not interesting, in view of the restriction g > k = 0 , , ωa considered in Tables 1 and 2, respectively. The first of conditions (26), γ = 0, gives µ = ν + ρ and, in view of (8), E = (2 ρ + 2 ν + 1 / ω. (29)The second condition, β = 0, gives ρ ( ωa + 2 ν + ρ/
2) = 0 , (30)that, besides the trivial one ρ = 0, has the solution ρ = − (2 ωa + 4 ν ) , i. e. g = (2 ωa + 4 ν )(2 ωa + 4 ν + 1) . (31) ν = 0 ). In the case of being ρ = − ωa , i. e. g = 2 ωa (2 ωa + 1) , and for the energy E = ( − ωa + 1 / ω, one has the quasi-polynomial solutionΨ( z ) ∝ ( z + 1) − ωa exp (cid:18) − ωa z (cid:19) or, in terms of the original notation,Ψ( x ) ∝ ( x + a ) − ωa exp (cid:16) − ω x (cid:17) . This is the solution mentioned in Eq. (2) of Ref. [1]. It can be recognized in ourTable 1 (ground state for g = 2) and in our Table 2 (ground state for g = 20). ν = 1 / ). For ρ = − ωa − , i. e. g = (2 ωa + 2)(2 ωa + 3) , E = ( − ωa − / ω, one finds the quasi-polynomial solutionΨ( z ) ∝ ( z + 1) − ωa − z / exp (cid:18) − ωa z (cid:19) or, in terms of the variable x ,Ψ( x ) ∝ ( x + a ) − ωa − x exp (cid:16) − ω x (cid:17) . Examples of the case under consideration appear in Tables 1 (first excited statefor g = 12) and 2 (first excited state for g = 42). k = 1 ν = 0 ). Now, from (27), we have E = (2 ρ + 5 / ω . and, from (28),14 ρ (cid:0) ρ + 4( ωa + 1) ρ + (4 ω a + 10 ωa + 1) ρ + 2 ωa (2 ωa + 5) (cid:1) = 0 , For ωa = 1 / ρ = − A / / − − A − / , with A = 3 (cid:16) − √ (cid:17) , corresponding to an intensity of the additional potential g = A / / A / / / A − / + 25 A − / . For ωa = 2 there is also one nontrivial real solution ρ = − B / / − − B − / , with B = 3 (cid:16) − √ (cid:17) , to which it corresponds g = B / / B / + 82 / B − / + 121 B − / . ν = 1 / ). For energy we have the value E = (2 ρ + 7 / ω . ρ the equation14 ρ (cid:0) ρ + 4( ωa + 2) ρ + (4 ω a + 18 ωa + 15) ρ + 2(2 ω a + 9 ωa + 3) (cid:1) = 0 , For ωa = 1 / ρ = − , ρ = − (cid:16) − √ (cid:17) , ρ = − (cid:16) √ (cid:17) , with corresponding intensities g = 2 , g = 29 − √ , g = 29 + 5 √ . For ωa = 2 one obtains also three nontrivial real solutions ρ = − (1 / (cid:18) − / · ℜ (cid:18)(cid:16) − i √ (cid:17) / (cid:19)(cid:19) ,ρ = − (1 / (cid:18)
16 + 55 / ℜ (cid:18)(cid:16) i √ (cid:17) (cid:16) − i √ (cid:17) / (cid:19)(cid:19) ,ρ = − (1 / (cid:18)
16 + 55 / ℜ (cid:18)(cid:16) − i √ (cid:17) (cid:16) − i √ (cid:17) / (cid:19)(cid:19) . The corresponding values of g are obtained by using g = ρ ( ρ − . (32) k = 2 ν = 0 ). The energy is now given by E = (2 ρ + 9 / ω , where ρ is a solution of18 ρ (cid:16) ρ + 6( ωa + 2) ρ + 3(4 ω a + 18 ωa + 13) ρ + 4(2 ω a + 18 ω a + 39 ωa + 8) ρ + 2(12 ω a + 82 ω a + 108 ωa + 3) ρ + 4(4 ω a + 28 ω a + 27 ωa ) (cid:17) = 0 , For ωa = 1 /
2, this equation has, besides the trivial solution, two complex andthe real ones ρ = − ,ρ = − (cid:18) C / − (cid:16) − C + 64 C − / (cid:17) / (cid:19) ,ρ = − (cid:18) C / + (cid:16) − C + 64 C − / (cid:17) / (cid:19) , C = 13 (cid:18)
37 + 5 / (cid:18)(cid:16) − √ (cid:17) / + (cid:16) √ (cid:17) / (cid:19)(cid:19) . The corresponding intensities of the additional potential are given by Eq. (32).For ωa = 2 there are also three non trivial real solutions, ρ ≃ − . , ρ ≃ − . , ρ ≃ − . , which correspond to g ≃ . , g ≃ . , g ≃ . . ν = 1 / ). We have for the energy E = (2 ρ + 11 / ω , and for ρ ρ (cid:16) ρ + 6( ωa + 3) ρ + 3(4 ω a + 26 ωa + 35) ρ + 4(2 ω a + 24 ω a + 81 ωa + 59) ρ + 2(12 ω a + 118 ω a + 276 ωa + 105) ρ + 4(4 ω a + 40 ω a + 75 ωa + 15) (cid:17) = 0 . For ωa = 1 / ν = − ,ν = 5 − (cid:18) D / − (cid:16) − D + 52 D − / (cid:17) / (cid:19) ,ν = − − (cid:18) D / + (cid:16) − D + 52 D − / (cid:17) / (cid:19) , where D = 13 (cid:18)
46 + (cid:16) − √ (cid:17) / + (cid:16) √ (cid:17) / (cid:19) . The corresponding values of g follow from Eq. (32).For ωa = 2 there are also three nontrivial real solutions, ρ ≃ − . , ρ ≃ − . , ρ ≃ − . , corresponding to intensities g ≃ . , g ≃ . , g ≃ . . E a = 5 / ωa = 1 / g = 12. It is not difficult to see that it corresponds to k = 4, ν = 0, and ρ = −
3. The wave function is in this caseΨ( z ) ∝ ( z + 1) − exp( − z/ (cid:0) − z − z − (2 / z − (1 / z (cid:1) , that is, in terms of x ,Ψ( x ) ∝ ( x + a ) − exp( − x / a ) (cid:18) − x a − x a − x a − x a (cid:19) . In the preceding Section we have found, whenever ωa = 1 /
2, a quasi-polynomialsolution with ρ = − k and in both casesof even or odd wave functions. There are only two exceptions, namely the caseof k = 0, ν = 1 / k = 1, ν = 0. Although we have not discussedthe cases of k = 3 , , . . . , one can easily check that the same value ρ = − ν = 0 and for ν = 1 / ωa = 1 / ρ = −
1, that is, g = 2, the generalized isotonic potential isexactly solvable [1]. In what follows, we show that the polynomials entering thesolutions found by Cari˜nena et al. are in fact confluent Heun polynomials.We assume from now on that ωa = 1 / ρ = −
1, and µ = k + ν − k beinga positive integer. According to Eqs. (21), (22) and (25), the quasi-polynomialsolutions are of the formΨ( z ) = ( z + 1) − z ν exp ( − z/ Q ( ν ) k ( z ) , ν = 0 , / , (33)the polynomial (of degree k ) Q ( ν ) k obeying the differential equation d Q ( ν ) k dz + (cid:18) −
12 + 2 ν + 1 / z − z + 1 (cid:19) d Q ( ν ) k dz + kz/ k/ − νz ( z + 1) Q ( ν ) k = 0 . (34)This is but a particular case of Eq. (22) that, as we have already mentioned, isa CHE. Confluent Heun polynomials can be written as linear combinations ofhypergeometric or confluent hypergeometric polynomials [5, Sec. 2.3]. This sec-ond possibility is more convenient for a comparison with the results of Cari˜nena et al. [1]. With this purpose, we introduce a new variable y = z x a , (35)in terms of which the differential equation reads( y +1 / y d Q ( ν ) k dy + (( y +1 / ν +1 / − y ) − y ) d Q ( ν ) k dy + (( y +1 / k − ν ) Q ( ν ) k = 0 , (36)14hat can be written in the form(( y + 1 / D + D ) Q ( ν ) k = 0 (37)with the differential operators D ≡ y d dy + (2 ν + 1 / − y ) ddy + k , (38) D ≡ − y ddy − ν . (39)Now we try in (37) the sum of confluent hypergeometric polynomials Q ( ν ) k = k X n =0 A ( ν ) n,k M ( − ( k − n ) , ν + 1 / , y ) , (40)with coefficients A ( ν ) n,k to be determined. According to Eqs. 13.1.1 and 13.4.10,respectively, of Ref [8], one has D M ( − ( k − n ) , ν + 1 / , y ) = n M ( − ( k − n ) , ν + 1 / , y ) , (41) D M ( − ( k − n ) , ν + 1 / , y ) = − k − n + ν ) M ( − ( k − n ) , ν + 1 / , y )+ 2( k − n ) M ( − ( k − n ) + 1 , ν + 1 / , y ) , (42)and Eq. (37) turns into k X n =0 A ( ν ) n,k (cid:16) ( n ( y + 5 / − k − ν ) M ( − ( k − n ) , ν + 1 / , y )+2( k − n ) M ( − ( k − n ) + 1 , ν + 1 / , y ) (cid:17) = 0 . (43)Cancelation of the coefficients of the successively decreasing powers of y in theleft hand side of the last equation gives, for k ≥ A ( ν )0 ,k arbitrary , A ( ν )1 ,k = − k + νk + 2 ν − / A ( ν )0 ,k , A ( ν )2 ,k = 2 k − k + 4 kν − ν k + 2 ν − / k + 2 ν − / A ( ν )0 ,k , A ( ν )3 ,k = A ( ν )4 ,k = . . . = 0 , in both cases of ν = 0 or ν = 1 /
2. Substitution of these expressions in (40)gives Q (0) k = A (0)0 ,k (cid:18) M ( − k, / , y ) − kk − M ( − k +1 , / , y ) + kk − M ( − k +2 , / , y ) (cid:19) , (44)15 (1 / k = A (1 / ,k (cid:18) M ( − k, / , y ) − M ( − k +1 , / , y ) + k − k − M ( − k +2 , / , y ) (cid:19) . (45)Now we can use the relations between confluent hypergeometric and Hermitepolynomials [8, Eqs. 22.5.56 and 22.5.57] [9, Sec. 10.13, Eqs. (17) and (18)] M ( − m, / , y ) = ( − m m !(2 m )! H m ( √ y ) , (46) √ y M ( − m, / , y ) = ( − m m !2(2 m + 1)! H m +1 ( √ y ) , (47)and choose A (0)0 ,k = ( − k (2 k )! k ! and A (1 / ,k = ( − k / (2 k + 1)! k ! (48)to get Q (0) k = H k ( √ y ) + 8 kH k − ( √ y ) + 8 k (2 k − H k − ( √ y ) , (49) z / Q (1 / k = H k +1 ( √ y ) + 4(2 k + 1) H k − ( √ y ) + 4(2 k + 1)(2 k − H k − ( √ y ) , (50)which are, respectively, the polynomials P k and P k +1 of Ref. [1], the variablebeing √ y = x √ a = √ ω x . (51)In the above discussion of Eq. (43) we have left aside the cases of k = 0 and k = 1. For k = 0 and ν = 0, Eq. (43) is satisfied irrespective of the value of A (0)0 , , whereas for ν = 1 / A (1 / , = 0, which implies thatthe resulting Q (1 / is identically equal to zero. For k = 1, instead, Eq. (43) issatisfied, for ν = 0, only if A (0)0 , = A (0)1 , = 0 and consequently Q (0)1 identicallyequal to zero, whereas for ν = 1 / A (1 / , = − A (1 / , . All thisis in accordance with the two exceptional cases of k = 0, ν = 1 /
2, and k = 1, ν = 0, encountered in Ref. [1] and mentioned at the beginning of this Section. Most of the quantum mechanical problems which admit a simple solution are ofthe hypergeometric class: the corresponding Schr¨odinger equation turns, by anadequate transformation, into a hypergeometric or a confluent hypergeometricone. In this paper we have shown that the generalized quantum isotonic oscil-lator belongs to the Heun class. Other examples of physical problems of thisclass can be found in Chapter 4 of Ref. [4].Up to now, the most appealing feature of the generalized quantum isotonicpotential was its exact solvability for certain values of its parameters [1]. The16act, shown in this paper, that it is quasi-exactly solvable when those parameterstake particular values spread over a wide range makes it to be especially suitedto serve as a workbench to test the accuracy of approximate (perturbative,variational, etc.) methods of solution of the Schr¨odinger equation.Quasi-exact solvability of quantum Hamiltonians is closely related to theirLie-algebraic properties [10]. A discussion of this topic with reference to thegeneralized isotonic oscillator would be necessary, but it lies out of the scope ofthis paper.The sequence {P n } arising in the exactly solvable case [1] does not includea linear ( n = 1) nor a quadratic ( n = 2) polynomials. Therefore, it cannot beused as a basis for an expansion. Similar sequences of polynomial eigenfunctionsof a Sturm-Liouville problem have been found by G´omez-Ullate, Kamran andMilson [11] and by Quesne [12]. In the case of the first authors, the sequencesdoes not include the constant ( n = 0) polynomial. Quesne has found sequenceswithout the constant polynomial and also sequences without the constant andthe linear polynomials. Nevertheless, the first authors have proved that suchsequences, which they denominate exceptional polynomial systems , are a basisin their corresponding L Hilbert spaces. Besides, G´omez-Ullate, Kamran andMilson [13] have given an extension of the Bochner’s theorem applicable to thosesequences of orthogonal polynomials, solution of a Sturm-Liouville problem, thatstart with a polynomial of degree one. It would be interesting to explore thepossibility of an analogous extension of the theorem for sequences of polynomialslike that discussed in Ref. [1], where the absent polynomials are not the lowestorder ones.
Acknowledgments
The author is greatly indebted to Professors J. F. Cari˜nena and M. F. Ra˜nadafor providing him with their paper and for plenty of fruitful comments. Thesuggestions of two anonymous referees have greatly contributed to improve thepresentation of this article. Financial aid of Comisi´on Interministerial de Cienciay Tecnolog´ıa and of Diputaci´on General de Arag´on is acknowledged.
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