Study of the generalized quantum isotonic nonlinear oscillator potential

We study the generalized quantum isotonic oscillator Hamiltonian given by H=-d^2/dr^2+l(l+1)/r^2+w^2r^2+2g(r^2-a^2)/(r^2+a^2)^2, g>0. Two approaches are explored. A method for finding the quasi-polynomial solutions is presented, and explicit expressions for these polynomials are given, along with the conditions on the potential parameters. By using the asymptotic iteration method we show how the eigenvalues of this Hamiltonian for arbitrary values of the parameters g, w and a may be found to high accuracy.

In a more recent work, Fellows and Smith [6] showed that the potential V (x) = x 2 + 8(2x 2 − 1)/(2x 2 + 1) 2 as well as, for certain values of the parameters w, g and a, the potential V (x) = w 2 x 2 +2g(x 2 − a 2 )/(x 2 + a 2 ) 2 of the Schrödinger equation are indeed supersymmetric partners of the harmonic oscillator potential. Using the supersymmetric approach, the authors were able to construct an infinite set of exact soluble potentials, along with their eigenfunctions and eigenvalues. Very recently, Sesma [9], using a Möbius transformation, was able to transform Eq.(4) into a confluent Heun equation [8] and thereby obtain an efficient algorithm to solve the Schrödinger equation (4) numerically.
The purpose of the present work is to provide a detailed solution, by means of the quasi-polynomial solutions and the application of the asymptotic iteration method [2][3][4][5], for the Schrödinger equation − d 2 dr 2 + l(l + 1) r 2 + w 2 r 2 + 2g r 2 − a 2 (r 2 + a 2 ) 2 ψ(r) = 2Eψ(r), (5) where l is the angular momentum number l = −1, 0, 1, . . . . Our results show that the quasi-exact solutions of Sesma [9] as well the results of Cariñena et al. [1] follow as special cases of our general approach. The present article is organized as follows. In the next section, some preliminary analysis of the Schrödinger equation (5) is presented. A general approach for finding polynomial solutions of Eq.(5), for certain values of parameters w and g, is presented, and is based on a recent work of Ciftci et al. [3] for solving the second-order linear differential equation More general quasi-exact solutions, including the results of Sesma [9], are discussed in section III. Unrestricted solutions of Eq.(5) based on the asymptotic iteration method are discussed in Section IV.

II. GENERALIZED QUANTUM ISOTONIC OSCILLATOR -PRELIMINARY RESULTS
A simple scaling argument, using r = a 2 x, allows us to write the equation (5) as A further substitution z = x 2 + 1 yields a differential equation with two regular singular points at z = 0, 1 and one irregular singular point of rank 2 at z = ∞. The roots µ's of the indicial equation for the regular singular point z = 0 reads µ ± = 1 2 (1 ± √ 1 + 4g), while the roots of the indicial equation at z = 1 are µ + = (l + 1)/2 and µ − = −l/2. Since the singularity for z → ∞ corresponds to that for x → ∞, it is necessary that the solution for z → ∞ behave as ψ(x) ∼ exp(−wa 2 x 2 /2). Consequently, we may assume the general solution of equation (7) which vanishes at the origin and at infinity takes the form A straightforward calculation shows that f n (x) are the solutions of the second-order homogeneous linear differential equation In the next sections, we attempt to give a general solution of this equation. For now, we assume that µ takes the value of the indicial root which allows us to write Eq.(9) as We now consider the cases where the following two equations are satisfied    2µ(2l + 3 + 2wa 2 ) + 2µ(µ − 1) = 0, The solutions of this system, for g and µ, are given explicitly by In the next, we consider each case of these two sets of solutions.
If l = −1, the determinant ∆ n+1 is identically zero for all n, which is equivalent to the exact solutions of the one-dimensional harmonic oscillator problem.
where the confluent hypergeometric function 1 F 1 (−n; a; z) defined, in terms of the Pochhammer symbol (or Gamma function Γ(a)) The polynomial solutions f n (x) = 1 F 1 (−n; l + 3 2 ; wa 2 x 2 ) are easily obtained by using the asymptotic iteration method (AIM), which is summarized by means of the following theorem.

B. Case 2
The second set of solutions allow us to write the differential equation (9) as A further change of variable z = x 2 + 1 allows us to write the differential equation (27) as Again, Eq.(28) is a special case of the differential equation (14) with a 3,0 = a 3,3 = τ 1,1 = 0, a 3,1 = 4, a 3,2 = −4, a 2,0 = −4wa 2 , a 2,1 = −2(6l + 5 + 6wa 2 ), a 2,2 = 16(l + 1 + wa 2 ) and τ 1,0 = −2Ea 2 − wa 2 (6l + 5 + 8wa 2 ). Consequently, the polynomial solutions f n (x) of (28) are subject to the following two conditions: the necessary condition (15) reads and the sufficient condition; namely, the vanishing of the tridiagonal determinant Eq(16), reads and n ′ = n is fixed for the given dimension of the determinant ∆ n+1 . From the sufficient condition (30) we obtain the following conditions on the parameters For a physically meaningful solution we must have a 2 w > 0. This is possible for a very restricted value of the angular momentum number l. Since β 0 = 0, we may observe that where Q l n−1 (a 2 w) are polynomials in the parameter product a 2 w. For physically acceptable solutions, we must have l = −1 and the factor (l + 1 + a 2 w) yields a 2 w = 0, which is not physically acceptable; so we ignore it. The second factor (1 + 2l + 2a 2 w) implies a special value of a 2 w = 1/2, for all n, which we will study shortly in full detail. Meanwhile, the polynomials Q l n (a 2 w) give new values, not reported before, of a 2 w that yield quasi-exact solutions of the Schrödinger equation (with one eigenstate) where and f n (x) are the solutions of For example, ∆ 4 = 0 implies, using (31), that a 2 w = 15 14 , and thus we have for the exact solution with a plot of the wave function and potential given in Figure 1.  Similar results can be obtained for ∆ n+1 = 0, for n ≥ 5.
C. Exactly solvable quantum isotonic nonlinear oscillator As mentioned above, for l = −1 and a 2 w = 1/2, it clear that ∆ n+1 = 0 for all n and the one-dimensional Schrödinger equation has the exact solutions where f n (x) are the polynomial solutions of the following second-order linear differential equation (z = x 2 + 1) By using AIM (Theorem 2, Eq.(21)), we find that the polynomial solutions f n (x) of Eq.(38) are given explicitly as a set of polynomial solutions that can be generated using f 0 (x) = 1, f n (x) = −3x(2n + 1) 1 F 1 (−n; 3 2 ; 1 2 (x − 1)) + 6((n + 1)x − 1) 1 F 1 (−n + 1; up to a constant factor, where, again, 1 F 1 refers to the confluent hypergeometric function defined by (20). Note that the polynomials f n (x) in equation (40) can be expressed in terms of the associated Laguerre polynomials [10] as
A. Particular Case: n = 0 For k (f ixed) ≡ n = 0, the differential equation (45) has the exact solution f 0 (t) = 1 if g and µ satisfies, simultaneously, the following system of equations Solving this system of equations for g and µ, we obtain the following values of g = 2(1 + l + a 2 w)(3 + 2l + 2a 2 w), and µ = −2(l + 1 + wa 2 ), and the ground-state energy, in this case, is given by Eq.(44), namely, which in complete agreement with the results of Section II.B.

IV. NUMERICAL COMPUTATION BY USE OF THE ASYMPTOTIC ITERATION METHOD
For the potential parameters w, a 2 and g, not necessarily obeying the conditions for quasi-polynomial solutions discussed in the previous sections, the asymptotic iteration method can be employed to compute the eigenvalues of Schrödinger equation (7) for arbitrary values w, a 2 and g. The functions λ 0 and s 0 , using Eqs.(43) and (44), are given by  n l wa 2 Conditions E wa 2 n,l ≡ E wa 2 n,l (µ, g) where t ∈ (0, 1). The AIM sequence λ n (x) and s n (x) can be calculated iteratively using the iterative sequences (23). The energy eigenvalues of the quantum nonlinear isotonic potential (7) are obtained from the roots of the termination condition (22). According to the asymptotic iteration method, in particular the study of Brodie et al [2], unless the differential equation is exactly solvable, the termination condition (22) produces for each iteration an expression that depends on both t and E (for given values of the parameters wa 2 , g and l). In such a case, one faces the problem of finding the best possible starting value t = t 0 that stabilizes the AIM process [2]. Fortunately, since t ∈ (0, 1), the starting value t 0 doesn't represent a serious issue in our eigenvalue calculation using (57) and the termination condition (22) in contrast to the case of computing the eigenvalues using λ 0 (x) and s 0 (x) as given by, for example, equation (9), where x ∈ (0, ∞). In Table IV, we report our numerical results for energies of the four lowest states of the generalized isotonic oscillator of parameters w and a such that wa 2 = 2 and for different values of g. In this table, we set l = −1 for computing the energies E 0 a 2 and E 2 a 2 , while we put l = 0 for computing the energies E 1 a 2 and E 3 a 2 , respectively. For most of these values, the starting value of t is t 0 = 0.5 and is shifted towards zero as g gets larger in value. For the values of g that admit a quasi-polynomial solution, the number of iteration doesn't exceed three. For most of the other values of g, the total number of iteration didn't exceed 65. We found that for wa 2 = 2 and the values of g reported in Table IV, the number of iteration is relatively small compared to the case of wa 2 = 1/2 and a large value of the parameter g. The numerical computations in the present work were done using Maple version 13 running on an IBM architecture personal computer in a high-precision environment. In order to accelerate our computation we have written our own code for a root-finding algorithm instead of using the default procedure Solve of Maple 13. These numerical results are accurate to the number of decimals reported.

V. CONCLUSION
We have provided a detailed solution of the eigenproblem posed by Schrödiger's equation with a generalized nonlinear isotonic oscillator potential. We have presented a method for computing the quasi-polynomial solutions in cases where the potential parameters satisfy certain conditions. In other more general cases we have used the asymptotic iteration method to find accurate numerical solutions for arbitrary values of the potential parameters g, w and a. IV: Energies of the four lowest states of the generalized isotonic oscillator of parameters w and a given for l = −1 as wa 2 = 2 and for different values of the parameter g. The subscript numbers represents the number of iterations used by AIM.