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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 750168, 20 pages
Study of the Generalized Quantum Isotonic Nonlinear Oscillator Potential
1Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PE, C1A 4P3, Canada
2Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montréal, QC, H3G 1M8, Canada
3Department of Physics, Faculty of Arts and Sciences, Gazi University, 06500 Ankara, Turkey
Received 11 March 2011; Accepted 13 April 2011
Academic Editor: B. G. Konopelchenko
Copyright © 2011 Nasser Saad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the generalized quantum isotonic oscillator Hamiltonian given by , . Two approaches are explored. A method for finding the quasipolynomial 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 , , and may be found to high accuracy.
Recently, Cariñena et al.  studied a quantum nonlinear oscillator potential whose Schrödinger equation reads The interest in this problem came from the fact that it is exactly solvable in a sense that the exact eigenenergies and eigenfunctions can be obtained explicitly. Indeed, Cariñena et al.  were able to show that where the polynomials factors are related to the Hermite polynomials by means of In a more recent work, Fellows and Smith  showed that the potential as well as, for certain values of the parameters , , and , the potential 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 , using a Möbius transformation, was able to transform (1.4) into a confluent Heun equation , and thereby obtain an efficient algorithm to solve the Schrödinger equation (1.4) numerically.
The purpose of the present work is to provide a detailed solution, by means of the quasipolynomial solutions and the application of the asymptotic iteration method [5–8], for the Schrödinger equation where is the angular momentum number . Our results show that the quasiexact solutions of Sesma  as well the results of Cariñena et al.  follow as special cases of our general approach. The present paper is organized as follows. In Section 2, some preliminary analysis of the Schrödinger equation (1.5) is presented. A general approach for finding polynomial solutions of (1.5), for certain values of parameters and , is presented and is based on a recent work of Ciftci et al.  for solving the second-order linear differential equation More general quasiexact solutions, including the results of Sesma , are discussed in Section 3. Unrestricted solutions of (1.5) based on the asymptotic iteration method are discussed in Section 4.
2. Generalized Quantum Isotonic Oscillator—Preliminary Results
A simple scaling argument, using , allows us to write (1.5) as A further substitution yields a differential equation with two regular singular points at and one irregular singular point of rank 2 at . The roots 's of the indicial equation for the regular singular point reads , while the roots of the indicial equation at are and . Since the singularity for corresponds to that for , it is necessary that the solution for behave as . Consequently, we may assume the general solution of (2.1) which vanishes at the origin and at infinity takes the form A straightforward calculation shows that 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 takes the value of the indicial root which allows us to write (2.3) as We now consider the cases where the following two equations are satisfied: The solutions of this system, for and , are given explicitly by Next, we consider each case of these two sets of solutions.
2.1. Case 1
The first set of solutions reduces the differential equation (2.3) to which is a special case of the general differential equation with , , , , and . The necessary and sufficient conditions for polynomial solutions of (2.9) are given by the following theorem .
Theorem 2.1. The second-order linear differential equation (2.9) has a polynomial solution of degree if along with the vanishing of -determinant given by where and is fixed for a given in the determinant .
If , the determinant is identically zero for all , which is equivalent to the exact solutions of the one-dimensional harmonic oscillator problem.
For , we have for , , and we obtain the exact solutions of the Gol'dman and Krivchenkov (or Isotonic) Hamiltonian , where These exact solutions are given by  where the confluent hypergeometric function defined in terms of the Pochhammer symbol (or Gamma function ) as The polynomial solutions are easily obtained by using the asymptotic iteration method (AIM), which is summarized by means of the following theorem.
Theorem 2.2 (Ciftci et al. [7, equations (2.13)-(2.14)]). Given and in , the differential equation has the general solution if for some where
For the differential equation (2.8) with the first few iterations with , using (2.20), imply which we may easily generalized using the definition of the confluent hypergeometric function, (2.18), as up to a constant.
2.2. Case 2
The second set of solutions allow us to write the differential equation (2.3) as A further change of variable allows us to write the differential equation (2.27) as Again, (2.28) is a special case of the differential equation (2.9) with , , , , , , and . Consequently, the polynomial solutions of (2.28) are subject to the following two conditions: the necessary condition (2.10) reads and the sufficient condition; namely, the vanishing of the tridiagonal determinant (2.12), reads where and is fixed for the given dimension of the determinant . From the sufficient condition (2.31), we obtain the following conditions on the parameters: For a physically meaningful solution, we must have . This is possible for a very restricted value of the angular momentum number . Since , we may observe that where are polynomials in the parameter product .
For physically acceptable solutions, we must have and the factor yields , which is not physically acceptable, so we ignore it. The second factor implies a special value of , for all , which we will study shortly in full detail. Meanwhile, the polynomials give new values, not reported before, of that yield quasiexact solutions of the Schrödinger equation (with one eigenstate) where and are the solutions of For example, implies, using (2.34), that , and thus, we have for the exact solution with a plot of the wave function and potential given in Figure 1.
Further, , equation (2.34) implies and we have for the exact solutions Similar results can be obtained for , for .
2.3. Exactly Solvable Quantum Isotonic Nonlinear Oscillator
As mentioned above, for and , it clear that for all and the one-dimensional Schrödinger equation has the exact solutions where are the polynomial solutions of the following second-order linear differential equation () By using AIM (Theorem 2.2, (2.20)), we find that the polynomial solutions of (2.45) are given explicitly as a set of polynomial solutions that can be generated using up to a constant factor, where, again, refers to the confluent hypergeometric function defined by (2.18). Note that the polynomials in (2.47) can be expressed in terms of the associated Laguerre polynomials  as
3. Quasipolynomial Solutions of the Generalized Quantum Isotonic Oscillator
In this section, we study the quasipolynomial solutions of the differential equation (2.3). We note first, using the change of variable , equation (2.3) can be written as By means of the Möbius transformation that maps the singular points into , we obtain where we assume that The differential equation (3.2) can be written as which we may now compare with equation (2.9) in Theorem 2.1 with , , , , , , , , . We, thus, conclude that the quasipolynomial solutions of (3.4) are subject to the following conditions: along with the vanishing of the tridiagonal determinant where Here, again, is fixed for given , the fixed size of the determinant .
3.1. Particular Case:
For , the differential equation (3.4) has the exact solution if and satisfy, simultaneously, the following system of equations: Solving this system of equations for and , we obtain the following values of and the ground-state energy, in this case, is given by (3.3); namely, which in complete agreement with the results of Section 2.2.
3.2. Particular Case:
For , the determinant of (3.7) yields where the energy is given by use of (3.3), for the computed values of and , by Further, (3.11) yields the solutions for as functions of and where the energy states are now given by (3.12) along with given by (3.13). We may also note that for Further, for , we obtain the unnormalized wave function (see (2.2)) Thus, we may summarize these results as follows. The exact solutions of the Schrödinger equation (2.1) are given by (3.15) and (3.16) only if and are the solutions of the system given by (3.11). In Tables 1 and 2, we report few quasiexact solutions that can be obtained using this approach.
3.2.1. Particular Case
For , the determinant along with the necessary condition (3.7) yields where, again, the energy is given, for the computed values of and using (3.3) and (3.17), by In Table 3, we report the numerical results for some of the exact solutions of and using (3.17) and the values of , , ,, , and , respectively. We have also computed the corresponding eigenvalues .
4. Numerical Computation by the Use of the Asymptotic Iteration Method
For the potential parameters , , and , not necessarily obeying the conditions for quasipolynomial solutions discussed in the previous sections, the asymptotic iteration method can be employed to compute the eigenvalues of Schrödinger equation (2.1) for arbitrary values , , and . The functions and , using (3.2) and (3.3), are given by where . The AIM sequence and can be calculated iteratively using the iterative sequences (2.22). The energy eigenvalues of the quantum nonlinear isotonic potential (2.1) are obtained from the roots of the termination condition (2.21). According to the asymptotic iteration method, in particular, the study of Champion et al. , unless the differential equation is exactly solvable, the termination condition (2.21) produces for each iteration an expression that depends on both and (for given values of the parameters , , and ). In such a case, one faces the problem of finding the best possible starting value that stabilizes the AIM process . Fortunately, since , the starting value does not represent a serious issue in our eigenvalue calculation using (4.1) and the termination condition (2.21) in contrast to the case of computing the eigenvalues using and as given by, for example, (2.3), where . In Table 4, we report our numerical results for energies of the four lowest states of the generalized isotonic oscillator of parameters and such that and for different values of . In this table, we set for computing the energies and , while we put for computing the energies and , respectively. For most of these values, the starting value of is and is shifted towards zero as gets larger in value. For the values of that admit a quasipolynomial solution, the number of iteration does not exceed three. For most of the other values of , the total number of iteration did not exceed 65. We found that for and the values of reported in Table 4, the number of iteration is relatively small compared to the case of and a large value of the parameter . 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.
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 quasipolynomial 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 , , and .
Partial financial support of this work under Grant nos. GP249507 and GP3438 from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by the authors.
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