Abstract

We study the generalized quantum isotonic oscillator Hamiltonian given by 𝐻=𝑑2/𝑑𝑟2+𝑙(𝑙+1)/𝑟2+𝑤2𝑟2+2𝑔(𝑟2𝑎2)/(𝑟2+𝑎2)2, 𝑔>0. 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 𝑔, 𝑤, and 𝑎 may be found to high accuracy.

1. Introduction

Recently, Cariñena et al. [1] studied a quantum nonlinear oscillator potential whose Schrödinger equation reads𝑑2𝑑𝑥2+𝑥2+82𝑥212𝑥2+12𝜓𝑛(𝑥)=𝐸𝑛𝜓(𝑥).(1.1) 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. [1] were able to show that𝜓𝑛𝑃(𝑥)=𝑛(𝑥)2𝑥2𝑒+1𝑥2/2,𝐸𝑛=3+2𝑛,𝑛=0,3,4,5,,(1.2) where the polynomials factors 𝑃𝑛(𝑥) are related to the Hermite polynomials by means of𝑃𝑛𝐻(𝑥)=1,if𝑛=0,𝑛(𝑥)+4𝑛𝐻𝑛2(𝑥)+4𝑛(𝑛3)𝐻𝑛4(𝑥),if𝑛=3,4,5,.(1.3) In a more recent work, Fellows and Smith [2] showed that the potential 𝑉(𝑥)=𝑥2+8(2𝑥21)/(2𝑥2+1)2 as well as, for certain values of the parameters 𝑤, 𝑔, and 𝑎, the potential 𝑉(𝑥)=𝑤2𝑥2+2𝑔(𝑥2𝑎2)/(𝑥2+𝑎2)2 of the Schrödinger equation𝑑2𝑑𝑥2+𝑤2𝑥2𝑥+2𝑔2𝑎2𝑥2+𝑎22𝜓𝑛(𝑥)=2𝐸𝑛𝜓(𝑥)(1.4) 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 [3], using a Möbius transformation, was able to transform (1.4) into a confluent Heun equation [4], 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 [58], for the Schrödinger equation𝑑2𝑑𝑟2+𝑙(𝑙+1)𝑟2+𝑤2𝑟2𝑟+2𝑔2𝑎2𝑟2+𝑎22𝜓(𝑟)=2𝐸𝜓(𝑟),(1.5) where 𝑙 is the angular momentum number 𝑙=1,0,1,. Our results show that the quasiexact solutions of Sesma [3] as well the results of Cariñena et al. [1] 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 𝑤 and 𝑔, is presented and is based on a recent work of Ciftci et al. [6] for solving the second-order linear differential equation3𝑖=0𝑎3,𝑖𝑥𝑖𝑦+2𝑖=0𝑎2,𝑖𝑥𝑖𝑦1𝑖=0𝜏1,𝑖𝑥𝑖𝑦=0.(1.6) More general quasiexact solutions, including the results of Sesma [3], 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 𝑟=𝑎2𝑥, allows us to write (1.5) as𝑑2𝑑𝑥2+𝑙(𝑙+1)𝑥2+𝑤𝑎22𝑥2𝑥+2𝑔21𝑥2+12𝜓(𝑥)=2𝐸𝑎2𝜓(𝑥).(2.1) A further substitution 𝑧=𝑥2+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(𝑤𝑎2𝑥2/2). Consequently, we may assume the general solution of (2.1) which vanishes at the origin and at infinity takes the form𝜓𝑛(𝑥)=𝑥𝑙+1𝑥2+1𝜇𝑒(𝑤𝑎2/2)𝑥2𝑓𝑛(𝑥).(2.2) A straightforward calculation shows that 𝑓𝑛(𝑥) are the solutions of the second-order homogeneous linear differential equation𝑓(𝑥)+2(𝑙+1)𝑥+4𝜇𝑥𝑥2+12𝑤𝑎2𝑥𝑓+(𝑥)2𝐸𝑎2𝑤𝑎2(2𝑙+3+4𝜇)+2𝜇2𝑙+3+2𝑤𝑎2+4𝜇(𝜇1)2𝑔𝑥2++14(𝑔𝜇(𝜇1))𝑥2+12×𝑓(𝑥)=0.(2.3) 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 root𝜇𝜇=121,1+4𝑔(2.4) which allows us to write (2.3) as𝑓𝑛(𝑥)+2(𝑙+1)𝑥+4𝜇𝑥𝑥2+12𝑤𝑎2𝑥𝑓𝑛+(𝑥)2𝐸𝑎2𝑤𝑎2(2𝑙+3+4𝜇)+2𝜇2𝑙+3+2𝑤𝑎2+2𝜇(𝜇1)𝑥2𝑓+1𝑛(𝑥)=0.(2.5) We now consider the cases where the following two equations are satisfied: 2𝜇2𝑙+3+2𝑤𝑎2+2𝜇(𝜇1)=0,𝑔=𝜇(𝜇1).(2.6) The solutions of this system, for 𝑔 and 𝜇, are given explicitly by𝑔=0,or𝑔=21+𝑙+𝑎2𝑤3+2𝑙+2𝑎2𝑤,𝜇=0,or𝜇=21+𝑙+𝑎2𝑤.(2.7) Next, we consider each case of these two sets of solutions.

2.1. Case  1

The first set of solutions (𝑔,𝜇)=(0,0) reduces the differential equation (2.3) to𝑥𝑓𝑛(𝑥)+2𝑤𝑎2𝑥2𝑓+2(𝑙+1)𝑛(𝑥)+2𝐸𝑎2𝑤𝑎2(2𝑙+3)𝑥𝑓𝑛(𝑥)=0,(2.8) which is a special case of the general differential equation𝑎3,0𝑥3+𝑎3,1𝑥2+𝑎3,2𝑥+𝑎3,3𝑦+𝑎2,0𝑥2+𝑎2,1𝑥+𝑎2,2𝑦𝜏1,0𝑥+𝜏1,1𝑦=0,(2.9) with 𝑎3,0=𝑎3,1=𝑎3,3=𝑎2,1=𝜏1,1=0, 𝑎3,2=1, 𝑎2,0=2𝑤𝑎2, 𝑎2,2=2(𝑙+1), and 𝜏1,0=2𝐸𝑎2+𝑤𝑎2(2𝑙+3). The necessary and sufficient conditions for polynomial solutions of (2.9) are given by the following theorem [6].

Theorem 2.1. The second-order linear differential equation (2.9) has a polynomial solution of degree 𝑛 if 𝜏1,0=𝑛(𝑛1)𝑎3,0+𝑛𝑎2,0,𝑛=0,1,2,,(2.10) along with the vanishing of (𝑛+1)×(𝑛+1)-determinant Δ𝑛+1 given by Δ𝑛+1=|||||||||||||||||𝛽0𝛼1𝜂1𝛾1𝛽1𝛼2𝜂2𝛾2𝛽2𝛼3𝜂3𝛾𝑛2𝛽𝑛2𝛼𝑛1𝜂𝑛1𝛾𝑛1𝛽𝑛1𝛼𝑛𝛾𝑛𝛽𝑛|||||||||||||||||=0,(2.11) where 𝛽𝑛=𝜏1,1𝑛(𝑛1)𝑎3,1+𝑎2,1,𝛼𝑛(=𝑛𝑛1)𝑎3,2+𝑎2,2,𝛾𝑛=𝜏1,0(𝑛1)(𝑛2)𝑎3,0+𝑎2,0,𝜂𝑛=𝑛(𝑛+1)𝑎3,3,(2.12) and 𝜏1,0 is fixed for a given 𝑛 in the determinant Δ𝑛+1=0.

Thus, the necessary condition for the differential equation (2.8) to have polynomial solutions 𝑓𝑛(𝑥)=𝑛𝑖=0𝑐𝑖𝑥𝑖 is2𝐸𝑛𝑎2=𝑤𝑎22𝑛+2𝑙+3,𝑛=0,1,2,,(2.13) while the sufficient condition, (2.12), isΔ𝑛+1=|||||||||||||||||0𝛼1𝛾0010𝛼20𝛾20𝛼30𝛾𝑛20𝛼𝑛10𝛾𝑛10𝛼𝑛𝛾𝑛0|||||||||||||||||=0,if𝑛=0,2,4,,(𝑛1)/2𝑗=0(1)2𝑗+1𝛼2𝑗+1𝛾2𝑗+1=0,if𝑛=1,3,5,.,(2.14) where 𝛽𝑛=0, 𝛼𝑛=𝑛(𝑛+2𝑙+1) and 𝛾𝑛=2𝑤𝑎2(𝑛𝑛1).

If 𝑙=1, the determinant Δ𝑛+1 is identically zero for all 𝑛, which is equivalent to the exact solutions of the one-dimensional harmonic oscillator problem.

For 𝑙1, we have for 𝑛=0,2,4,, Δ𝑛+10, and we obtain the exact solutions of the Gol'dman and Krivchenkov (or Isotonic) Hamiltonian 𝐻0, where𝐻0𝜓𝑛𝑙𝑑(𝑥)2𝑑𝑥2+𝑙(𝑙+1)𝑥2+𝑤2𝑎4𝑥2𝜓𝑛𝑙(𝑥)=2𝐸𝑔=0𝑛𝑙𝑎2𝜓𝑛𝑙(𝑥),0𝑥<.(2.15) These exact solutions are given by [9]2𝑎2𝐸𝑔=0𝑛𝑙=𝑤𝑎2𝜓(4𝑛+2𝑙+3),𝑛=0,1,2,,𝑛𝑙(𝑥)=𝑥𝑙+1𝑒𝑤𝑎2𝑥21/2𝐹13𝑛;𝑙+2;𝑤𝑎2𝑥2,𝑛=0,1,2,,(2.16) where the confluent hypergeometric function 1𝐹1(𝑛;𝑎;𝑧) defined in terms of the Pochhammer symbol (or Gamma function Γ(𝑎)) (𝑎)𝑘=Γ(𝑎+𝑘)=Γ(𝑎)1,if(𝑘=0,𝑎{0}),𝑎(𝑎+1)(𝑎+2)(𝑎+𝑘1),if(𝑘=,𝑎),(2.17) as1𝐹1(𝑛;𝑎;𝑧)=𝑛𝑘=0(𝑛)𝑘𝑧𝑘(𝑎)𝑘𝑘!.(2.18) The polynomial solutions 𝑓𝑛(𝑥)=1𝐹1(𝑛;𝑙+(3/2);𝑤𝑎2𝑥2) 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 𝜆0𝜆0(𝑥) and 𝑠0𝑠0(𝑥) in 𝐶, the differential equation 𝑓(𝑥)=𝜆0(𝑥)𝑓(𝑥)+𝑠0(𝑥)𝑓(𝑥)(2.19) has the general solution 𝑓(𝑥)=exp𝑥𝐶𝛼(𝑡)𝑑𝑡2+𝐶1𝑥exp𝑡𝜆0(𝜏)+2𝛼(𝜏)𝑑𝜏𝑑𝑡,(2.20) if for some 𝑛+={1,2,}𝑠𝑛𝜆𝑛=𝑠𝑛1𝜆𝑛1=𝛼(𝑥),or𝛿𝑛(𝑥)=𝜆𝑛𝑠𝑛1𝜆𝑛1𝑠𝑛=0,(2.21) where 𝜆𝑛=𝜆𝑛1+𝑠𝑛1+𝜆0𝜆𝑛,𝑠𝑛=𝑠𝑛1+𝑠0𝜆𝑛.(2.22)

For the differential equation (2.8) with𝜆0(𝑥)=2𝑤𝑎2𝑥2+2(𝑙+1)𝑥,𝑠0(𝑥)=2𝐸𝑎2𝑤𝑎2,(2𝑙+3)(2.23) the first few iterations with 𝛿𝑛=𝜆𝑛𝑠𝑛1𝜆𝑛1𝑠𝑛=0, using (2.20), imply𝑓0𝑓(𝑥)=1,1(𝑥)=2𝑤𝑎2𝑥2𝑓(2𝑙+3),2(𝑥)=4𝑤2𝑎4𝑥44𝑤𝑎2(2𝑙+5)𝑥2+(2𝑙+3)(2𝑙+5),(2.24) which we may easily generalized using the definition of the confluent hypergeometric function, (2.18), as𝑓𝑛(𝑥)=1𝐹13𝑛;𝑙+2;𝑤𝑎2𝑥2,(2.25) up to a constant.

2.2. Case  2

The second set of solutions 2(𝑔,𝜇)=1+𝑙+𝑎2𝑤3+2𝑙+2𝑎2𝑤,21+𝑙+𝑎2𝑤(2.26) allow us to write the differential equation (2.3) as𝑓𝑛(𝑥)+2(𝑙+1)𝑥8𝑙+1+𝑎2𝑤𝑥𝑥2+12𝑤𝑎2𝑥𝑓𝑛+(𝑥)2𝐸𝑎2+𝑤𝑎26𝑙+5+8𝑤𝑎2𝑓𝑛(𝑥)=0.(2.27) A further change of variable 𝑧=𝑥2+1 allows us to write the differential equation (2.27) as4𝑧(𝑧1)𝑓(𝑧)4𝑎2𝑤𝑧2+26𝑙+5+6𝑤𝑎2𝑧16𝑙+1+𝑤𝑎2𝑓+(𝑧)2𝐸𝑎2+𝑤𝑎26𝑙+5+8𝑤𝑎2𝑧𝑓(𝑧)=0,(2.28) Again, (2.28) is a special case of the differential equation (2.9) with 𝑎3,0=𝑎3,3=𝜏1,1=0, 𝑎3,1=4, 𝑎3,2=4, 𝑎2,0=4𝑤𝑎2, 𝑎2,1=2(6𝑙+5+6𝑤𝑎2), 𝑎2,2=16(𝑙+1+𝑤𝑎2), and 𝜏1,0=2𝐸𝑎2𝑤𝑎2(6𝑙+5+8𝑤𝑎2). Consequently, the polynomial solutions 𝑓𝑛(𝑥) of (2.28) are subject to the following two conditions: the necessary condition (2.10) reads2𝐸𝑛𝑎2=𝑤𝑎24𝑛6𝑙58𝑤𝑎2,𝑛=0,1,2,,(2.29) and the sufficient condition; namely, the vanishing of the tridiagonal determinant (2.12), reads Δ𝑛+1=|||||||||||||||||𝛽0𝛼1𝛾1𝛽1𝛼2𝛾2𝛽2𝛼3𝛾𝑛2𝛽𝑛2𝛼𝑛1𝛾𝑛1𝛽𝑛1𝛼𝑛𝛾𝑛𝛽𝑛|||||||||||||||||=0,(2.30) where𝛽𝑛=2𝑛2𝑛6𝑙76𝑤𝑎2,𝛼𝑛=4𝑛𝑛4𝑙54𝑎2𝑤,𝛾𝑛=4𝑤𝑎2𝑛𝑛,1(2.31) and 𝑛=𝑛 is fixed for the given dimension of the determinant Δ𝑛+1. From the sufficient condition (2.31), we obtain the following conditions on the parameters: Δ2=0𝑎2𝑤𝑙+1+𝑎2𝑤Δ=0,3=0𝑎2𝑤𝑙+1+𝑎2𝑤1+2𝑙+2𝑎2𝑤Δ=0,4=0𝑎2𝑤𝑙+1+𝑎2𝑤1+2𝑙+2𝑎2𝑤3(1+6𝑙)+14𝑎2𝑤Δ=0,5=0𝑎2𝑤𝑙+1+𝑎2𝑤1+2𝑙+2𝑎2𝑤3(6𝑙1)(6𝑙+1)+4(38𝑙+1)𝑎2𝑤+44𝑎4𝑤2Δ=0,6=0𝑎2𝑤𝑙+1+𝑎2𝑤1+2𝑙+2𝑎2𝑤×3(2𝑙1)(6𝑙1)(6𝑙+1)+2208𝑙2𝑎54𝑙52𝑤+200𝑙𝑎4𝑤2=0,=.(2.32) For a physically meaningful solution, we must have 𝑎2𝑤>0. This is possible for a very restricted value of the angular momentum number 𝑙. Since 𝛽0=0, we may observe that Δ𝑛+1=𝑙+1+𝑎2𝑤1+2𝑙+2𝑎2𝑤×|||||||||||||||||𝛽2𝛼3𝛾3𝛽3𝛼4𝛾4𝛽4𝛼5𝛾𝑛2𝛽𝑛2𝛼𝑛1𝛾𝑛1𝛽𝑛1𝛼𝑛𝛾𝑛𝛽𝑛|||||||||||||||||=𝑙+1+𝑎2𝑤1+2𝑙+2𝑎2𝑤×𝑄𝑙𝑛1𝑎2𝑤,(2.33) where 𝑄𝑙𝑛1(𝑎2𝑤) are polynomials in the parameter product 𝑎2𝑤.

For physically acceptable solutions, we must have 𝑙=1 and the factor (𝑙+1+𝑎2𝑤) yields 𝑎2𝑤=0, which is not physically acceptable, so we ignore it. The second factor (1+2𝑙+2𝑎2𝑤) implies a special value of 𝑎2𝑤=1/2, for all 𝑛, which we will study shortly in full detail. Meanwhile, the polynomials 𝑄𝑙𝑛(𝑎2𝑤)𝑄𝑙=1𝑛1𝑎2𝑤=1,if𝑛=2,14𝑎2𝑤15,if𝑛=3,44𝑎4𝑤2148𝑎2𝑤+105,if𝑛=4,200𝑎4𝑤2514𝑎2𝑤+315,if𝑛=5,(2.34) give new values, not reported before, of 𝑎2𝑤 that yield quasiexact solutions of the Schrödinger equation (with one eigenstate)𝜓𝑛(𝑥)+𝑤𝑎22𝑥2+4𝑎2𝑤1+2a2𝑤𝑥21𝑥2+12𝜓𝑛(𝑥)=𝑤𝑎24𝑛+18𝑎2𝑤𝜓𝑛(𝑥),(2.35) where 𝜓𝑛𝑥(𝑥)=2+12𝑎2𝑤𝑒𝑤𝑎2𝑥2/2𝑓𝑛(𝑥),(2.36) and 𝑓𝑛(𝑥) are the solutions of4𝑧(𝑧1)𝑓(𝑧)4𝑎2𝑤𝑧2+21+6𝑤𝑎2𝑧16𝑤𝑎2𝑓(𝑧)+4𝑛𝑤𝑎2𝑧𝑓(𝑧)=0,𝑧=𝑥2+1.(2.37) For example, Δ4=0 implies, using (2.34), that 𝑎2𝑤=15/14, and thus, we have for𝜓3(𝑥)+225𝑥1962+660𝑥4921𝑥2+12𝜓3(𝑥)=465𝜓983(𝑥),(2.38) the exact solution 𝜓3𝑥(𝑥)=2+115/7𝑒(15/28)𝑥245𝑥6+225𝑥4+315𝑥2,49(2.39) with a plot of the wave function and potential given in Figure 1.

Further, Δ5=0, equation (2.34) implies 𝑎2𝑤=37±2221422,(2.40) and we have for𝜓4(𝑥)+37±22214222𝑥2+237±112141148±11214𝑥1121𝑥2+12𝜓4=(𝑥)37±222142239411214𝜓114(𝑥),(2.41) the exact solutions 𝜓±4(𝑥𝑥)=2+1((37/11)±(214/11))𝑒((37/44)±(214/44))𝑥2×1575𝑥8+9660±420𝑥2146+26250±2100𝑥2144+29820±2940𝑥21421129±188.214(2.42) Similar results can be obtained for Δ𝑛+1=0, for 𝑛5.

2.3. Exactly Solvable Quantum Isotonic Nonlinear Oscillator

As mentioned above, for 𝑙=1 and 𝑎2𝑤=1/2, it clear that Δ𝑛+1=0 for all 𝑛 and the one-dimensional Schrödinger equation𝑑2𝑑𝑥2+𝑥24+4𝑥21𝑥2+12𝜓𝑛3(𝑥)=2𝑛2𝜓𝑛(𝑥),𝑛=0,1,2,(2.43) has the exact solutions𝜓𝑛𝑥(𝑥)=2+11𝑒𝑥2/4𝑓𝑛(𝑥),(2.44) where 𝑓𝑛(𝑥) are the polynomial solutions of the following second-order linear differential equation (𝑧=𝑥2+1)4𝑧(𝑧1)𝑓𝑛(𝑧)2𝑧2𝑓+4𝑧8𝑛(𝑧)+2𝑛𝑧𝑓𝑛(𝑧)=0.(2.45) By using AIM (Theorem 2.2, (2.20)), we find that the polynomial solutions 𝑓𝑛(𝑥) of (2.45) are given explicitly as𝑓0𝑓(𝑥)=1,1(𝑥)=𝑥2𝑓2,2(𝑥)=𝑥36𝑥2𝑓+8,3(𝑥)=𝑥416𝑥3+52𝑥2𝑓52,4(𝑥)=𝑥530𝑥4+250𝑥3580𝑥2+464,(2.46) a set of polynomial solutions that can be generated using𝑓0𝑓(𝑥)=1,𝑛(𝑥)=3𝑥(2𝑛+1)1𝐹13𝑛;2;12(𝑥1)+6((𝑛+1)𝑥1)1𝐹13𝑛+1;2;12,(𝑥1)(2.47) up to a constant factor, where, again, 1𝐹1 refers to the confluent hypergeometric function defined by (2.18). Note that the polynomials 𝑓𝑛(𝑥) in (2.47) can be expressed in terms of the associated Laguerre polynomials [10] as𝑓0𝑓(𝑥)=1,𝑛3(𝑥)=(1)𝑛𝜋Γ(𝑛)×2Γ(𝑛+3/2)(1+𝑛)(𝑥1)2𝐿+𝑛𝑛1/2𝑥12(𝑥1)((1+𝑛)𝑥1)𝐿𝑛3/2𝑥12.(2.48)

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 𝑧=𝑥2, equation (2.3) can be written as𝑓𝑛(𝑧)+2𝑙+3+2𝑧2𝜇𝑧+1𝑤𝑎2𝑓𝑛+(𝑧)2𝐸𝑎2𝑤𝑎2(2𝑙+3+4𝜇)+𝜇4𝑧2𝑙+3+2𝑤𝑎2𝑔2𝑧(𝑧+1)2(𝑧1)𝑧(𝑧+1)2+𝜇(𝜇1)(𝑧+1)2𝑓𝑛(𝑧)=0.(3.1) By means of the Möbius transformation 𝑧=𝑡/(1𝑡) that maps the singular points {1,0,} into {0,1,}, we obtain𝑓𝑛(𝑡)+2𝑙+3+2𝑡(1𝑡)2(𝜇1)1𝑡𝑤𝑎2(1𝑡)2𝑓𝑛+𝜇(𝑡)2𝑙+3+2𝑤𝑎22𝑡(1𝑡)2𝑔2(2𝑡1)𝑡(1𝑡)2+𝜇(𝜇1)(1𝑡)2𝑓𝑛(𝑡)=0,(3.2) where we assume that2𝐸𝑎2(2𝑙+3+4𝜇)𝑤𝑎2=0.(3.3) The differential equation (3.2) can be written as𝑡32𝑡2𝑓+𝑡𝑛(𝑡)+2(𝜇1)𝑡2+2𝜇𝑤𝑎27𝑙23𝑡+𝑙+2𝑓𝑛+𝑔(𝑡)(𝜇(𝜇1)𝑔)𝑡+23+𝜇𝑙+2+𝑤𝑎2𝑓𝑛(𝑡)=0,(3.4) which we may now compare with equation (2.9) in Theorem 2.1 with 𝑎3,0=1, 𝑎3,1=2, 𝑎3,2=1, 𝑎3,3=0, 𝑎2,0=2(𝜇1), 𝑎2,1=(2𝜇𝑤𝑎2𝑙7/2), 𝑎2,2=(𝑙+3/2), 𝜏1,0=(𝜇(𝜇1)𝑔), 𝜏1,1=𝑔/2𝜇(𝑙+3/2+𝑤𝑎2). We, thus, conclude that the quasipolynomial solutions 𝑓𝑛(𝑡) of (3.4) are subject to the following conditions:𝑔=(𝜇𝑘)(𝜇𝑘1),𝑘=0,1,2,,(3.5) along with the vanishing of the tridiagonal determinant Δ𝑛+1=0||||||||||||||𝛽0𝛼1𝛾1𝛽1𝛼2𝛾2𝛽2𝛼3𝛾𝑛1𝛽𝑛1𝛼𝑛𝛾𝑛𝛽𝑛||||||||||||||=0,(3.6) where𝛽𝑛1=2𝑔+(𝜇𝑛)3+2𝑙+4𝑛+2𝑎2𝑤,𝛼𝑛1=𝑛𝑛+𝑙+2,𝛾𝑛=𝑔(𝜇𝑛+1)(𝜇𝑛).(3.7) Here, again, 𝑔=(𝜇𝑘)(𝜇𝑘1) is fixed for given 𝑘=𝑛, the fixed size of the determinant Δ𝑛+1.

3.1. Particular Case: 𝑛=0

For 𝑘(𝑓𝑖𝑥𝑒𝑑)𝑛=0, the differential equation (3.4) has the exact solution 𝑓0(𝑡)=1 if 𝑔 and 𝜇 satisfy, simultaneously, the following system of equations: 𝑔+𝜇3+2𝑙+2𝑎2𝑤=0,𝑔=𝜇(𝜇1).(3.8) Solving this system of equations for 𝑔 and 𝜇, we obtain the following values of𝑔=21+𝑙+𝑎2𝑤3+2𝑙+2𝑎2𝑤,𝜇=2𝑙+1+𝑤𝑎2,(3.9) and the ground-state energy, in this case, is given by (3.3); namely,𝐸𝑎21=2𝑎2𝑤5+6𝑙+8𝑎2𝑤,(3.10) which in complete agreement with the results of Section 2.2.

3.2. Particular Case: 𝑛=1

For 𝑘(𝑓𝑖𝑥𝑒𝑑)𝑛=1, the determinant Δ2=0 of (3.7) yields𝑔2+𝑔1+10𝜇+2𝑙(2𝜇+1)+2𝑎2𝑤(2𝜇1)+𝜇(𝜇1)15+4𝑙2+8𝑙2+𝑎2𝑤+4𝑎2𝑤5+𝑎2𝑤=0,𝑔(𝜇1)(𝜇2)=0,(3.11) where the energy is given by use of (3.3), for the computed values of 𝜇 and 𝑔, by3𝐸=𝑙+2+2𝜇𝑤.(3.12) Further, (3.11) yields the solutions for 𝑙 as functions of 𝜇 and 𝑎2𝑤𝑙=25+4𝑎2𝑤𝜇2𝜇2±443+8𝑎2𝑤𝜇+9𝜇24𝜇1,(3.13) where the energy states are now given by (3.12) along with 𝑙 given by (3.13). We may also note that for𝑎21𝑤=2𝑎(𝑘+1),𝑘=0,1,2,,(3.14)2𝐸±1=8𝜇(𝑘+1)2+(2𝑘+1)𝜇6𝜇2±44(4𝑘+7)𝜇+9𝜇2.(3.15) Further, for 𝑔=(𝜇1)(𝜇2), we obtain the unnormalized wave function (see (2.2))𝜓1,𝑙(𝑥)=𝑥𝑙+11+𝑥2𝜇1𝑒𝑤𝑎2𝑥2/21+1+2𝑙+𝜇+2𝑎2𝑤5+2𝑙+𝜇+2𝑎2𝑤𝑥2.(3.16) 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 𝑔 and 𝜇 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 𝑛=2

For 𝑘(𝑓𝑖𝑥𝑒𝑑)𝑛=2, the determinant Δ3=0 along with the necessary condition (3.7) yields𝑔3+3𝑔27𝜇1+2𝑙(1+𝜇)+2𝑎2𝑤(𝜇1)𝑔18+56𝑙+8𝑙2+18(7+2𝑙)𝜇3(5+2𝑙)(7+2𝑙)𝜇212𝑎2𝑤(𝜇1)((7+2𝑙)𝜇4)4𝑎4𝑤2(2+3(𝜇2)𝜇)+𝜇(𝜇2)(𝜇1)105+142𝑙+60𝑙2+8𝑙3+6𝑎2𝑤(5+2𝑙)(7+2𝑙)+12𝑎4𝑤2(7+2𝑙)+8𝑎6𝑤3=0,𝑔(𝜇2)(𝜇3)=0,(3.17) where, again, the energy is given, for the computed values of 𝜇 and 𝑔 using (3.3) and (3.17), by 3𝐸=𝑙+2+2𝜇𝑤.(3.18) In Table 3, we report the numerical results for some of the exact solutions of 𝜇 and 𝑔 using (3.17) and the values of (l,𝑤𝑎2)=(1,1/2), (𝑙,𝑤𝑎2)=(1,1), (𝑙,𝑤𝑎2)=(1,3/2),(𝑙,𝑤𝑎2)=(1,2), (𝑙,𝑤𝑎2)=(0,1/2), and (𝑙,𝑤𝑎2)=(0,2), respectively. We have also computed the corresponding eigenvalues 𝐸𝑤𝑎22,𝑙𝐸𝑤𝑎22,𝑙(𝜇,𝑔).

4. Numerical Computation by the Use of the Asymptotic Iteration Method

For the potential parameters 𝑤, 𝑎2, and 𝑔, 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 𝑤, 𝑎2, and 𝑔. The functions 𝜆0 and 𝑠0, using (3.2) and (3.3), are given by𝜆0(𝑡)=2𝑙+3+22𝑡(1𝑡)𝐸𝑎2/2𝑤𝑎2((2𝑙+3)/4)1(1𝑡)𝑤𝑎2(1𝑡)2,𝑠0(𝑡)=𝐸𝑎2/2𝑤𝑎2((2𝑙+3)/4)2𝑙+3+2w𝑎22𝑡(1𝑡)2𝑔2(2𝑡1)𝑡(1𝑡)2+𝐸𝑎2/2𝑤𝑎2((2𝑙+3)/4)𝐸𝑎2/2𝑤𝑎2((2𝑙+3)/4)1(1𝑡)2,(4.1) where 𝑡(0,1). The AIM sequence 𝜆𝑛(𝑥) and 𝑠𝑛(𝑥) 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. [5], unless the differential equation is exactly solvable, the termination condition (2.21) produces for each iteration an expression that depends on both 𝑡 and 𝐸 (for given values of the parameters 𝑤𝑎2, 𝑔, and 𝑙). In such a case, one faces the problem of finding the best possible starting value 𝑡=𝑡0 that stabilizes the AIM process [5]. Fortunately, since 𝑡(0,1), the starting value 𝑡0 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 𝜆0(𝑥) and 𝑠0(𝑥) as given by, for example, (2.3), where 𝑥(0,). In Table 4, we report our numerical results for energies of the four lowest states of the generalized isotonic oscillator of parameters 𝑤 and 𝑎 such that 𝑤𝑎2=2 and for different values of 𝑔. In this table, we set 𝑙=1 for computing the energies 𝐸0𝑎2 and 𝐸2𝑎2, while we put 𝑙=0 for computing the energies 𝐸1𝑎2 and 𝐸3𝑎2, respectively. For most of these values, the starting value of 𝑡 is 𝑡0=0.5 and is shifted towards zero as 𝑔 gets larger in value. For the values of 𝑔 that admit a quasipolynomial solution, the number of iteration does not exceed three. For most of the other values of 𝑔, the total number of iteration did not exceed 65. We found that for 𝑤𝑎2=2 and the values of 𝑔 reported in Table 4, the number of iteration is relatively small compared to the case of 𝑤𝑎2=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.

5. 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 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 𝑔, 𝑤, and 𝑎.

Acknowledgment

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.