# Study of the Generalized Quantum Isotonic Nonlinear Oscillator Potential

Authors: Nasser Saad; Richard L. Hall; Hakan Çiftçi; Özlem Yeşiltaş
Journal: Advances in Mathematical Physics
 (2011)

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## Abstract

We study the generalized quantum isotonic oscillator Hamiltonian given byH=−d2/dr2+l(l+1)/r2+w2r2+2g(r2−a2)/(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.


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## Body

## 1. Introduction

Recently, Cariñena et al. [1] studied a quantum nonlinear oscillator potential whose Schrödinger equation reads(1.1)[-d2dx2+x2+82x2-1(2x2+1)2]ψn(x)=Enψ(x).
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 that(1.2)ψn(x)=Pn(x)(2x2+1)e-x2/2,En=-3+2n,n=0,3,4,5,…,
where the polynomials factors Pn(x) are related to the Hermite polynomials by means of(1.3)Pn(x)={1,ifn=0,Hn(x)+4nHn-2(x)+4n(n-3)Hn-4(x),ifn=3,4,5,….
In a more recent work, Fellows and Smith [2] showed that the potential V(x)=x2+8(2x2-1)/(2x2+1)2 as well as, for certain values of the parameters w, g, and a, the potential V(x)=w2x2+2g(x2-a2)/(x2+a2)2 of the Schrödinger equation(1.4)[-d2dx2+w2x2+2gx2-a2(x2+a2)2]ψn(x)=2Enψ(x)
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 [5–8], for the Schrödinger equation(1.5)[-d2dr2+l(l+1)r2+w2r2+2gr2-a2(r2+a2)2]ψ(r)=2Eψ(r),
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 equation(1.6)(∑i=03a3,ixi)y′′+(∑i=02a2,ixi)y′-(∑i=01τ1,ixi)y=0.
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, usingr=a2x, allows us to write (1.5) as(2.1)[-d2dx2+l(l+1)x2+(wa2)2x2+2gx2-1(x2+1)2]ψ(x)=2Ea2ψ(x).
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 form(2.2)ψn(x)=xl+1(x2+1)μe-(wa2/2)x2fn(x).
A straightforward calculation shows that fn(x) are the solutions of the second-order homogeneous linear differential equation(2.3)f′′(x)+(2(l+1)x+4μxx2+1-2wa2x)f′(x)+[2Ea2-wa2(2l+3+4μ)+2μ(2l+3+2wa2)+4μ(μ-1)-2gx2+1+4(g-μ(μ-1))(x2+1)2]×f(x)=0.
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(2.4)μ≡μ-=12(1-1+4g),
which allows us to write (2.3) as(2.5)fn′′(x)+(2(l+1)x+4μxx2+1-2wa2x)fn′(x)+[2Ea2-wa2(2l+3+4μ)+2μ(2l+3+2wa2)+2μ(μ-1)x2+1]fn(x)=0.
We now consider the cases where the following two equations are satisfied: (2.6)2μ(2l+3+2wa2)+2μ(μ-1)=0,g=μ(μ-1).
The solutions of this system, for g and μ, are given explicitly by(2.7)g=0,org=2(1+l+a2w)(3+2l+2a2w),μ=0,orμ=-2(1+l+a2w).
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) to(2.8)xfn′′(x)+[-2wa2x2+2(l+1)]fn′(x)+(2Ea2-wa2(2l+3))xfn(x)=0,
which is a special case of the general differential equation(2.9)(a3,0x3+a3,1x2+a3,2x+a3,3)y′′+(a2,0x2+a2,1x+a2,2)y′-(τ1,0x+τ1,1)y=0,
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
(2.10)τ1,0=n(n-1)a3,0+na2,0,n=0,1,2,…,
along with the vanishing of (n+1)×(n+1)-determinant Δn+1 given by
(2.11)Δn+1=|β0α1η1γ1β1α2η2γ2β2α3η3⋱⋱⋱⋱γn-2βn-2αn-1ηn-1γn-1βn-1αnγnβn|=0,
where
(2.12)βn=τ1,1-n((n-1)a3,1+a2,1),αn=-n((n-1)a3,2+a2,2),γn=τ1,0-(n-1)((n-2)a3,0+a2,0),ηn=-n(n+1)a3,3,
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)=∑i=0ncixi is(2.13)2Ena2=wa2(2n′+2l+3),n′=0,1,2,…,
while the sufficient condition, (2.12), is(2.14)Δn+1=|0α100γ10α20γ20α30⋱⋱⋱⋱γn-20αn-10γn-10αnγn0|={0,ifn=0,2,4,…,∏j=0(n-1)/2(-1)2j+1α2j+1γ2j+1=0,ifn=1,3,5,….,
where βn=0, αn=-n(n+2l+1) and γn=2wa2(n-n′-1).Ifl=-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.Forl≠-1, we have for n=0,2,4,…, Δn+1≡0, and we obtain the exact solutions of the Gol'dman and Krivchenkov (or Isotonic) Hamiltonian H0, where(2.15)H0ψnl(x)≡[-d2dx2+l(l+1)x2+w2a4x2]ψnl(x)=2Enlg=0a2ψnl(x),0≤x<∞.
These exact solutions are given by [9](2.16)2a2Enlg=0=wa2(4n+2l+3),n=0,1,2,…,ψnl(x)=xl+1e-wa2x2/21F1(-n;l+32;wa2x2),n=0,1,2,…,
where the confluent hypergeometric function 1F1(-n;a;z) defined in terms of the Pochhammer symbol (or Gamma function Γ(a)) (2.17)(a)k=Γ(a+k)Γ(a)={1,if(k=0,a∈C∖{0}),a(a+1)(a+2)⋯(a+k-1),if(k=N,a∈C),
as(2.18)1F1(-n;a;z)=∑k=0n(-n)kzk(a)kk!.
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)]).
Givenλ0≡λ0(x) and  s0≡s0(x) in  C∞, the differential equation
(2.19)f′′(x)=λ0(x)f′(x)+s0(x)f(x)
has the general solution
(2.20)f(x)=exp⁡⁡(-∫xα(t)dt)[C2+C1∫xexp⁡⁡(∫t(λ0(τ)+2α(τ))dτ)dt],
if for some n∈ℕ+={1,2,…}(2.21)snλn=sn-1λn-1=α(x),orδn(x)=λnsn-1-λn-1sn=0,
where
(2.22)λn=λn-1′+sn-1+λ0λn,sn=sn-1′+s0λn.For the differential equation (2.8) with(2.23)λ0(x)=-(-2wa2x2+2(l+1))x,s0(x)=-(2Ea2-wa2(2l+3)),
the first few  iterations with δn=λnsn-1-λn-1sn=0, using (2.20), imply(2.24)f0(x)=1,f1(x)=2wa2x2-(2l+3),f2(x)=4w2a4x4-4wa2(2l+5)x2+(2l+3)(2l+5),⋮
which we may easily generalized using the definition of the confluent hypergeometric function, (2.18), as(2.25)fn(x)=1F1(-n;l+32;wa2x2),
up to a constant.


### 2.2. Case  2

The second set of solutions(2.26)(g,μ)=(2(1+l+a2w)(3+2l+2a2w),-2(1+l+a2w))
allow us to write the differential equation (2.3) as(2.27)fn′′(x)+(2(l+1)x-8(l+1+a2w)xx2+1-2wa2x)fn′(x)+(2Ea2+wa2(6l+5+8wa2))fn(x)=0.
A further change of variable z=x2+1 allows us to write the differential equation (2.27) as(2.28)4z(z-1)f′′(z)-(4a2wz2+2(6l+5+6wa2)z-16(l+1+wa2))f′(z)+(2Ea2+wa2(6l+5+8wa2))zf(z)=0,
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=-2Ea2-wa2(6l+5+8wa2). Consequently, the polynomial solutions fn(x) of (2.28) are subject to the following two conditions: the necessary condition (2.10) reads(2.29)2Ena2=wa2(4n′-6l-5-8wa2),n′=0,1,2,…,
and the sufficient condition; namely, the vanishing of the tridiagonal determinant (2.12), reads (2.30)Δn+1=|β0α1γ1β1α2γ2β2α3⋱⋱⋱γn-2βn-2αn-1γn-1βn-1αnγnβn|=0,
where(2.31)βn=-2n(2n-6l-7-6wa2),αn=4n(n-4l-5-4a2w),γn=4wa2(n-n′-1),
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: (2.32)Δ2=0⟹a2w(l+1+a2w)=0,Δ3=0⟹a2w(l+1+a2w)(1+2l+2a2w)=0,Δ4=0⟹a2w(l+1+a2w)(1+2l+2a2w)(3(1+6l)+14a2w)=0,Δ5=0⟹a2w(l+1+a2w)(1+2l+2a2w)(3(6l-1)(6l+1)+4(38l+1)a2w+44a4w2)=0,Δ6=0⟹a2w(l+1+a2w)(1+2l+2a2w)×(3(2l-1)(6l-1)(6l+1)+2(208l2-54l-5)a2w+200la4w2)=0,⋮=⋮.
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 (2.33)Δn+1=(l+1+a2w)(1+2l+2a2w)×|β2α3γ3β3α4γ4β4α5⋱⋱⋱γn-2βn-2αn-1γn-1βn-1αnγnβn|=(l+1+a2w)(1+2l+2a2w)×Qn-1l(a2w),
where Qn-1l(a2w) are polynomials in the parameter product a2w.For physically acceptable solutions, we must havel=-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 Qnl(a2w)(2.34)Qn-1l=-1(a2w)={1,ifn=2,14a2w-15,ifn=3,44a4w2-148a2w+105,ifn=4,200a4w2-514a2w+315,ifn=5,⋮
give new values, not reported before, of a2w that yield quasiexact solutions of the Schrödinger equation (with one eigenstate)(2.35)-ψn′′(x)+[(wa2)2x2+4a2w(1+2a2w)(x2-1)(x2+1)2]ψn(x)=wa2(4n+1-8a2w)ψn(x),
where (2.36)ψn(x)=(x2+1)-2a2we-wa2x2/2fn(x),
and fn(x) are the solutions of(2.37)4z(z-1)f′′(z)-(4a2wz2+2(-1+6wa2)z-16wa2)f′(z)+4nwa2zf(z)=0,z=x2+1.
For example, Δ4=0 implies, using (2.34), that a2w=15/14, and thus, we have for(2.38)-ψ3′′(x)+[225196x2+66049(x2-1)(x2+1)2]ψ3(x)=46598ψ3(x),
the exact solution (2.39)ψ3(x)=(x2+1)-15/7e-(15/28)x2(45x6+225x4+315x2-49),
with a plot of the wave function and potential given in Figure 1.Figure 1
Plot of the unnormalized wave functionψ3(x) and the potential V3=(225/196)x2+(660/49)(x2-1)/(x2+1)2.Further,Δ5=0, equation (2.34) implies (2.40)a2w=3722±21422,
and we have for(2.41)-ψ4′′(x)+[(3722±21422)2x2+2(3711±21411)(4811±21411)(x2-1)(x2+1)2]ψ4(x)=(3722±21422)(3911∓421411)ψ4(x),
the exact solutions (2.42)ψ4±(x)=(x2+1)-((37/11)±(214/11))e-((37/44)±(214/44))x2×(1575x8+(9660±420214)x6+(26250±2100214)x4+(29820±2940214)x2-(1129±188214)).
Similar results can be obtained for Δn+1=0, for n≥5.


### 2.3. Exactly Solvable Quantum Isotonic Nonlinear Oscillator

As mentioned above, forl=-1 and a2w=1/2, it clear that Δn+1=0 for all n and the one-dimensional Schrödinger equation(2.43)[-d2dx2+x24+4(x2-1)(x2+1)2]ψn(x)=(2n-32)ψn(x),n=0,1,2,…
has the exact solutions(2.44)ψn(x)=(x2+1)-1e-x2/4fn(x),
where fn(x) are the polynomial solutions of the following second-order linear differential equation (z=x2+1)(2.45)4z(z-1)fn′′(z)-(2z2+4z-8)fn′(z)+2nzfn(z)=0.
By using AIM (Theorem 2.2, (2.20)), we find that the polynomial solutions fn(x) of (2.45) are given explicitly as(2.46)f0(x)=1,f1(x)=x2-2,f2(x)=x3-6x2+8,f3(x)=x4-16x3+52x2-52,f4(x)=x5-30x4+250x3-580x2+464,⋮
a set of polynomial solutions that can be generated using(2.47)f0(x)=1,fn(x)=-3x(2n+1)1F1(-n;32;12(x-1))+6((n+1)x-1)1F1(-n+1;32;12(x-1)),
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] as(2.48)f0(x)=1,fn(x)=3(-1)nπΓ(n)2Γ(n+3/2)×[((1+n)(x-1)2+n)Ln1/2(x-12)-(x-1)((1+n)x-1)Ln3/2(x-12)].


## 2.1. Case  1

The first set of solutions(g,μ)=(0,0) reduces the differential equation (2.3) to(2.8)xfn′′(x)+[-2wa2x2+2(l+1)]fn′(x)+(2Ea2-wa2(2l+3))xfn(x)=0,
which is a special case of the general differential equation(2.9)(a3,0x3+a3,1x2+a3,2x+a3,3)y′′+(a2,0x2+a2,1x+a2,2)y′-(τ1,0x+τ1,1)y=0,
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
(2.10)τ1,0=n(n-1)a3,0+na2,0,n=0,1,2,…,
along with the vanishing of (n+1)×(n+1)-determinant Δn+1 given by
(2.11)Δn+1=|β0α1η1γ1β1α2η2γ2β2α3η3⋱⋱⋱⋱γn-2βn-2αn-1ηn-1γn-1βn-1αnγnβn|=0,
where
(2.12)βn=τ1,1-n((n-1)a3,1+a2,1),αn=-n((n-1)a3,2+a2,2),γn=τ1,0-(n-1)((n-2)a3,0+a2,0),ηn=-n(n+1)a3,3,
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)=∑i=0ncixi is(2.13)2Ena2=wa2(2n′+2l+3),n′=0,1,2,…,
while the sufficient condition, (2.12), is(2.14)Δn+1=|0α100γ10α20γ20α30⋱⋱⋱⋱γn-20αn-10γn-10αnγn0|={0,ifn=0,2,4,…,∏j=0(n-1)/2(-1)2j+1α2j+1γ2j+1=0,ifn=1,3,5,….,
where βn=0, αn=-n(n+2l+1) and γn=2wa2(n-n′-1).Ifl=-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.Forl≠-1, we have for n=0,2,4,…, Δn+1≡0, and we obtain the exact solutions of the Gol'dman and Krivchenkov (or Isotonic) Hamiltonian H0, where(2.15)H0ψnl(x)≡[-d2dx2+l(l+1)x2+w2a4x2]ψnl(x)=2Enlg=0a2ψnl(x),0≤x<∞.
These exact solutions are given by [9](2.16)2a2Enlg=0=wa2(4n+2l+3),n=0,1,2,…,ψnl(x)=xl+1e-wa2x2/21F1(-n;l+32;wa2x2),n=0,1,2,…,
where the confluent hypergeometric function 1F1(-n;a;z) defined in terms of the Pochhammer symbol (or Gamma function Γ(a)) (2.17)(a)k=Γ(a+k)Γ(a)={1,if(k=0,a∈C∖{0}),a(a+1)(a+2)⋯(a+k-1),if(k=N,a∈C),
as(2.18)1F1(-n;a;z)=∑k=0n(-n)kzk(a)kk!.
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)]).
Givenλ0≡λ0(x) and  s0≡s0(x) in  C∞, the differential equation
(2.19)f′′(x)=λ0(x)f′(x)+s0(x)f(x)
has the general solution
(2.20)f(x)=exp⁡⁡(-∫xα(t)dt)[C2+C1∫xexp⁡⁡(∫t(λ0(τ)+2α(τ))dτ)dt],
if for some n∈ℕ+={1,2,…}(2.21)snλn=sn-1λn-1=α(x),orδn(x)=λnsn-1-λn-1sn=0,
where
(2.22)λn=λn-1′+sn-1+λ0λn,sn=sn-1′+s0λn.For the differential equation (2.8) with(2.23)λ0(x)=-(-2wa2x2+2(l+1))x,s0(x)=-(2Ea2-wa2(2l+3)),
the first few  iterations with δn=λnsn-1-λn-1sn=0, using (2.20), imply(2.24)f0(x)=1,f1(x)=2wa2x2-(2l+3),f2(x)=4w2a4x4-4wa2(2l+5)x2+(2l+3)(2l+5),⋮
which we may easily generalized using the definition of the confluent hypergeometric function, (2.18), as(2.25)fn(x)=1F1(-n;l+32;wa2x2),
up to a constant.


## 2.2. Case  2

The second set of solutions(2.26)(g,μ)=(2(1+l+a2w)(3+2l+2a2w),-2(1+l+a2w))
allow us to write the differential equation (2.3) as(2.27)fn′′(x)+(2(l+1)x-8(l+1+a2w)xx2+1-2wa2x)fn′(x)+(2Ea2+wa2(6l+5+8wa2))fn(x)=0.
A further change of variable z=x2+1 allows us to write the differential equation (2.27) as(2.28)4z(z-1)f′′(z)-(4a2wz2+2(6l+5+6wa2)z-16(l+1+wa2))f′(z)+(2Ea2+wa2(6l+5+8wa2))zf(z)=0,
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=-2Ea2-wa2(6l+5+8wa2). Consequently, the polynomial solutions fn(x) of (2.28) are subject to the following two conditions: the necessary condition (2.10) reads(2.29)2Ena2=wa2(4n′-6l-5-8wa2),n′=0,1,2,…,
and the sufficient condition; namely, the vanishing of the tridiagonal determinant (2.12), reads (2.30)Δn+1=|β0α1γ1β1α2γ2β2α3⋱⋱⋱γn-2βn-2αn-1γn-1βn-1αnγnβn|=0,
where(2.31)βn=-2n(2n-6l-7-6wa2),αn=4n(n-4l-5-4a2w),γn=4wa2(n-n′-1),
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: (2.32)Δ2=0⟹a2w(l+1+a2w)=0,Δ3=0⟹a2w(l+1+a2w)(1+2l+2a2w)=0,Δ4=0⟹a2w(l+1+a2w)(1+2l+2a2w)(3(1+6l)+14a2w)=0,Δ5=0⟹a2w(l+1+a2w)(1+2l+2a2w)(3(6l-1)(6l+1)+4(38l+1)a2w+44a4w2)=0,Δ6=0⟹a2w(l+1+a2w)(1+2l+2a2w)×(3(2l-1)(6l-1)(6l+1)+2(208l2-54l-5)a2w+200la4w2)=0,⋮=⋮.
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 (2.33)Δn+1=(l+1+a2w)(1+2l+2a2w)×|β2α3γ3β3α4γ4β4α5⋱⋱⋱γn-2βn-2αn-1γn-1βn-1αnγnβn|=(l+1+a2w)(1+2l+2a2w)×Qn-1l(a2w),
where Qn-1l(a2w) are polynomials in the parameter product a2w.For physically acceptable solutions, we must havel=-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 Qnl(a2w)(2.34)Qn-1l=-1(a2w)={1,ifn=2,14a2w-15,ifn=3,44a4w2-148a2w+105,ifn=4,200a4w2-514a2w+315,ifn=5,⋮
give new values, not reported before, of a2w that yield quasiexact solutions of the Schrödinger equation (with one eigenstate)(2.35)-ψn′′(x)+[(wa2)2x2+4a2w(1+2a2w)(x2-1)(x2+1)2]ψn(x)=wa2(4n+1-8a2w)ψn(x),
where (2.36)ψn(x)=(x2+1)-2a2we-wa2x2/2fn(x),
and fn(x) are the solutions of(2.37)4z(z-1)f′′(z)-(4a2wz2+2(-1+6wa2)z-16wa2)f′(z)+4nwa2zf(z)=0,z=x2+1.
For example, Δ4=0 implies, using (2.34), that a2w=15/14, and thus, we have for(2.38)-ψ3′′(x)+[225196x2+66049(x2-1)(x2+1)2]ψ3(x)=46598ψ3(x),
the exact solution (2.39)ψ3(x)=(x2+1)-15/7e-(15/28)x2(45x6+225x4+315x2-49),
with a plot of the wave function and potential given in Figure 1.Figure 1
Plot of the unnormalized wave functionψ3(x) and the potential V3=(225/196)x2+(660/49)(x2-1)/(x2+1)2.Further,Δ5=0, equation (2.34) implies (2.40)a2w=3722±21422,
and we have for(2.41)-ψ4′′(x)+[(3722±21422)2x2+2(3711±21411)(4811±21411)(x2-1)(x2+1)2]ψ4(x)=(3722±21422)(3911∓421411)ψ4(x),
the exact solutions (2.42)ψ4±(x)=(x2+1)-((37/11)±(214/11))e-((37/44)±(214/44))x2×(1575x8+(9660±420214)x6+(26250±2100214)x4+(29820±2940214)x2-(1129±188214)).
Similar results can be obtained for Δn+1=0, for n≥5.


## 2.3. Exactly Solvable Quantum Isotonic Nonlinear Oscillator

As mentioned above, forl=-1 and a2w=1/2, it clear that Δn+1=0 for all n and the one-dimensional Schrödinger equation(2.43)[-d2dx2+x24+4(x2-1)(x2+1)2]ψn(x)=(2n-32)ψn(x),n=0,1,2,…
has the exact solutions(2.44)ψn(x)=(x2+1)-1e-x2/4fn(x),
where fn(x) are the polynomial solutions of the following second-order linear differential equation (z=x2+1)(2.45)4z(z-1)fn′′(z)-(2z2+4z-8)fn′(z)+2nzfn(z)=0.
By using AIM (Theorem 2.2, (2.20)), we find that the polynomial solutions fn(x) of (2.45) are given explicitly as(2.46)f0(x)=1,f1(x)=x2-2,f2(x)=x3-6x2+8,f3(x)=x4-16x3+52x2-52,f4(x)=x5-30x4+250x3-580x2+464,⋮
a set of polynomial solutions that can be generated using(2.47)f0(x)=1,fn(x)=-3x(2n+1)1F1(-n;32;12(x-1))+6((n+1)x-1)1F1(-n+1;32;12(x-1)),
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] as(2.48)f0(x)=1,fn(x)=3(-1)nπΓ(n)2Γ(n+3/2)×[((1+n)(x-1)2+n)Ln1/2(x-12)-(x-1)((1+n)x-1)Ln3/2(x-12)].


## 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 as(3.1)fn′′(z)+(2l+32z+2μz+1-wa2)fn′(z)+[2Ea2-wa2(2l+3+4μ)4z+μ(2l+3+2wa2)2z(z+1)-g2(z-1)z(z+1)2+μ(μ-1)(z+1)2]fn(z)=0.
By means of the Möbius transformation z=t/(1-t) that maps the singular points {-1,0,∞} into {0,1,∞}, we obtain(3.2)fn′′(t)+(2l+32t(1-t)+2(μ-1)1-t-wa2(1-t)2)fn′(t)+[μ(2l+3+2wa2)2t(1-t)2-g2(2t-1)t(1-t)2+μ(μ-1)(1-t)2]fn(t)=0,
where we assume that(3.3)2Ea2-(2l+3+4μ)wa2=0.
The differential equation (3.2) can be written as(3.4)(t3-2t2+t)fn′′(t)+[-2(μ-1)t2+(2μ-wa2-l-72)t+(l+32)]fn′(t)+[(μ(μ-1)-g)t+g2+μ(l+32+wa2)]fn(t)=0,
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(μ-1), a2,1=(2μ-wa2-l-7/2), a2,2=(l+3/2), τ1,0=-(μ(μ-1)-g), τ1,1=-g/2-μ(l+3/2+wa2). We, thus, conclude that the quasipolynomial solutions fn(t) of (3.4) are subject to the following conditions:(3.5)g=(μ-k)(μ-k-1),k=0,1,2,…,
along with the vanishing of the tridiagonal determinant Δn+1=0(3.6)|β0α1γ1β1α2γ2β2α3⋱⋱⋱γn-1βn-1αnγnβn|=0,
where(3.7)βn=-12(g+(μ-n)(3+2l+4n+2a2w)),αn=-n(n+l+12),γn=g-(μ-n+1)(μ-n).
Here, again, g=(μ-k)(μ-k-1) is fixed for given k=n, the fixed size of the determinant Δn+1.


### 3.1. Particular Case:n=0

Fork(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: (3.8)g+μ(3+2l+2a2w)=0,g=μ(μ-1).
Solving this system of equations for g and μ, we obtain the following values of(3.9)g=2(1+l+a2w)(3+2l+2a2w),μ=-2(l+1+wa2),
and the ground-state energy, in this case, is given by (3.3); namely,(3.10)Ea2=-12a2w(5+6l+8a2w),
which in complete agreement with the results of Section 2.2.


### 3.2. Particular Case:n=1

Fork(fixed)≡n=1, the determinant Δ2=0 of (3.7) yields(3.11)g2+g(-1+10μ+2l(2μ+1)+2a2w(2μ-1))+μ(μ-1)(15+4l2+8l(2+a2w)+4a2w(5+a2w))=0,g-(μ-1)(μ-2)=0,
where the energy is given by use of (3.3), for the computed values of μ and g, by(3.12)E=(l+32+2μ)w.
Further, (3.11) yields the solutions for l as functions of μ and a2w(3.13)l=2-(5+4a2w)μ-2μ2±4-4(3+8a2w)μ+9μ24μ≥-1,
where the energy states are now given by (3.12) along with l given by (3.13). We may also note that for(3.14)a2w=12(k+1),k=0,1,2,…,(3.15)a2E±=-18μ(k+1)(-2+(2k+1)μ-6μ2±4-4(4k+7)μ+9μ2).
Further, for g=(μ-1)(μ-2), we obtain the unnormalized wave function (see (2.2))(3.16)ψ1,l(x)=xl+1(1+x2)μ-1e-wa2x2/2(1+1+2l+μ+2a2w5+2l+μ+2a2wx2).
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 ifg 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.Table 1
Conditions on the value of the parametersg and μ for the quasipolynomial solutions in the case of Δ2=0 with different values of wa2 and l.
nlwa2ConditionsEn,lwa2≡En,lwa2(μ,g)1−11/2μ=1/3(-3-15A-1/3-A1/3),A=3(36-961)E1,-11/2=-w(3/2+2/3A1/3+10A-1/3)g=1/9A-2/3(15+6A1/3+A2/3)(15+9A1/3+A2/3)1μ=1/3(-5-19A-1/3-A1/3),A=161-32118E1,-11=-w(17/6+2/3A1/3+38/3A-1/3)g=1/9A-2/3(19+8A1/3+A2/3)(19+11A1/3+A2/3)3/2μ=1/3(-7-25A-1/3-A1/3),A=199-1874E1,-13/2=-w(25/6+2/3A1/3+50/3A-1/3)g=1/9A-2/3(25+10A1/3+A2/3)(25+13A1/3+A2/3)2μ=1/3(-9-33A-1/3-A1/3),A=3(72-1191)E1,-12=-w(11/2+2/3A1/3+22A-1/3)g=1/9A-2/3(33+12A1/3+A2/3)(33+15A1/3+A2/3)01/2μ=0E1,01/2=3/2wg=2μ=-1/2(7+17)E1,01/2=-1/2(11+217)wg=29+517μ=-1/2(7-17)E1,01/2=-1/2(11-217)wg=29-5171μ=-3+BE1,01=-(9/2-2B)wg=(-4+B)(-5+B)B=1/3ℜ(A1/3+33A-1/3),A=-108+3i2697μ=-3-B,E1,01=-(9/2+2B)wg=(5+B)(4+B)B=ℜ((11(1+i3)A-1/3/2)+((1-i3)A1/3/6)),A=-108+3i2697μ=-3-B,E1,01=-(9/2+2B)wg=(5+B)(4+B)B=ℜ(11(1-i3)A-1/3/2+((1+i3)A1/3/6)),A=-108+3i2697Table 2
Conditions on the value of the parametersg and μ for the quasipolynomial solutions in the case of Δ2=0 with different values of wa2 and l.
nlwa2ConditionsEn,lwa2≡En,lwa2(μ,g)03/2μ=-11/3+BE1,03/2=-1/6(35-12B)wg=(-14/3+B)(-17/3+B)B=1/3ℜ(A1/3+43A-1/3),A=-98+9i863μ=-11/3-B,E1,01=-1/6(35+12B)wg=1/9(17+3B)(14+3B)B=1/6ℜ(43(1+i3)A-1/3+(1-i3)A1/3),A=-98+9i863μ=-11/3-B,E1,01=-1/6(35+12B)wg=1/9(17+3B)(14+3B)B=1/6ℜ(43(1-i3)A-1/3+(1+i3)A1/3),A=-98+9i8632μ=-13/3+BE1,03/2=-1/6(43-12B)wg=1/9(-16+3B)(-19+3B)B=1/3ℜ(A1/3+55A-1/3),A=-55+165i6μ=-13/3-B,E1,01=-1/6(43+12B)wg=1/9(16+3B)(19+3B)B=1/6ℜ(55(1+i3)A-1/3+(1-i3)A1/3),A=-55+165i6μ=-13/3-B,E1,01=-1/6(43+12B)wg=1/9(16+3B)(19+3B)B=1/6ℜ(55(1-i3)A-1/3+(1+i3)A1/3),A=-55+165i6


#### 3.2.1. Particular Casen=2

Fork(fixed)≡n=2, the determinant Δ3=0 along with the necessary condition (3.7) yields(3.17)g3+3g2(7μ-1+2l(1+μ)+2a2w(μ-1))-g[18+56l+8l2+18(7+2l)μ-3(5+2l)(7+2l)μ2-12a2w(μ-1)((7+2l)μ-4)-4a4w2(2+3(μ-2)μ)]+μ(μ-2)(μ-1)(105+142l+60l2+8l3+6a2w(5+2l)(7+2l)+12a4w2(7+2l)+8a6w3)=0,g-(μ-2)(μ-3)=0,
where, again, the energy is given, for  the computed values of μ and g using (3.3) and (3.17), by (3.18)E=(l+32+2μ)w.
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 E2,lwa2≡E2,lwa2(μ,g).Table 3
Exact eigenvalues for different values ofl and wa2 in the case Δ3=0.
nlwa2ConditionsEn,l≡En,lwa2(μ,g)2−11/2μ1=-6.301870878994198E2,-11/2=-6.051870878994198g1=77.22293097048609μ2=-2.4855365082108594E2,-11/2=-2.2355365082108594g2=24.6055742747033331μ1=-7.398182984326876E2,-11=-7.148182984326876g1=97.7240263912181μ2=-3.3550579014968194E2,-11=-3.1050579014968194g2=34.03170302988033μ3=0.9498105417574756E2,-12=1.1998105417574756g3=2.15308735644625143/2μ1=-8.469623341124414E2,-13/2=-8.219623341124414g1=120.08263624614156μ2=-4.27750521216504E2,-11=-4.02750521216504g2=45.684576900924284μ3=0.9282653601757613E2,-11=1.1782653601757613g3=2.22034977802342942μ1=-9.525122115065386E2,-12=-9.275122115065383g1=144.35356188223463μ2=-5.226942179911145E2,-12=-4.976942179911145g2=59.45563545168999μ3=0.9186508169859244E2,-12=1.1686508169859244g3=2.250665238619284201/2μ1=-8.032243023438463E2,-12=-7.282243023438463g1=110.67814310476818μ2=-4.32825470612182E2,-1=-3.57825470612182g2=46.375062331674782μ1=-11.307737259773461E2,-12=-10.557737259773461g1=190.4036082349363μ2=-7.180564905703867E2,-12=-6.430564905703867g2=93.46333689354533μ3=0.9472009101393033E2,-12=1.6972009101393033g3=2.1611850134722084


## 3.1. Particular Case:n=0

Fork(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: (3.8)g+μ(3+2l+2a2w)=0,g=μ(μ-1).
Solving this system of equations for g and μ, we obtain the following values of(3.9)g=2(1+l+a2w)(3+2l+2a2w),μ=-2(l+1+wa2),
and the ground-state energy, in this case, is given by (3.3); namely,(3.10)Ea2=-12a2w(5+6l+8a2w),
which in complete agreement with the results of Section 2.2.


## 3.2. Particular Case:n=1

Fork(fixed)≡n=1, the determinant Δ2=0 of (3.7) yields(3.11)g2+g(-1+10μ+2l(2μ+1)+2a2w(2μ-1))+μ(μ-1)(15+4l2+8l(2+a2w)+4a2w(5+a2w))=0,g-(μ-1)(μ-2)=0,
where the energy is given by use of (3.3), for the computed values of μ and g, by(3.12)E=(l+32+2μ)w.
Further, (3.11) yields the solutions for l as functions of μ and a2w(3.13)l=2-(5+4a2w)μ-2μ2±4-4(3+8a2w)μ+9μ24μ≥-1,
where the energy states are now given by (3.12) along with l given by (3.13). We may also note that for(3.14)a2w=12(k+1),k=0,1,2,…,(3.15)a2E±=-18μ(k+1)(-2+(2k+1)μ-6μ2±4-4(4k+7)μ+9μ2).
Further, for g=(μ-1)(μ-2), we obtain the unnormalized wave function (see (2.2))(3.16)ψ1,l(x)=xl+1(1+x2)μ-1e-wa2x2/2(1+1+2l+μ+2a2w5+2l+μ+2a2wx2).
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 ifg 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.Table 1
Conditions on the value of the parametersg and μ for the quasipolynomial solutions in the case of Δ2=0 with different values of wa2 and l.
nlwa2ConditionsEn,lwa2≡En,lwa2(μ,g)1−11/2μ=1/3(-3-15A-1/3-A1/3),A=3(36-961)E1,-11/2=-w(3/2+2/3A1/3+10A-1/3)g=1/9A-2/3(15+6A1/3+A2/3)(15+9A1/3+A2/3)1μ=1/3(-5-19A-1/3-A1/3),A=161-32118E1,-11=-w(17/6+2/3A1/3+38/3A-1/3)g=1/9A-2/3(19+8A1/3+A2/3)(19+11A1/3+A2/3)3/2μ=1/3(-7-25A-1/3-A1/3),A=199-1874E1,-13/2=-w(25/6+2/3A1/3+50/3A-1/3)g=1/9A-2/3(25+10A1/3+A2/3)(25+13A1/3+A2/3)2μ=1/3(-9-33A-1/3-A1/3),A=3(72-1191)E1,-12=-w(11/2+2/3A1/3+22A-1/3)g=1/9A-2/3(33+12A1/3+A2/3)(33+15A1/3+A2/3)01/2μ=0E1,01/2=3/2wg=2μ=-1/2(7+17)E1,01/2=-1/2(11+217)wg=29+517μ=-1/2(7-17)E1,01/2=-1/2(11-217)wg=29-5171μ=-3+BE1,01=-(9/2-2B)wg=(-4+B)(-5+B)B=1/3ℜ(A1/3+33A-1/3),A=-108+3i2697μ=-3-B,E1,01=-(9/2+2B)wg=(5+B)(4+B)B=ℜ((11(1+i3)A-1/3/2)+((1-i3)A1/3/6)),A=-108+3i2697μ=-3-B,E1,01=-(9/2+2B)wg=(5+B)(4+B)B=ℜ(11(1-i3)A-1/3/2+((1+i3)A1/3/6)),A=-108+3i2697Table 2
Conditions on the value of the parametersg and μ for the quasipolynomial solutions in the case of Δ2=0 with different values of wa2 and l.
nlwa2ConditionsEn,lwa2≡En,lwa2(μ,g)03/2μ=-11/3+BE1,03/2=-1/6(35-12B)wg=(-14/3+B)(-17/3+B)B=1/3ℜ(A1/3+43A-1/3),A=-98+9i863μ=-11/3-B,E1,01=-1/6(35+12B)wg=1/9(17+3B)(14+3B)B=1/6ℜ(43(1+i3)A-1/3+(1-i3)A1/3),A=-98+9i863μ=-11/3-B,E1,01=-1/6(35+12B)wg=1/9(17+3B)(14+3B)B=1/6ℜ(43(1-i3)A-1/3+(1+i3)A1/3),A=-98+9i8632μ=-13/3+BE1,03/2=-1/6(43-12B)wg=1/9(-16+3B)(-19+3B)B=1/3ℜ(A1/3+55A-1/3),A=-55+165i6μ=-13/3-B,E1,01=-1/6(43+12B)wg=1/9(16+3B)(19+3B)B=1/6ℜ(55(1+i3)A-1/3+(1-i3)A1/3),A=-55+165i6μ=-13/3-B,E1,01=-1/6(43+12B)wg=1/9(16+3B)(19+3B)B=1/6ℜ(55(1-i3)A-1/3+(1+i3)A1/3),A=-55+165i6


### 3.2.1. Particular Casen=2

Fork(fixed)≡n=2, the determinant Δ3=0 along with the necessary condition (3.7) yields(3.17)g3+3g2(7μ-1+2l(1+μ)+2a2w(μ-1))-g[18+56l+8l2+18(7+2l)μ-3(5+2l)(7+2l)μ2-12a2w(μ-1)((7+2l)μ-4)-4a4w2(2+3(μ-2)μ)]+μ(μ-2)(μ-1)(105+142l+60l2+8l3+6a2w(5+2l)(7+2l)+12a4w2(7+2l)+8a6w3)=0,g-(μ-2)(μ-3)=0,
where, again, the energy is given, for  the computed values of μ and g using (3.3) and (3.17), by (3.18)E=(l+32+2μ)w.
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 E2,lwa2≡E2,lwa2(μ,g).Table 3
Exact eigenvalues for different values ofl and wa2 in the case Δ3=0.
nlwa2ConditionsEn,l≡En,lwa2(μ,g)2−11/2μ1=-6.301870878994198E2,-11/2=-6.051870878994198g1=77.22293097048609μ2=-2.4855365082108594E2,-11/2=-2.2355365082108594g2=24.6055742747033331μ1=-7.398182984326876E2,-11=-7.148182984326876g1=97.7240263912181μ2=-3.3550579014968194E2,-11=-3.1050579014968194g2=34.03170302988033μ3=0.9498105417574756E2,-12=1.1998105417574756g3=2.15308735644625143/2μ1=-8.469623341124414E2,-13/2=-8.219623341124414g1=120.08263624614156μ2=-4.27750521216504E2,-11=-4.02750521216504g2=45.684576900924284μ3=0.9282653601757613E2,-11=1.1782653601757613g3=2.22034977802342942μ1=-9.525122115065386E2,-12=-9.275122115065383g1=144.35356188223463μ2=-5.226942179911145E2,-12=-4.976942179911145g2=59.45563545168999μ3=0.9186508169859244E2,-12=1.1686508169859244g3=2.250665238619284201/2μ1=-8.032243023438463E2,-12=-7.282243023438463g1=110.67814310476818μ2=-4.32825470612182E2,-1=-3.57825470612182g2=46.375062331674782μ1=-11.307737259773461E2,-12=-10.557737259773461g1=190.4036082349363μ2=-7.180564905703867E2,-12=-6.430564905703867g2=93.46333689354533μ3=0.9472009101393033E2,-12=1.6972009101393033g3=2.1611850134722084


## 3.2.1. Particular Casen=2

Fork(fixed)≡n=2, the determinant Δ3=0 along with the necessary condition (3.7) yields(3.17)g3+3g2(7μ-1+2l(1+μ)+2a2w(μ-1))-g[18+56l+8l2+18(7+2l)μ-3(5+2l)(7+2l)μ2-12a2w(μ-1)((7+2l)μ-4)-4a4w2(2+3(μ-2)μ)]+μ(μ-2)(μ-1)(105+142l+60l2+8l3+6a2w(5+2l)(7+2l)+12a4w2(7+2l)+8a6w3)=0,g-(μ-2)(μ-3)=0,
where, again, the energy is given, for  the computed values of μ and g using (3.3) and (3.17), by (3.18)E=(l+32+2μ)w.
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 E2,lwa2≡E2,lwa2(μ,g).Table 3
Exact eigenvalues for different values ofl and wa2 in the case Δ3=0.
nlwa2ConditionsEn,l≡En,lwa2(μ,g)2−11/2μ1=-6.301870878994198E2,-11/2=-6.051870878994198g1=77.22293097048609μ2=-2.4855365082108594E2,-11/2=-2.2355365082108594g2=24.6055742747033331μ1=-7.398182984326876E2,-11=-7.148182984326876g1=97.7240263912181μ2=-3.3550579014968194E2,-11=-3.1050579014968194g2=34.03170302988033μ3=0.9498105417574756E2,-12=1.1998105417574756g3=2.15308735644625143/2μ1=-8.469623341124414E2,-13/2=-8.219623341124414g1=120.08263624614156μ2=-4.27750521216504E2,-11=-4.02750521216504g2=45.684576900924284μ3=0.9282653601757613E2,-11=1.1782653601757613g3=2.22034977802342942μ1=-9.525122115065386E2,-12=-9.275122115065383g1=144.35356188223463μ2=-5.226942179911145E2,-12=-4.976942179911145g2=59.45563545168999μ3=0.9186508169859244E2,-12=1.1686508169859244g3=2.250665238619284201/2μ1=-8.032243023438463E2,-12=-7.282243023438463g1=110.67814310476818μ2=-4.32825470612182E2,-1=-3.57825470612182g2=46.375062331674782μ1=-11.307737259773461E2,-12=-10.557737259773461g1=190.4036082349363μ2=-7.180564905703867E2,-12=-6.430564905703867g2=93.46333689354533μ3=0.9472009101393033E2,-12=1.6972009101393033g3=2.1611850134722084


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

For the potential parametersw, 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 by(4.1)λ0(t)=-(2l+32t(1-t)+2((Ea2/2wa2)-((2l+3)/4)-1)(1-t)-wa2(1-t)2),s0(t)=-(((Ea2/2wa2)-((2l+3)/4))(2l+3+2wa2)2t(1-t)2-g2(2t-1)t(1-t)2+((Ea2/2wa2)-((2l+3)/4))((Ea2/(2wa2))-((2l+3)/4)-1)(1-t)2),
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.Table 4
Energies of the four lowest states of the generalized isotonic oscillator of parametersw and a given for l=-1 as wa2=2 and for different values of the parameter g. The subscript numbers represents the number of iterations used by AIM.
wa2gE0a2E1a2E2a2E3a220.000 010.999 993 709 536(39)2.999 997 742 768(25)4.999 998 464 613(32)6.999 998 987 906(23)0.10.936 865 790 085(43)2.977 274 273 728(33)4.984 713 354 070(45)6.989 892 949 082(32)10.349 595 330 721(51)2.758 891 177 876(36)4.851 946 642 761(42)6.900 301 395 128(35)2−0.337 237 264 447(51)2.487 025 791 777(38)4.709 976 255 628(42)6.803 992 334 705(34)5−2.549 035 191 007(53)1.494 183 218 341(39)4.268 043 172 724(45)6.534 685 249 316(35)10−6.529 142 779 202(60)−0.660 939 314 881(40)3.318 493 978 272(46)6.100 400 048 017(38)12−8.182 546 155 166(65)−1.659 292 230 771(44)2.838 014 627 229(48)5.905 881 549 211(39)50−41.876 959 736 225(37)−26.863 072 307 493(33)−14.310 287 343 156(28)−4.206 192 073 796(31)


## 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 parametersg, w, and a.



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*Source: 750168-2011-06-13.xml*