Volume 2011, Article ID 750168,20pages doi:10.1155/2011/750168
Research Article
Study of the Generalized Quantum Isotonic Nonlinear Oscillator Potential
Nasser Saad,
1Richard L. Hall,
2Hakan C ¸ iftc¸i,
3and ¨ Ozlem Yes¸iltas¸
31Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PE, Canada C1A 4P3
2Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montr´eal, QC, Canada H3G 1M8
3Department of Physics, Faculty of Arts and Sciences, Gazi University, 06500 Ankara, Turkey
Correspondence should be addressed to Nasser Saad,nsaad@upei.ca Received 11 March 2011; Accepted 13 April 2011
Academic Editor: B. G. Konopelchenko
Copyrightq2011 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 byH −d2/dr2ll 1/r2w2r22gr2−a2/r2a22,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 parametersg,w, and amay be found to high accuracy.
1. Introduction
Recently, Cari ˜nena et al. 1 studied a quantum nonlinear oscillator potential whose Schr ¨odinger equation reads
− d2
dx2 x28 2x2−1 2x212
ψnx Enψx. 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, Cari ˜nena et al.1 were able to show that
ψnx Pnx
2x21e−x2/2, En−32n, n0,3,4,5, . . . ,
1.2
where the polynomials factorsPnxare related to the Hermite polynomials by means of
Pnx
⎧⎨
⎩
1, ifn0,
Hnx 4nHn−2x 4nn−3Hn−4x, ifn3,4,5, . . . .
1.3
In a more recent work, Fellows and Smith2showed that the potentialVx x282x2− 1/2x212as well as, for certain values of the parametersw,g, anda, the potentialVx w2x22gx2−a2/x2a22of the Schr ¨odinger equation
− d2
dx2 w2x22g x2−a2 x2a22
ψnx 2Enψx 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, Sesma3, using a M ¨obius transformation, was able to transform1.4into a confluent Heun equation4, and thereby obtain an efficient algorithm to solve the Schr ¨odinger equation1.4numerically.
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 method5–8, for the Schr ¨odinger equation
−d2
dr2 ll1
r2 w2r22g r2−a2 r2a22
ψr 2Eψr, 1.5
wherelis the angular momentum numberl−1,0,1, . . .. Our results show that the quasiexact solutions of Sesma3as well the results of Cari ˜nena et al.1follow as special cases of our general approach. The present paper is organized as follows. In Section 2, some preliminary analysis of the Schr ¨odinger equation 1.5 is presented. A general approach for finding polynomial solutions of 1.5, for certain values of parametersw and g, is presented and is based on a recent work of Ciftci et al.6for solving the second-order linear differential equation
3
i0
a3,ixi y 2
i0
a2,ixi y− 1
i0
τ1,ixi y0. 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 Section4.
2. Generalized Quantum Isotonic Oscillator—Preliminary Results
A simple scaling argument, usingra2x, allows us to write1.5as
− d2
dx2 ll1 x2
wa22
x22g x2−1 x212
ψx 2Ea2ψx. 2.1
A further substitutionzx21 yields a differential equation with two regular singular points atz 0,1 and one irregular singular point of rank 2 atz ∞. The rootsμ’s of the indicial equation for the regular singular pointz 0 readsμ± 1/21±
14g, while the roots of the indicial equation atz 1 areμ l1/2 andμ− −l/2. Since the singularity for z → ∞corresponds to that forx → ∞, it is necessary that the solution forz → ∞behave asψx∼exp−wa2x2/2. Consequently, we may assume the general solution of2.1which vanishes at the origin and at infinity takes the form
ψnx xl1
x21μ
e−wa2/2x2fnx. 2.2
A straightforward calculation shows thatfnxare the solutions of the second-order homo- geneous linear differential equation
fx
2l1
x 4μx
x21−2wa2x
fx
2Ea2−wa2
2l34μ 2μ
2l32wa2 4μ
μ−1
−2g
x21 4
g−μ μ−1 x212
×fx 0.
2.3
In the next sections, we attempt to give a general solution of this equation. For now, we assume thatμtakes the value of the indicial root
μ≡μ− 1 2
1−
14g
, 2.4
which allows us to write2.3as
fnx
2l1
x 4μx
x21 −2wa2x
fnx
2Ea2−wa2
2l34μ 2μ
2l32wa2 2μ
μ−1 x21
fnx 0.
2.5
We now consider the cases where the following two equations are satisfied:
2μ
2l32wa2 2μ
μ−1 0, g μ
μ−1 .
2.6
The solutions of this system, forgandμ, are given explicitly by g0, or g 2
1la2w
32l2a2w , μ0, or μ −2
1la2w .
2.7
Next, we consider each case of these two sets of solutions.
2.1. Case 1
The first set of solutionsg, μ 0,0reduces the differential equation2.3to xfnx
−2wa2x22l1
fnx
2Ea2−wa22l3
xfnx 0, 2.8 which is a special case of the general differential equation
a3,0x3a3,1x2a3,2xa3,3 y
a2,0x2a2,1xa2,2
y−τ1,0xτ1,1y0, 2.9 with a3,0 a3,1 a3,3 a2,1 τ1,1 0, a3,2 1,a2,0 −2wa2,a2,2 2l1, and τ1,0
−2Ea2wa22l3. The necessary and sufficient conditions for polynomial solutions of2.9 are given by the following theorem6.
Theorem 2.1. The second-order linear differential equation2.9has a polynomial solution of degree nif
τ1,0nn−1a3,0na2,0, n0,1,2, . . . , 2.10 along with the vanishing ofn1×n1-determinantΔn1given by
Δn1
β0 α1 η1
γ1 β1 α2 η2
γ2 β2 α3 η3
. .. ... ... ...
γn−2 βn−2 αn−1 ηn−1 γn−1 βn−1 αn
γn βn
0, 2.11
where
βnτ1,1−nn−1a3,1a2,1, αn−nn−1a3,2a2,2, γnτ1,0−n−1n−2a3,0a2,0,
ηn−nn1a3,3,
2.12
andτ1,0is fixed for a givennin the determinantΔn10.
Thus, the necessary condition for the differential equation 2.8to have polynomial solutionsfnx n
i0cixiis
2Ena2wa2
2n2l3
, n0,1,2, . . ., 2.13
while the sufficient condition,2.12, is
Δn1
0 α1 0 0 γ1 0 α2 0
γ2 0 α3 0 . .. ... ... ...
γn−2 0 αn−1 0 γn−1 0 αn
γn 0
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
0, ifn0,2,4, . . . ,
n−1/2
j0
−12j1α2j1γ2j10, ifn1,3,5, . . . .,
2.14
whereβn0,αn−nn2l1andγn2wa2n−n−1.
Ifl −1, the determinantΔn1 is identically zero for alln, which is equivalent to the exact solutions of the one-dimensional harmonic oscillator problem.
Forl / −1, we have forn 0,2,4, . . .,Δn1 ≡ 0, and we obtain the exact solutions of the Gol’dman and Krivchenkovor IsotonicHamiltonianH0, where
H0ψnlx≡
− d2
dx2 ll1
x2 w2a4x2
ψnlx 2Eg0nl a2ψnlx, 0≤x <∞. 2.15
These exact solutions are given by9
2a2Eg0nl wa24n2l3, n0,1,2, . . . , ψnlx xl1e−wa2x2/21F1
−n;l3 2;wa2x2
, n0,1,2, . . . ,
2.16
where the confluent hypergeometric function1F1−n;a;zdefined in terms of the Pochham- mer symbolor Gamma functionΓa
ak Γak Γa
⎧⎨
⎩
1, ifk0, a∈ \ {0},
aa1a2· · ·ak−1, ifk, a∈ , 2.17 as
1F1−n;a;z n
k0
−nkzk
akk! . 2.18
The polynomial solutions fnx 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 2.2Ciftci et al.7, equations2.13-2.14. Givenλ0 ≡λ0xands0≡s0xinC∞, the differential equation
fx λ0xfx s0xfx 2.19
has the general solution
fx exp
− x
αtdt
C2C1
x exp
t
λ0τ 2ατdτ dt
, 2.20
if for somen∈ {1,2, . . .}
sn λn sn−1
λn−1 αx, or δnx λnsn−1−λn−1sn0, 2.21 where
λnλn−1sn−1λ0λn, snsn−1s0λn.
2.22
For the differential equation2.8with
λ0x −
−2wa2x22l1
x ,
s0x −
2Ea2−wa22l3 ,
2.23
the first few iterations withδnλnsn−1−λn−1sn0, using2.20, imply f0x 1,
f1x 2wa2x2−2l3,
f2x 4w2a4x4−4wa22l5x2 2l32l5, ...
2.24
which we may easily generalized using the definition of the confluent hypergeometric function,2.18, as
fnx 1F1
−n;l3 2;wa2x2
, 2.25
up to a constant.
2.2. Case 2
The second set of solutions g, μ
2
1la2w
32l2a2w ,−2
1la2w
2.26
allow us to write the differential equation2.3as
fnx
2l1 x −8
l1a2w x
x21 −2wa2x fnx
2Ea2wa2
6l58wa2
fnx 0.
2.27
A further change of variablezx21 allows us to write the differential equation2.27as 4zz−1fz−
4a2wz22
6l56wa2 z−16
l1wa2 fz
2Ea2wa2
6l58wa2
zfz 0,
2.28
Again, 2.28is a special case of the differential equation2.9with a3,0 a3,3 τ1,1 0, a3,1 4,a3,2 −4,a2,0 −4wa2,a2,1 −26l56wa2,a2,2 16l1wa2, andτ1,0
−2Ea2−wa26l58wa2. Consequently, the polynomial solutionsfnxof2.28are subject to the following two conditions: the necessary condition2.10reads
2Ena2wa2
4n−6l−5−8wa2
, n0,1,2, . . ., 2.29
and the sufficient condition; namely, the vanishing of the tridiagonal determinant 2.12, reads
Δn1
β0 α1
γ1 β1 α2 γ2 β2 α3
. .. ... ...
γn−2 βn−2 αn−1 γn−1 βn−1 αn
γn βn
0, 2.30
where
βn−2n
2n−6l−7−6wa2 , αn 4n
n−4l−5−4a2w , γn4wa2
n−n−1 ,
2.31
and n n is fixed for the given dimension of the determinantΔn1. From the sufficient condition2.31, we obtain the following conditions on the parameters:
Δ20⇒a2w
l1a2w 0,
Δ30⇒a2w
l1a2w
12l2a2w 0, Δ40⇒a2w
l1a2w
12l2a2w
316l 14a2w 0,
Δ50⇒a2w
l1a2w
12l2a2w
36l−16l1 438l1a2w44a4w2 0, Δ60⇒a2w
l1a2w
12l2a2w
×
32l−16l−16l1 2
208l2−54l−5
a2w200la4w2 0, ....
2.32
For a physically meaningful solution, we must havea2w > 0. This is possible for a very restricted value of the angular momentum numberl. Sinceβ00, we may observe that
Δn1
l1a2w
12l2a2w
×
β2 α3
γ3 β3 α4 γ4 β4 α5
. .. ... ...
γn−2 βn−2 αn−1 γn−1 βn−1 αn
γn βn
l1a2w
12l2a2w
×Qln−1 a2w
,
2.33
whereQln−1a2ware polynomials in the parameter producta2w.
For physically acceptable solutions, we must havel −1 and the factorl1a2w yieldsa2w 0, which is not physically acceptable, so we ignore it. The second factor1 2l2a2wimplies a special value ofa2w1/2, for alln, which we will study shortly in full detail. Meanwhile, the polynomialsQnla2w
Ql−1n−1
a2w
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩
1, ifn2,
14a2w−15, ifn3,
44a4w2−148a2w105, ifn4, 200a4w2−514a2w315, ifn5, ...
2.34
give new values, not reported before, of a2w that yield quasiexact solutions of the Schr ¨odinger equationwith one eigenstate
−ψnx wa22
x24a2w
12a2w x2−1 x212
ψnx wa2
4n1−8a2w ψnx,
2.35
where
ψnx
x21−2a2w
e−wa2x2/2fnx, 2.36
−8 −6 −4 −2 0 2 4 6 8 x
−40
−20 0 20 40 60
Ψ
Wave function Potential
Figure 1: Plot of the unnormalized wave functionψ3xand the potentialV3 225/196x2660/49x2− 1/x212.
andfnxare the solutions of 4zz−1fz−
4a2wz22
−16wa2
z−16wa2
fz 4nwa2zfz 0, zx21.
2.37
For example,Δ40 implies, using2.34, thata2w15/14, and thus, we have for
−ψ3x 225
196x2660 49
x2−1 x212
ψ3x 465
98ψ3x, 2.38
the exact solution
ψ3x
x21−15/7
e−15/28x2
45x6225x4315x2−49
, 2.39
with a plot of the wave function and potential given in Figure1.
Further,Δ50, equation2.34implies
a2w 37 22±
√214
22 , 2.40
and we have for
−ψ4x
⎡
⎣ 37
22±
√214 22
2
x22 37
11±
√214 11
48 11±
√214 11
x2−1 x212
⎤
⎦ψ4x
37
22±
√214 22
39 11∓4√
214
11 ψ4x,
2.41
the exact solutions
ψ4±x
x21−37/11±√214/11
e−37/44±√214/44x2
×
1575x8
9660±420√ 214
x6
26250±2100√ 214
x4
29820±2940√ 214
x2
−
1129±188√ 214
.
2.42
Similar results can be obtained forΔn10, forn≥5.
2.3. Exactly Solvable Quantum Isotonic Nonlinear Oscillator
As mentioned above, forl −1 anda2w 1/2, it clear thatΔn1 0 for allnand the one- dimensional Schr ¨odinger equation
− d2 dx2 x2
4 4 x2−1 x212
ψnx
2n−3
2
ψnx, n0,1,2, . . . 2.43
has the exact solutions
ψnx
x21−1
e−x2/4fnx, 2.44
where fnx are the polynomial solutions of the following second-order linear differential equationzx21
4zz−1fnz−
2z24z−8
fnz 2nzfnz 0. 2.45
By using AIMTheorem2.2,2.20, we find that the polynomial solutionsfnxof2.45are given explicitly as
f0x 1, f1x x2−2, f2x x3−6x28,
f3x x4−16x352x2−52,
f4x x5−30x4250x3−580x2464, ...
2.46
a set of polynomial solutions that can be generated using
f0x 1, fnx −3x2n11F1
−n;3 2;1
2x−1
6n1x−11F1
−n1;3 2;1
2x−1
, 2.47
up to a constant factor, where, again, 1F1 refers to the confluent hypergeometric function defined by2.18. Note that the polynomialsfnxin2.47can be expressed in terms of the associated Laguerre polynomials10as
f0x 1, fnx 3−1n√
πΓn 2Γn3/2
×
1nx−12n L1/2n
x−1 2
−x−11nx−1L3/2n x−1
2
.
2.48
3. Quasipolynomial Solutions of the Generalized Quantum Isotonic Oscillator
In this section, we study the quasipolynomial solutions of the differential equation2.3. We note first, using the change of variablezx2, equation2.3can be written as
fnz
2l3
2z 2μ
z1−wa2
fnz
2Ea2−wa2
2l34μ
4z μ
2l32wa2
2zz1 −g
2
z−1 zz12 μ
μ−1 z12
fnz 0.
3.1
By means of the M ¨obius transformationzt/1−tthat maps the singular points{−1,0,∞}
into{0,1,∞}, we obtain
fnt
2l3 2t1−t2
μ−1
1−t − wa2
1−t2 fnt
μ
2l32wa2 2t1−t2 −g
2
2t−1 t1−t2 μ
μ−1 1−t2
fnt 0,
3.2
where we assume that
2Ea2−
2l34μ
wa20. 3.3
The differential equation3.2can be written as
t3−2t2t fnt
−2 μ−1
t2
2μ−wa2−l−7 2
t
l3
2
fnt
μ
μ−1
−g t g
2 μ
l3 2 wa2
fnt 0,
3.4
which we may now compare with equation2.9in Theorem2.1with a3,0 1, a3,1 −2, a3,2 1,a3,3 0,a2,0−2μ−1,a2,1 2μ−wa2−l−7/2,a2,2 l3/2,τ1,0−μμ−1−g, τ1,1−g/2−μl3/2wa2. We, thus, conclude that the quasipolynomial solutionsfntof 3.4are subject to the following conditions:
g μ−k
μ−k−1
, k0,1,2, . . ., 3.5
along with the vanishing of the tridiagonal determinantΔn10
β0 α1
γ1 β1 α2
γ2 β2 α3 . .. ... ...
γn−1 βn−1 αn
γn βn
0, 3.6
where
βn−1 2
g μ−n
32l4n2a2w , αn−n
nl1 2
, γng−
μ−n1 μ−n
.
3.7
Here, again,g μ−kμ−k−1is fixed for givenk n, the fixed size of the determinant Δn1.
3.1. Particular Case:n0
Forkfixed≡n0, the differential equation3.4has the exact solutionf0t 1 ifgandμ satisfy, simultaneously, the following system of equations:
gμ
32l2a2w
0, gμ μ−1
. 3.8
Solving this system of equations forgandμ, we obtain the following values of g2
1la2w
32l2a2w
, μ−2
l1wa2
, 3.9
and the ground-state energy, in this case, is given by3.3; namely,
Ea2 −1 2a2w
56l8a2w
, 3.10
which in complete agreement with the results of Section2.2.
3.2. Particular Case:n1
Forkfixed≡n1, the determinantΔ20 of3.7yields g2g
−110μ2l 2μ1
2a2w
2μ−1 μ
μ−1
154l28l
2a2w
4a2w
5a2w 0, g−
μ−1 μ−2
0,
3.11
where the energy is given by use of3.3, for the computed values ofμandg, by E
l 3
22μ
w. 3.12
Further,3.11yields the solutions forlas functions ofμanda2w
l 2−
54a2w
μ−2μ2±
4−438a2wμ9μ2
4μ ≥ −1, 3.13
where the energy states are now given by3.12along withlgiven by3.13. We may also note that for
a2w 1
2k1, k0,1,2, . . ., 3.14
a2E±− 1 8μk1
−2 2k1μ−6μ2±
4−44k7μ9μ2
. 3.15
Further, forg μ−1μ−2, we obtain the unnormalized wave functionsee2.2
ψ1,lx xl1
1x2μ−1
e−wa2x2/2
112lμ2a2w
52lμ2a2wx2 . 3.16 Thus, we may summarize these results as follows. The exact solutions of the Schr ¨odinger equation2.1are given by3.15and3.16only if g andμare the solutions of the system given by3.11. In Tables1and 2, we report few quasiexact solutions that can be obtained using this approach.
3.2.1. Particular Casen2
Forkfixed≡n2, the determinantΔ30 along with the necessary condition3.7yields g33g2
7μ−12l 1μ
2a2w μ−1
−g
1856l8l21872lμ−352l72lμ2−12a2w μ−1
72lμ−4
−4a4w2 23
μ−2 μ μ
μ−2 μ−1
105142l60l28l36a2w52l72l 12a4w272l 8a6w3
0, g−
μ−2 μ−3
0,
3.17
where, again, the energy is given, for the computed values ofμandgusing3.3and3.17, by
E
l 3 22μ
w. 3.18
Table1:ConditionsonthevalueoftheparametersgandμforthequasipolynomialsolutionsinthecaseofΔ20withdifferentvaluesofwa2andl. nlwa2ConditionsEwa2 n,l≡Ewa2 n,lμ,g 1−11/2μ1/3−3−15A−1/3−A1/3,A336−√ 961E1/2 1,−1−w3/22/3A1/310A−1/3 g1/9A−2/3156A1/3A2/3159A1/3A2/3 1μ1/3−5−19A−1/3−A1/3,A161−3√ 2118 E1 1,−1−w17/62/3A1/338/3A−1/3 g1/9A−2/3198A1/3A2/31911A1/3A2/3 3/2μ1/3−7−25A−1/3−A1/3,A199−18√ 74 E3/2 1,−1−w25/62/3A1/350/3A−1/3 g1/9A−2/32510A1/3A2/32513A1/3A2/3 2μ1/3−9−33A−1/3−A1/3,A372−√ 1191 E2 1,−1−w11/22/3A1/322A−1/3 g1/9A−2/33312A1/3A2/33315A1/3A2/3 01/2μ0 E1/2 1,03/2w g2 μ−1/27√ 17 E1/2 1,0−1/2112√ 17w g295√ 17 μ−1/27−√ 17 E1/2 1,0−1/211−2√ 17w g29−5√ 17 1μ−3B E1 1,0−9/2−2Bwg−4B−5B B1/3ÊA1/333A−1/3,A−1083i√ 2697 μ−3−B, E1 1,0−9/22Bwg5B4B BÊ111i√ 3A−1/3/21−i√ 3A1/3/6,A−1083i√ 2697 μ−3−B, E1 1,0−9/22Bwg5B4B BÊ111−i√ 3A−1/3/21i√ 3A1/3/6,A−1083i√ 2697
Table 2: Conditions on the value of the parametersgandμfor the quasipolynomial solutions in the case ofΔ20 with different values ofwa2andl.
n l wa2 Conditions Ewan,l2 ≡Ewan,l2μ, g
0 3/2
μ−11/3B
E3/21,0 −1/635−12Bw g −14/3B−17/3B
B1/3ÊA1/343A−1/3, A−989i√ 863
μ−11/3−B,
E11,0−1/63512Bw g1/9173B143B
B1/6Ê431i√
3A−1/3 1−i√
3A1/3, A−989i√ 863
μ−11/3−B,
E11,0−1/63512Bw g1/9173B143B
B1/6Ê431−i√
3A−1/3 1i√
3A1/3, A−989i√ 863
2
μ−13/3B
E3/21,0 −1/643−12Bw g1/9−163B−193B
B1/3ÊA1/355A−1/3, A−55165i√ 6
μ−13/3−B,
E11,0−1/64312Bw g1/9163B193B
B1/6Ê551i√
3A−1/3 1−i√
3A1/3, A−55165i√ 6
μ−13/3−B,
E11,0−1/64312Bw g1/9163B193B
B1/6Ê551−i√
3A−1/3 1i√
3A1/3, A−55165i√ 6
In Table3, we report the numerical results for some of the exact solutions ofμandg using 3.17and the values ofl, wa2 −1,1/2,l, wa2 −1,1,l, wa2 −1,3/2,l, wa2
−1,2, l, wa2 0,1/2, andl, wa2 0,2, respectively. We have also computed the corresponding eigenvaluesEwa2,l2≡Ewa2,l2μ, g.
4. Numerical Computation by the Use of the Asymptotic Iteration Method
For the potential parametersw,a2, andg, not necessarily obeying the conditions for quasi- polynomial solutions discussed in the previous sections, the asymptotic iteration method can
Table 3: Exact eigenvalues for different values oflandwa2in the caseΔ30.
n l wa2 Conditions En,l≡Ewan,l2μ, g
2 −1 1/2 μ1−6.301870878994198 E1/22,−1−6.051870878994198 g177.22293097048609
μ2−2.4855365082108594 E1/22,−1−2.2355365082108594 g224.605574274703333
1 μ1−7.398182984326876 E12,−1−7.148182984326876 g197.7240263912181
μ2−3.3550579014968194 E12,−1−3.1050579014968194 g234.03170302988033
μ30.9498105417574756 E2,−12 1.1998105417574756 g32.1530873564462514
3/2 μ1−8.469623341124414
E3/22,−1−8.219623341124414 g1120.08263624614156
μ2−4.27750521216504 E12,−1−4.02750521216504 g245.684576900924284
μ30.9282653601757613 E2,−11 1.1782653601757613 g32.2203497780234294
2 μ1−9.525122115065386 E22,−1−9.275122115065383 g1144.35356188223463
μ2−5.226942179911145 E22,−1−4.976942179911145 g259.45563545168999
μ30.9186508169859244
E2,−12 1.1686508169859244 g32.250665238619284
2 0 1/2 μ1−8.032243023438463 E22,−1−7.282243023438463 g1110.67814310476818
μ2−4.32825470612182 E2,−1−3.57825470612182 g246.37506233167478
2 μ1−11.307737259773461 E22,−1−10.557737259773461 g1190.4036082349363
μ2−7.180564905703867 E22,−1−6.430564905703867 g293.46333689354533
μ30.9472009101393033 E2,−12 1.6972009101393033 g32.1611850134722084
be employed to compute the eigenvalues of Schr ¨odinger equation2.1for arbitrary values w,a2, andg. The functionsλ0ands0, using3.2and3.3, are given by
λ0t −
2l3 2t1−t2
Ea2/2wa2
−2l3/4−1
1−t − wa2 1−t2 , s0t −
Ea2/2wa2
−2l3/4
2l32wa2
2t1−t2 −g
2
2t−1 t1−t2
Ea2/2wa2
−2l3/4 Ea2/
2wa2
−2l3/4−1 1−t2 ,
4.1
Table 4: Energies of the four lowest states of the generalized isotonic oscillator of parameterswanda given forl−1 aswa22 and for different values of the parameterg. The subscript numbers represents the number of iterations used by AIM.
wa2 g E0a2 E1a2 E2a2 E3a2
2 0.000 01 0.999 993 709 53639 2.999 997 742 76825 4.999 998 464 61332 6.999 998 987 90623 0.1 0.936 865 790 08543 2.977 274 273 72833 4.984 713 354 07045 6.989 892 949 08232 1 0.349 595 330 72151 2.758 891 177 87636 4.851 946 642 76142 6.900 301 395 12835 2 −0.337 237 264 44751 2.487 025 791 77738 4.709 976 255 62842 6.803 992 334 70534 5 −2.549 035 191 00753 1.494 183 218 34139 4.268 043 172 72445 6.534 685 249 31635
10 −6.529 142 779 20260 −0.660 939 314 88140 3.318 493 978 27246 6.100 400 048 01738 12 −8.182 546 155 16665 −1.659 292 230 77144 2.838 014 627 22948 5.905 881 549 21139 50 −41.876 959 736 22537 −26.863 072 307 49333 −14.310 287 343 15628 −4.206 192 073 79631
where t ∈ 0,1. The AIM sequence λnx and snx can be calculated iteratively using the iterative sequences2.22. 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 condition2.21produces for each iteration an expression that depends on bothtandEfor given values of the parameterswa2, g, andl. In such a case, one faces the problem of finding the best possible starting valuett0
that stabilizes the AIM process5. Fortunately, since t ∈ 0,1, the starting valuet0 does not represent a serious issue in our eigenvalue calculation using4.1 and the termination condition2.21in contrast to the case of computing the eigenvalues usingλ0xands0xas given by, for example,2.3, wherex∈0,∞. In Table4, we report our numerical results for energies of the four lowest states of the generalized isotonic oscillator of parameterswand asuch thatwa2 2 and for different values ofg. In this table, we setl −1 for computing the energiesE0a2 andE2a2, while we putl 0 for computing the energiesE1a2 and E3a2, respectively. For most of these values, the starting value oftist00.5 and is shifted towards zero asg gets larger in value. For the values ofgthat admit a quasipolynomial solution, the number of iteration does not exceed three. For most of the other values ofg, the total number of iteration did not exceed 65. We found that for wa2 2 and the values ofg reported in Table4, the number of iteration is relatively small compared to the case ofwa2 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 Schr ¨odiger’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 cer- tain 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, anda.
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.
References
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