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Bound states for spiked harmonic oscillators and truncated Coulomb potentials

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1999 J. Phys. B: At. Mol. Opt. Phys. 32 3055

(http://iopscience.iop.org/0953-4075/32/12/321)

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J. Phys. B: At. Mol. Opt. Phys.32 (1999) 3055–3063. Printed in the UK PII: S0953-4075(99)02178-1

Bound states for spiked harmonic oscillators and truncatedCoulomb potentials

Omar Mustafa and Maen OdehDepartment of Physics, Eastern Mediterranean University, G Magusa, North Cyprus, Mersin 10,Turkey

E-mail: omustafa.as@mozart.emu.edu.tr

Received 25 February 1999

Abstract. We propose a new analytical method to solve for the nonexactly solvable Schrodingerequation. Successfully, it is applied to a class of spiked harmonic oscillators and truncated Coulombpotentials. The utility of this method could be extended to study other systems of atomic, molecularand nuclear physics interest.

In atomic, molecular and nuclear physics, spiked harmonic oscillators and truncated Coulombpotentials are of significant interest. Realistic interaction potentials often have a repulsive core[1–5]. The simplest model of such a core is provided by the spiked harmonic oscillators

V (q) = c1q2 + c2q

−b, c1, c2, b > 0. (1)

On the other hand, the truncated Coulomb potential has been found to be pertinent to the studyof the energy levels of the hydrogen-like atoms exposed to intense laser radiation [6–11]. Ithas been shown [8, 10] that under Kramers–Henneberger transformation [12] the laser-dressedbinding potential for the hydrogenic system, often called the laser-dressed Coulomb potential,may be well simulated by

V (q) = − e2

(q2 + c2)1/2; c > 0, (2)

where the truncation parameterc is related to the strength of the irradiating laser field.Thus, it is interesting to carry out systematic studies of the bound states of these potentials.

Hall and Saad [4] have studied the spiked harmonic oscillator potentials via the smoothtransformation method (STM) of the exactly solvable potentialV (q) = c2q

2 + c2q−2 to

obtain lower and/or upper energy bounds. They have also calculated the energy eigenvaluesusing direct numerical integrations of Schrodinger’s equation [4]. Duttet al [6] have useda shifted 1/N expansion technique (SLNT) to obtain the energy levels of the laser-dressedCoulomb potential and compared their results with those of direct numerical integration (DNI)[7]. Nevertheless, neither SLNT nor STM is utilitarian in terms of calculating the eigenvaluesand eigenfunctions in one batch. Because of the complexity in handling large-order correctionsof the standard Rayleigh–Schrodinger perturbation theory, only low-order calculations havebeen reported for SLNT [6, 13] and large-order calculations have been neglected. Eventually,the results of SLNT are not as accurate as sought after.

0953-4075/99/123055+09$19.50 © 1999 IOP Publishing Ltd 3055

3056 O Mustafa and M Odeh

In this paper we formulate a method for solving the Schrodinger equation. In one batch,one should be able to study not only the eigenvalues but also the eigenfunctions. It simplyconsists of using 1/l as a perturbation expansion parameter, wherel = l − β, l is a quantumnumber, andβ is a suitable shift introduced to avoid the trivial casel = 0. Hence, hereafter,it will be referred to as the pseudoperturbative shifted-l expansion technique (PSLET).

The construction of our method starts with the time-independent one-dimensional formof the Schrodinger equation, in ¯h = m = 1 units,[

−1

2

d2

dq2+l(l + 1)

2q2+ V (q)

]9nr,l(q) = Enr,l9nr ,l(q), (3)

where the quantum numberl may specify parity,(−1)l+1, in one dimension (l = −1 or l = 0,andq ∈ (−∞,∞)) or angular momentum in three dimensions (l = 0, 1, . . . , andq ∈ (0,∞)),andnr = 0, 1, . . . counts the nodal zeros [6, 14–17].

To avoid the trivial casel = 0, the quantum numberl is shifted through the relationl = l − β. Equation (3) thus becomes{

−1

2

d2

dq2+ V (q)

}9nr,l(q) = Enr,l9nr ,l(q), (4)

V (q) = l2 + (2β + 1)l + β(β + 1)

2q2+l2

QV (q). (5)

Note thatQ is a constant that scales the potentialV (q) at large-l limit and is set, for any specificchoice ofl andnr , equal tol2 at the end of the calculations [13, 14].β is to be determinedlater.

Our systematic procedure begins with shifting the origin of the coordinate through

x = l1/2(q − q0)/q0, (6)

whereq0 is currently an arbitrary point to perform Taylor expansion about, with its particularvalue to be determined. Expansions about this point,x = 0 (i.e.q = q0), yield

1

q2=∞∑n=0

(−1)n(n + 1)

q20

xnl−n/2, (7)

V (x(q)) =∞∑n=0

(dnV (q0)

dqn0

)(q0x)

n

n!l−n/2. (8)

Obviously, the expansions in (7) and (8) centre the problem at an arbitrary pointq0 and thederivatives, in effect, contain information not only atq0 but also at any point on the axis, inaccordance with Taylor’s theorem. Also, it should be mentioned that the scaled coordinate,equation (6), has no effect on the energy eigenvalues, which are coordinate-independent. Itjust facilitates the calculations of both the energy eigenvalues and eigenfunctions. It is alsoconvenient to expandE as

Enr,l =∞∑

n=−2

E(n)nr ,ll−n. (9)

Equation (4) thus becomes[−1

2

d2

dx2+q2

0

lV (x(q))

]9nr,l(x) =

q20

lEnr ,l9nr ,l(x), (10)

withq2

0

lV (x(q)) = q2

0 l

[1

2q20

+V (q0)

Q

]+ l1/2

[−x +

V ′(q0)q30x

Q

]

Spiked harmonic oscillators and truncated Coulomb potentials 3057

+

[3

2x2 +

V ′′(q0)q40x

2

2Q

]+ (2β + 1)

∞∑n=1

(−1)n(n + 1)

2xnl−n/2

+q20

∞∑n=3

[(−1)n

(n + 1)

2q20

xn +

(dnV (q0)

dqn0

)(q0x)

n

n!Q

]l−(n−2)/2

+β(β + 1)∞∑n=0

(−1)n(n + 1)

2xnl−(n+2)/2 +

(2β + 1)

2, (11)

where the prime ofV (q0) denotes a derivative with respect toq0. Equation (10) is exactly ofthe type of Schrodinger equation for the one-dimensional anharmonic oscillator[

−1

2

d2

dx2+

1

2w2x2 + ε0 + P(x)

]Xnr (x) = λnrXnr (x), (12)

whereP(x) is a perturbation-like term andε0 is a constant. A simple comparison betweenequations (10)–(12) implies

ε0 = l[

1

2+q2

0V (q0)

Q

]+

2β + 1

2+β(β + 1)

2l, (13)

λnr = l[

1

2+q2

0V (q0)

Q

]+

[2β + 1

2+

(nr +

1

2

)w

]+

1

l

[β(β + 1)

2+ λ(0)nr

]+∞∑n=2

λ(n−1)nr

l−n,

(14)

and

λnr = q20

∞∑n=−2

E(n)nr ,ll−(n+1). (15)

Equations (14) and (15) yield

E(−2)nr ,l= 1

2q20

+V (q0)

Q(16)

E(−1)nr ,l= 1

q20

[2β + 1

2+

(nr +

1

2

)w

](17)

E(0)nr ,l= 1

q20

[β(β + 1)

2+ λ(0)nr

](18)

E(n)nr ,l= λ(n)nr /q2

0; n > 1. (19)

Hereq0 is chosen to minimizeE(−2)nr ,l

, i.e.

dE(−2)nr ,l

dq0= 0 and

d2E(−2)nr ,l

dq20

> 0. (20)

Hereby,V (q) is assumed to be well behaved so thatE(−2) has a minimumq0 and there arewell-defined bound states. Equation (20) in turn gives, withl = √Q,

l − β =√q3

0V′(q0). (21)

Consequently, the second term in equation (11) vanishes and the first term adds a constant tothe energy eigenvalues. It should be noted that the energy terml2E

(−2)nr ,l

has its counterpart inclassical mechanics. It corresponds roughly to the energy of a classical particle with angularmomentumLz = l executing circular motion of radiusq0 in the potentialV (q0). This termthus identifies the leading-order approximation, to all eigenvalues, as a classical approximation

3058 O Mustafa and M Odeh

and the higher-order corrections as quantum fluctuations around the minimumq0, organizedin inverse powers ofl.

The next leading correction to the energy series,lE(−1)nr ,l

, consists of a constant term andthe exact eigenvalues of the unperturbed harmonic oscillator potentialw2x2/2. The shiftingparameterβ is determined by choosinglE(−1)

nr ,l= 0. This choice is physically motivated. It

requires agreement not only between the PSLET eigenvalues and the exact known ones for theharmonic oscillator and Coulomb potentials but also between the eigenfunctions. Hence

β = −[ 12 + (nr + 1

2)w], (22)

where

w =√

3 +q0V ′′(q0)

V ′(q0). (23)

Then equation (11) reduces to

q20

lV (x(q)) = q2

0 l

[1

2q20

+V (q0)

Q

]+∞∑n=0

v(n)(x)l−n/2, (24)

where

v(0)(x) = 1

2w2x2 +

2β + 1

2, (25)

v(1)(x) = −(2β + 1)x − 2x3 +q5

0V′′′(q0)

6Qx3, (26)

and forn > 2

v(n)(x) = (−1)n(2β + 1)(n + 1)

2xn + (−1)n

β(β + 1)

2(n− 1)x(n−2)

+

[(−1)n

(n + 3)

2+

q(n+4)0

Q(n + 2)!

dn+2V (q0)

dqn+20

]xn+2. (27)

Equation (10) thus becomes[−1

2

d2

dx2+∞∑n=0

v(n)l−n/2]9nr,l(x) =

[1

l

(β(β + 1)

2+ λ(0)nr

)+∞∑n=2

λ(n−1)nr

l−n]9nr,l(x). (28)

When setting the nodeless,nr = 0, wavefunctions as

90,l(x(q)) = exp(U0,l(x)), (29)

equation (28) is readily transformed into the following Riccati equation:

− 12[U ′′(x) +U ′(x)U ′(x)] +

∞∑n=0

v(n)(x)l−n/2 = 1

l

(β(β + 1)

2+ λ(0)0

)+∞∑n=2

λ(n−1)0 l−n. (30)

Hereafter, we shall useU(x) instead ofU0,l(x) for simplicity, and the prime ofU(x) denotesits derivative with respect tox. It is evident that this equation admits solution of the form

U ′(x) =∞∑n=0

U(n)(x)l−n/2 +∞∑n=0

G(n)(x)l−(n+1)/2, (31)

where

U(n)(x) =n+1∑m=0

Dm,nx2m−1; D0,n = 0, (32)

G(n)(x) =n+1∑m=0

Cm,nx2m. (33)

Spiked harmonic oscillators and truncated Coulomb potentials 3059

Substituting equations (31)–(33) into equation (30) implies

− 12

∞∑n=0

[U(n)′ l−n/2 +G(n)′ l−(n+1)/2]

− 12

∞∑n=0

∞∑p=0

[U(n)U(p)l−(n+p)/2 +G(n)G(p)l−(n+p+2)/2 + 2U(n)G(p)l−(n+p+1)/2]

+∞∑n=0

v(n)l−n/2

= 1

l

(β(β + 1)

2+ λ(0)0

)+∞∑n=2

λ(n−1)0 l−n, (34)

where primes ofU(n)(x) andG(n)(x) denote derivatives with respect tox. Equating thecoefficients of the same powers ofl andx, respectively, (of course, the other way aroundwould work equally well) one obtains

− 12U

(0)′ − 12U

(0)U(0) + v(0) = 0, (35)

U(0)′(x) = D1,0; D1,0 = −w, (36)

and integration over dx yields

U(0)(x) = −wx. (37)

Similarly,

− 12[U(1)′ +G(0)′ ] − U(0)U(1) − U(0)G(0) + v(1) = 0, (38)

U(1)(x) = 0, (39)

G(0)(x) = C0,0 +C1,0x2, (40)

C1,0 = −B1

w, (41)

C0,0 = 1

w(C1,0 + 2β + 1), (42)

B1 = −2 +q5

0

6Q

d3V (q0)

dq30

, (43)

− 12[U(2)′ +G(1)′ ] − 1

2

2∑n=0

U(n)U(2−n) − 12G

(0)G(0) −1∑n=0

U(n)G(1−n) + v(2)

= β(β + 1)

2+ λ(0)0 , (44)

U(2)(x) = D1,2x +D2,2x3, (45)

G(1)(x) = 0, (46)

D2,2 = 1

w

(C2

1,0

2− B2

)(47)

D1,2 = 1

w

(3

2D2,2 +C0,0C1,0 − 3

2(2β + 1)

), (48)

B2 = 5

2+q6

0

24Q

d4V (q0)

dq40

, (49)

λ(0)0 = − 1

2(D1,2 +C20,0), (50)

3060 O Mustafa and M Odeh

and so on. Thus, one can calculate the energy eigenvalue and the eigenfunctions fromthe knowledge ofCm,n andDm,n in a hierarchical manner. Nevertheless, the procedurejust described is suitable for systematic calculations using software packages (such asMATHEMATICA, MAPLE, or REDUCE) to determine the energy eigenvalue and eigenfunctioncorrections up to any order of the pseudoperturbation series.

Although the energy series, equation (9), could appear divergent, or, at best, asymptoticfor small l, one can still calculate the eigenenergies to a very good accuracy by forming thesophisticated Pade approximation to the energy series. The energy series, equation (9), iscalculated up toE(4)0,l /l

4 by

E0,l = l2E(−2)0,l +E(0)0,l + · · · +E(4)0,l /l

4 + O(1/l5), (51)

and with theP 33 (1/l) andP 4

3 (1/l) Pade approximants it becomes

E0,l [3, 3] = l2E(−2)0,l + P 3

3 (1/l), (52)

and

E0,l [3, 4] = l2E(−2)0,l + P 4

3 (1/l). (53)

Hereby, an ‘if’ statement is in point. If the energy series, equation (9), is a Stieltjes series,though it is difficult to prove, thenE0,l [3, 3] andE0,l [3, 4] provide upper and lower bounds tothe energy [18, 19]. Our strategy is therefore clear.

Let us begin with the spiked harmonic oscillators

V (q) = 12(q

2 + aq−b) (54)

for which equation (22), withnr = 0, implies

β = − 12(1 +w); w =

√√√√8q0 + ab(b − 2)q−(b+1)0

2q0 − abq−(b+1)0

. (55)

In turn equation (21) reads

l +1

2

1 +

√√√√8q0 + ab(b − 2)q−(b+1)0

2q0 − abq−(b+1)0

= q20

√1− ab

2q−(b+2)0 . (56)

Equation (56) is explicit inq0 and evidently a closed form solution forq0 is hard to find, infact almost impossible. However, numerical solutions are feasible. Onceq0 is determined thecoefficientsCm,n andDm,n are obtained in a sequential manner. Consequently, the eigenvalues,equation (51), and eigenfunctions, equations (31)–(33), are calculated in the same batch foreach value ofa, b, andl. In tables 1 and 2 we list PSLET resultsEP , equation (51), alongwith [3, 3] and [3, 4] Pade approximants, equations (52) and (53), respectively. The results ofthe STM [4] and DNI [4] are also displayed for comparison purposes.

Our calculated values of the bound-state energies,EP , compare well with those fromdirect numerical integrations [4]. In table 1 the Pade approximantsE[3, 3] andE[3, 4] arein almost total agreement with those of Hall and Saad [4] from DNI of the Schrodingerequation. Moreover, it is evident thatE[3, 3] andE[3, 4] have provided upper and lowerbounds, respectively, to the energy series. However, the same cannot be concluded fromtable 2. Eventually, our computed values of the bound-state energies,EP , do not contradictwith the upper and/or lower bounds reported by Hall and Saad [4] from the STM.

Moreover, our result forb = 2 listed in table 1 is in excellent agreement with theexact one, 65.253 4584, obtained from equation (2) of [4]. On the other hand, one couldrewrite the centrifugal term in (3) plus the potential (54) asl′(l′ + 1)/(2q2) + q2/2, where

Spiked harmonic oscillators and truncated Coulomb potentials 3061

Table 1. 1s-state energies, in ¯h = m = 1 units, of the potentialV (q) = (q2 + 1000/qb)/2. EPrepresents PSLET results, equation (51),E

U,LS with U andL denote upper and lower bounds from

STM [4], andEN from DNI [4]. E[3, 4] is the [3, 4] Pade approximant obtained by replacing thelastj digits ofE[3, 3] with thej digits in parentheses.

b EP E[3, 3] and(E[3, 4]) ES EN

0.5 415.889 78 415.889 786 (86) 416.309 77U 415.889 791.0 190.723 30 190.723 308 (07) 190.992 13U 190.723 311.5 104.410 22 104.410 224 (24) 104.539 93U 104.410 221.9 71.061 57 71.061 5789 (87) 71.086 86U 71.061 582.0 65.253 45 65.253 4589 (86) 65.253 46 65.253 462.1 60.152 00 60.152 0114 (11) 60.127 04L 60.152 012.5 44.955 47 44.955 4855 (50) 44.833 49L 44.955 493.0 33.316 75 33.316 7621 (18) 33.079 40L 33.316 763.5 26.108 84 26.108 8462 (48) 25.762 04L 26.108 854.0 21.369 50 21.369 4640 (14) 20.918 65L 21.369 644.5 18.101 94 18.101 8377 (10) 17.552 18L 18.101 835.0 15.761 34 15.761 144 (25) 15.117 58L 15.761 135.5 14.031 38 14.031 12 (07) 13.298 42L 14.031 076.0 12.718 86 12.718 79 (61) 11.901 53L 12.718 62

Table 2. 1s-state energies, in ¯h = m = 1 units, of the potentialV (q) = (q2 + a/q5/2)/2. EPrepresents PSLET results, equation (51),ES denotes the lower bounds from STM [4], andEN fromDNI [4]. E[3, 4] is the [3, 4] Pade approximant obtained by replacing the lastj digits ofE[3, 3]with thej digits in parentheses.

a EP E[3, 3] and(E[3, 4]) ES EN

1000 44.955 47 44.955 4855 (50) 44.833 49 44.955 49100 17.541 68 17.541 911 (899) 17.419 00 17.541 8910 7.734 23 7.736 06 (548) 7.611 69 7.735 115 6.296 79 6.299 88 (756) 6.173 94 6.296 471 4.328 61 4.528 (290) 4.204 53 4.317 310.5 3.857 40 3.8308 (289) 3.746 11 3.848 550.05 3.134 31 3.1606 (893) 3.109 54 3.152 430.005 3.014 45 3.0199 (201) 3.011 78 3.019 05

l′ = − 12 +

√(l + 1

2)2 + a, and proceed by shifting the irrational quantum numberl′ through

l = l′ − β. In this case, one obtains the known exact resultEP = (l′ + 32) for the

harmonic oscillatorq2/2 from the leading terml2E(−2) and the remainder energy correctionsare identically zero.

Next, we consider the laser-dressed Coulomb potential

V (q) = − 1√q2 + c2

, c > 0. (57)

In this case

w =√q2

0 + 4c2

q20 + c2

, (58)

and

l +1

2

(1 +

√q2

0 + 4c2

q20 + c2

)= q2

0[q20 + c2]−3/4. (59)

3062 O Mustafa and M Odeh

Table 3. Bound-state energies, in ¯h = m = 1 units, of the potentialV (q) = −(q2 + c2)−1/2 forthe 1s, 2p, 3d, and 4f states.EP represents PSLET results, equation (51),ESLNT is from SLNT[6], andEN from DNI [7]. E[3, 4] is the [3, 4] Pade approximant obtained by replacing the lastj

digits ofE[3, 3] with thej digits in parentheses.

c State −EP −E[3, 3] and(−E[3, 4]) −ESLNT −EN1 1s 0.274 12 0.274 78 (62) 0.275 96 0.274 39

2p 0.113 087 0.112 96 (303) 0.112 826 0.113 0243d 0.054 435 7 0.054 437 1 (82) 0.054 442 0.054 436 24f 0.031 068 45 0.031 068 46 (47) 0.031 069 0.031 068 46

5 1s 0.107 083 6 0.107 081 3 (10) 0.107 396 0.107 081 42p 0.068 191 40 0.068 186 67 (33) 0.068 233 0.068 187 163d 0.043 255 86 0.043 257 30 (20) 0.043 247 0.043 257 554f 0.028 105 34 0.028 105 20 (25) 0.028 101 0.028 105 24

10 1s 0.063 738 31 0.063 738 17 (21) 0.063 820 0.063 738 92p 0.046 200 43 0.046 199 03 (00) 0.046 228 0.046 199 043d 0.033 158 68 0.033 158 55 (53) 0.033 164 0.033 158 594f 0.023 806 62 0.023 806 72 (71) 0.023 806 0.023 806 74

Table 4. Bound-state energies, in ¯h = m = 1 units, of the potentialV (q) = −(q2 + c2)−1/2 forthe 1s, 2p, 3d, and 4f states.EP represents PSLET results, equation (51),ESLNT is from SLNT[6], andEN from DNI [7]. E[3, 4] is the [3, 4] Pade approximant obtained by replacing the lastj

digits ofE[3, 3] with thej digits in parentheses.

c State −EP −E[3, 3] and(−E[3, 4]) −ESLNT −EN50 1s 0.016 260 71 0.016 260 72 (71) 0.016 263 0.016 260 72

2p 0.014 088 37 0.014 088 37 (37) 0.014 090 0.014 088 383d 0.012 158 71 0.012 158 71 (71) 0.012 160 0.012 158 714f 0.010 458 42 0.010 458 42 (42) 0.010 459 0.010 458 42

100 1s 0.008 629 78 0.008 629 78 (78) 0.008 630 0.008 629 782p 0.007 800 13 0.007 800 13 (13) 0.007 800 0.007 800 133d 0.007 035 19 0.007 035 19 (19) 0.007 035 0.007 035 194f 0.006 332 73 0.006 332 73 (73) 0.006 333 0.006 332 73

200 1s 0.004 502 85 0.004 502 85 (85) 0.004 503 0.004 502 862p 0.004 193 07 0.004 193 07 (07) 0.004 193 0.004 193 073d 0.003 900 20 0.003 900 20 (20) 0.003 900 0.003 900 204f 0.003 623 85 0.003 623 85 (85) 0.003 624 0.003 623 85

Again, we numerically solve forq0 and proceed exactly as above to calculate the energyeigenvalues and eigenfunctions in the same batch. In tables 3 and 4 we collect the results forthe truncation parameterc = 1, 5, 10, 50, 100, 200 based on our approach. The energiesEP ,equation (51), compare well with those of Singhet al [7] from numerical integrations. ThePade approximantsE[3, 3] andE[3, 4] are in almost complete accord with those of Singhet al[7]. However, they do not provide upper and lower bounds to the energy series, equation (51).Perhaps, it should be mentioned that the approximate binding potential, equation (57), is validfor a hydrogen atom in a laser field which corresponds to a truncation parameterc in the range20–60 [6]. Higher and lower values ofc have been considered for academic interest only.

Before we conclude, some remarks deserve to be mentioned.

Spiked harmonic oscillators and truncated Coulomb potentials 3063

For the two problems discussed in this paper, we have shown that it is an easy taskto implement PSLET without having to worry about the ranges of couplings and forms ofperturbations in the potential involved. In contrast to the textbook Rayleigh–Schrodingerperturbation theory, an easy feasibility of computation of the eigenvalues and eigenfunctions, inone batch, has been demonstrated, and satisfactory accuracies have been obtained. Moreover,a nice numerical trend of convergence has been achieved. Nevertheless, another suitablecriterion for choosing the value of the shiftβ, reported in [14], is also feasible. This referenceshould be consulted for more details.

It is not easy to prove that the energy series, equation (51), is a Stieltjes series. But, if it is aStieltjes series, the [N,N ] and [N,N +1] Pade approximants provide upper and lower boundsto the energy series. Table 1 bears this out. Moreover, in view of the results listed in tables 1–4one can confidently conclude that the [3, 3] and [3, 4] Pade approximants to the energy seriesequation (51) can be used to determine the energy eigenvalues to a very satisfactory accuracy.

From the knowledge ofCm,n and Dm,n one can calculate, in the same batch, thewavefunctions to study electronic transitions and multiphoton emission occurring in atomicsystems in the presence of intense laser fields, for example. Such studies already lie beyondthe scope of our present methodical proposal.

Finally, the attendant technique PSLET could be applied to the Schrodinger equation withrational potentials, such as the nonpolynomial oscillatorV (q) = q2+λq2/(1+gq2). This typeof potential is an interesting model in laser and quantum field theories [20]. The feasibility ofPSLET extends also to a class of screened Coulomb potentials, which have relevance in atomicand plasma physics, and to some other models of interest ([21–26] and references therein).

References

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