# Supersymmetry and the relationship between a class of singular potentials in arbitrary dimensions

**ABSTRACT** The eigenvalues of the potentials $V_{1}(r)=\frac{A_{1}}{r}+\frac{A_{2}}{r^{2}}+\frac{A_{3}}{r^{3}}+\frac{A_{4 }}{r^{4}}$ and $V_{2}(r)=B_{1}r^{2}+\frac{B_{2}}{r^{2}}+\frac{B_{3}}{r^{4}}+\frac{B_{4}}{r^ {6}}$, and of the special cases of these potentials such as the Kratzer and Goldman-Krivchenkov potentials, are obtained in N-dimensional space. The explicit dependence of these potentials in higher-dimensional space is discussed, which have not been previously covered.

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**ABSTRACT:**Exact solution of Schrödinger equation for the Mie potential is obtained for an arbitrary angular momentum. The energy eigenvalues and the corresponding wavefunctions are calculated by the use of the Nikiforov–Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The bound states are calculated numerically for some values of ℓ and n with n≤ 5. They are applied to several diatomic molecules.Journal of Mathematical Chemistry 01/2008; 43(2):749-755. · 1.23 Impact Factor - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]

**ABSTRACT:**The three-body Schr\"{o}dinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length $\kappa+1$, $\kappa=0,1,...$ As a result, the solutions to the three-body Schr\"{o}dinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve - in these subalgebras - the radial Schr\"{o}dinger equation in three dimensions with the inverse power potential of the form $r^{-\kappa-1}$. As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with $\kappa=0$, are reduced to the well-known eigenvalues of bound states at threshold.05/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**The three-body Schrödinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length ${\kappa + 1, \kappa = 0, 1, \ldots}$ As a result, the solutions to the three-body Schrödinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve—in these subalgebras—the radial Schrödinger equation in three dimensions with the inverse power potential of the form ${r^{-{\kappa}-1}}$ . As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with ${\kappa = 0}$ , are reduced to the well-known eigenvalues of bound states at threshold.Few-Body Systems 11/2013; · 1.05 Impact Factor

Page 1

arXiv:quant-ph/0106142v3 27 Nov 2002

Supersymmetry and the relationship between

a class of singular potentials in arbitrary

dimensions

B. G¨ on¨ ul, O.¨Ozer, M. Ko¸ cak, D. Tutcu and Y. Can¸ celik

Department of Engineering Physics, University of Gaziantep,

27310 Gaziantep-T¨ urkiye

February 9, 2008

Abstract

The eigenvalues of the potentials V1(r) =

V2(r) = B1r2+B2

as the Kratzer and Goldman-Krivchenkov potentials, are obtained in N-

dimensional space. The explicit dependence of these potentials in higher-

dimensional space is discussed, which have not been previously covered.

A1

r+A2

r2 +A3

r3 +A4

r4 and

r2+B3

r4+B4

r6, and of the special cases of these potentials such

Pacs Numbers: 03.65.Fd, 03.65.Ge

1Introduction

Singular potentials have attracted much attention in recent years for a variety of

reasons, two of them being that (i) the ordinary perturbation theory fails badly

for such potentials, and (ii) in physics, one often encounters phenomenological

potentials that are strongly singular at the origin such as certain type of nucleon-

nucleon potentials, singular models of fields in zero dimensions, etc. Thus a study

of such potentials is of interest, both from the fundamental and applied point of

view.

One of the challenging problems in non-relativistic quantum mechanics is to

find exact solutions to the Schr¨ odinger equation for potentials that can be used

in different field of physics. Recently, several authors obtained exact solutions for

the fourth-order inverse-power potential

V1(r) =A1

r

+A2

r2+A3

r3+A4

r4

(1)

1

Page 2

using analytical methods [1-3]. These methods yield exact solutions for a single

state only for a potential of type (1) with restrictions on the coupling constants.

The interest is mainly due to the wide applicability of these type inverse-power

potentials. Some areas of interest are ion-atom scattering [4], several interac-

tions between the atoms [5], low-energy physics [6], interatomic interactions in

molecular physics [7] and solid-state physics [8].

The advent of supersymmetry has had a significant impact on theoretical

physics in a number of distinct disciplines. One subfield that has been receiving

much attention is supersymmetric quantum mechanics [9] in which the Hamilto-

nians of distinct systems are related by a supersymmetry algebra. In this work,

we are concerned with, via supersymmetric quantum mechanics, clarifying the

relationship between two distinct systems having an interaction potential of type

(1) and interacting through

V2(r) = B1r2+B2

r2+B3

r4+B4

r6

(2)

singular even-power potentials which have been widely used in a variety of fields,

e.g. see [6,10]. In recent years, the higher order anharmonic potentials have drawn

more attentions of physicists and mathematicians in order to partly understand

a newly discovered phenomena such as the structural phase transitions [11], the

polaron formation in solids [12], the concept of false vacuo in field theory [13], fibre

optics [14], and molecular physics [15]. In addition, some 60 years ago Michels

et al.. [16] proposed the idea of simulating the effect of pressure on an atom by

enclosing it in a impenetrable spherical box. Since that time there have been a

large number of publications, for an overview see [17], dealing with studies on

quantum systems enclosed in boxes, which involve an interaction potential that

is a special case (B2= 0) of (2). This field has received added impetus in recent

years because of the fabrication of semiconductor quantum dots [18].

The main motivation behind this work is to reveal the existence of a link

between potentials of type (1) and (2) in N-dimensional space, and between their

special cases such as a Mie-type potential (or Kratzer) [19] and pseudoharmonic-

like (or Goldman-Krivchenkov) potential [20] in higher dimensions, which to our

knowledge has never been appeared in the literature. On the other hand, with

the advent of growth technique for the realization of the semiconductor quantum

wells, the quantum mechanics of low-dimensional systems has become a major

research field. The work presented in this letter would also be helpful to the

literature in this respect as the results can readily be extended to lower dimensions

as well.

2

Page 3

2The Schr¨ odinger equation in N-dimensional

space

It is well known that the general framework of the non-relativistic quantum me-

chanics is by now well understood and its predictions have been carefully proved

against observations. Physics is permanently developing in a tight interplay with

mathematics. It is of importance to know therefore whether some familiar prob-

lems are a particular case of a more general scheme or to search if a map between

the radial equations of two different systems exists. It is hence worthwhile to

study the Schr¨ odinger equation in the arbitrary dimensional spaces which has

attracted much more attention to many authors. Many efforts have in particu-

lar been produced in the literature over several decades to study the stationary

Schr¨ odinger equation in various dimensions with a central potential containing

negative powers of the radial coordinates [21, and the references therein].

The radial Schr¨ odinger equation for a spherically symmetric potential in N-

dimensional space (we shall use through this paper the natural units such that

¯ h = m = 1)

−1

2

?d2R

dr2+N − 1

r

dR

dr

?

+ℓ(ℓ + N − 2)

2r2

R = [E − V (r)]R(3)

is transformed to

−d2Ψ

dr2+

?(M − 1)(M − 3)

4r2

+ 2V (r)

?

Ψ = 2EΨ (4)

where Ψ, the reduced radial wave function, is defined by

Ψ(r) = r(N−1)/2R(r)(5)

and

M = N + 2ℓ (6)

Eq.(4) can also be written as

−1

2

d2Ψ

dr2+

?Λ(Λ + 1)

2r2

+ V (r)

?

Ψ = EΨ(7)

where Λ = (M − 3)/2.

dimensions has the same form as the three-dimensional one. Consequently, given

that the potential has the same form in any dimension, the solution in three

dimensions can be used to obtain the solution in any dimension simply by using

the substitution ℓ → Λ. It should be noted that N and ℓ enter into expressions

(4) and (7) in the form of the combinations N + 2ℓ. Hence, the solutions for a

particular central potential V (r) are the same as long as M(= N + 2ℓ) remains

We see that the radial Schr¨ odinger equation in N-

3

Page 4

unaltered. Therefore the s-wave eigensolutions (Ψℓ=0) and eigenvalues in four-

dimensional space are identical to the p-wave solutions (Ψℓ=1) in two-dimensions.

The technique of changing the independent coordinate has always been use-

ful tool in the solution of the Schr¨ odinger equation. For instance, this allows

something of a systematic approach enabling to recognize the equivalence of su-

perficially unrelated quantum mechanical problems. Many recent papers have

addressed this old subject. In the light of these works we proceed by substituting

r = αρ2/2 and R = F(ρ)/ρλ, λ an integer, suggested by the known transfor-

mations between Coulomb and harmonic oscillator problems [22] and used to

show the relation between the perturbed Coulomb problem and the sextic anhar-

monic oscillator in arbitrary dimensions [23,24], we transform Eq. (3) to another

Schr¨ odinger-like equation in N′= 2N − 2 − 2λ dimensional space with angular

momentum L = 2ℓ + λ,

−1

2

?d2F

dρ2+N′− 1

ρ

dF

dρ

?

+L(L + N′− 2)

2ρ2

F = [ˆE −ˆV (ρ)]F (8)

where

ˆE −ˆV (ρ) = Eα2ρ2− α2ρ2V (αρ2/2)(9)

and α is a parameter to be adjusted properly. Note that leaving re-scaling con-

stant α arbitrary for now gives us an additional degree of freedom. When we

discuss bound state eigenvalues later, we can use this to allow the values of the

potential coefficients to be completely independent of each other. Thus, by this

transformation, in general, the N-dimensional radial wave Schr¨ odinger equation

with angular momentum ℓ can be transformed to a N′= 2N−2−2λ dimensional

equation with angular momentum L = 2ℓ + λ. If we choose α2= 1/|E|, with E

corresponding the eigenvalue for the inverse power potential of Eq. (1), then Eq.

(8) corresponds to the Schr¨ odinger equation of an singular even-power potential

ˆV (ρ) = ρ2+4A2

ρ2+8A3

ρ4|E|1/2+16A4

ρ6|E|(10)

with eigenvalue

ˆE =−2A1

|E|1/2

(11)

Thus, the system given by Eq. (1) in N-dimensional space is reduced to

another system defined by Eq. (2) in N′= 2N − 2 − 2λ dimensional space.

In other words, by changing the independent variable in the radial Schr¨ odinger

equation, we have been able to demonstrate a close equivalence between singular

potentials of type (1) and (2). Note that when N = 3 and λ = 0 one finds

N′= 4, and when λ = 1 we get N′= 2. It is also easy to see that N′+ 2L

does not depend on λ, which leads to map two distinct problems in three- and

four-dimensional space [24].

4

Page 5

3Mappings between two distinct systems

3.1Quasi-exactly solvable case

Since Eq. (4) for the reduced radial wave Ψ(r) in the N-dimensional space has the

structure of the one-dimensional Schr¨ odinger equation for a spherically symmetric

potential V (r), we may define the supersymmetric partner potentials [9]

V±(r) = W2(r) ± W′(r)(12)

which has a zero-energy solution, and the corresponding eigenfunction is given

by

Ψn=0(r) ∝ exp[±

In constructing these potentials one should be careful about the behaviour of

the wave function Ψ(r) near r = 0 and r → ∞. It may be mentioned that Ψ(r)

behaves like r(M−1)/2near r = 0 and it should be normalizable. For the inverse

power potential of Eq. (1) we set

?r

W(r)dr] (13)

W(r) =−a

r2+c

r− b,b,c > 0(14)

and identify V+(r)with the effective potential so that

V+(r) =

?2A4

r4+2A3

r3+2A2

r2+2A1

r

?

+(M − 1)(M − 3)

4r2

− 2En=0

(15)

and substituting Eq. (14) into Eq. (12) we obtain

V+(r) =a2

r4+2a(1 − c)

r3

+c(c − 1) + 2ab

r2

−2bc

r

+ b2

(16)

and the relations between the parameters satisfy the supersymmetric constraints

a = ±

?

2A4

;c = 1 −

A3

±√2A4

(17)

The potential (1) admits the exact solutions

Ψn=0(r) = N0rcexp(a

r− br)(18)

where N0is the normalization constant, with the physically acceptable eigenval-

ues

En=0= −b2

2= −

1

16A4

?

A3

√2A4(1 +

A3

√2A4) −1

4(M − 1)(M − 3) − 2A2

?2

(19)

5

Page 6

in the case of a < 0 and under the constraints

A1= −(1 +

A3

√2A4)

?

−2En=0

(20)

The results obtained agree with those in Refs. [2,3,21] for three-dimensions. Note

that in order to retain the well-behaved solution at r → 0 and at r → ∞ we have

chosen a = −√2A4.

The expressions obtained above can easily be extended to the lower dimen-

sions. For example, one can readily check that our two-dimensional solutions

(N = 2,ℓ → ℓ − 1/2) for the inverse power potential considered are in excellent

agreement with the literature [21]. The ground state solutions in arbitrary dimen-

sions for the Coulomb (A2= A3= A4= 0), and for a the Kratzer (A3= A4= 0)

[19], and for an inverse-power A3= 0 [1,2] potentials can also be found from the

above prescriptions.

For the singular even-power anharmonic oscillator potential of Eq. (2), we set

W(r) = µr +δ

r+η

r3, δ > 0(21)

which leads to

Ψn=0(r) = C0rδexp(µr2

2

−

η

2r2) (22)

with C0being the corresponding normalization constant, and identify V+(r) with

the effective potential so that

V+(r) =

?2B4

r6+2B3

r4+2B2

r2+ 2B1r2

?

+(M − 1)(M − 3)

4r2

− 2˜En=0

= W2(r) + W′(r) =η2

r6+η(2δ − 3)

r4

+δ(δ − 1) + 2ηµ

r2

+ µ2r2+ µ(2δ + 1) (23)

and the relations between the potential parameters satisfy the supersymmetric

constraints

η = ±

As we are dealing with a confined particle system, the positive values for η and

the negative values for µ would of course be the right choice to ensure the well

behaved nature of the wave function behaviour at the origin and at infinity.

Hence, physically meaningful ground state energy eigenvalues for the potential of

interest are

?

2B4 ; δ =3

2+B3

η

; µ = ±

?

2B1

(24)

˜En=0= −µ

2(2δ + 1) =

?

B1

2

?

2 +

?

1 − 16

?

B1B4+ 8B2+ (M − 1)(M − 3)

?

(25)

At this point we should report that our results reproduce those obtained by

[17,25,26] when potential (2) (in case B2 = 0) is confined to an impenetrable

6

Page 7

spherical box in 2- and 3-dimensions. It is also not difficult to see that if one

takes η = 0 in Eq. (23), then Eq. (25) becomes the exact energy spectra of

N-dimensional harmonic oscillator. Further, one easily check that in case B4=

B3= 0, the above energy expression correctly reproduce the eigenvalues of the

pseudo-type potential in 3-dimension [27] which is the subject of the next section.

Finally, we wish to discuss briefly the explicit mapping between the singular

potentials given by Eqs. (1) and (2). If one consider the transformed anharmonic

oscillator potential of Eq. (10) and repeat the above mathematical procedure

carried out through Eqs. (21-25), then the corresponding eigenvalue equation

reads

ˆEn=0= −2µ(1 +

A3

√2A4)(26)

Using the physically acceptable definition of A1in Eq. (20), the above equation

can be rearranged as

ˆEn=0= −

2A1

|En=0|1/2

(27)

where En=0has been described in Eq. (19). This brief discussion shows explicitly

the relation between the two singular potentials in higher dimensions and verifies

Eq. (11).

3.2Exactly solvable case

Kasap [27] and his co-workers used supersymmetric quantum mechanics to find

exact results for the special cases of the singular potentials of (1) and (2), more

precisely the solutions of the Kratzer and pseudoharmonic potentials in three

dimensions. Their results can be easily generalized to N-dimensions by the sub-

stitution ℓ → Λ = (M−3)/2 as indicated in section II. This extension to arbitrary

dimensions helps us in constructing the map between these two distinct systems.

The study of anharmonic oscillators has raised a considerable amount of inter-

est because of its various applications especially in molecular physics. The Morse

potential is commonly used for anharmonic oscillator. However, its wave func-

tion does not vanish at the origin, but those for Mie-type and pseudoharmonic

potentials do. The Mie-type potential possesses the general features of the true

interaction energy, inter-atomic and inter-molecular, and dynamical properties of

solids [28]. On the other hand, the pseudoharmonic potential may be used for the

energy spectrum of linear and non-linear systems [20]. The Mie-type and pseudo-

harmonic potentials are two special kinds of analytically solvable singular-power

potentials as they have the property of shape-invariance.

Starting with the general form of the Mie-type potential

V (r) = D0

?

p

q − p(σ

r)q−

q

q − p(σ

r)p

?

(28)

7

Page 8

where D0is the interaction energy between two atoms in a molecular system at

r = σ, and q > p is always satisfied. If we take q = 2p and p = 1, we arrive at a

special case of the potential in Eq. (28), which is exactly solvable

V (r) =A

r2−B

r

(29)

where A = D0σ2and B = 2D0σ. The above potential, the so-called Kratzer

potential, includes the terms which give the representation of both the steep

repulsive branch and the long-range attraction. A single minimum occurs at r = σ

where the energy is −D0. Considerable interest has recently been shown in this

potential as a model to describe inter-nucleon vibration [29] and, in applications

this Mie type potential offers one of the most important exactly solvable models

of atomic and molecular physics and quantum chemistry [30].

We set the superpotential for the Kratzer effective potential

W(r) =

B/2

β + (β2+ C)1/2−β + (β2+ C)1/2

r

(30)

where

C =Λ(Λ + 1)

2

+ A , Λ = ℓ +1

2(N − 3) , β =

1

2√2

(31)

and obtained the exact spectrum in N-dimensional space as

En=

?

B/2β

2n + 1 + [(2Λ + 1)2+ A/β2]1/2

?2

, n = 0,1,2,...(32)

and from Eq. (13) the exact unnormalized ground state wavefunction can be

expressed as

Ψn=0(r) = r1/2{1+[(2Λ+1)2+A/β2]1/2} × exp(−

Br/4β2

1 + [(2Λ + 1)2+ A/β2]1/2) (33)

The excited state wavefunctions can be easily determined from the usual approach

in supersymmetric quantum mechanics [9] and the normalization coefficients for

each quantum state wave function can be analytically worked out using the ex-

plicit recurrence relation given in a recent work [31].

As a second application, we consider the general form of the pseudoharmonic

potential

˜V (r) = V0(r

r0

which can be used to calculate the vibrational energies of diatomic molecules

with the equilibrium bond length r0and force constant k = 8V0/r2

corresponding superpotential as

−r0

r)2=˜Br2+

˜A

r2− 2V0

(34)

0, and set the

W(r) =

?

˜Br −β + (β2+˜C)1/2

r

(35)

8

Page 9

where˜B = V0/r2

the potential in arbitrary dimensions is

0,˜C = [Λ(Λ + 1) + 2˜A]/2,˜A = V0r2

0. The exact full spectrum of

˜ En= 2β

?

˜B{4n + 2 + [(2Λ + 1)2+˜A/β2]1/2} − 2V0

(36)

and the unnormalized exact ground state wave function is

Ψn=0(r) = r1/2{1+[(2Λ+1)2+˜ A/β2]1/2} × exp(−

√˜Br2

4β

)(37)

Using the discussion in section 2, one can transform the Kratzer potential

in Eq. (29) to its dual potential- shifted (by 2V0) pseudoharmonic-like potential

in Eq. (34) with some restrictions in potential parameters. In the light of Eqs.

(9-11), the transformed potential reads

ˆV (ρ) = ρ2+4A

ρ2

(38)

which is in the form of the Goldman-Krivchenkov potential. Here A(= D0σ2)is

the Kratzer potential parameter and, considering Eqs. (34) through Eq. 36,

constraints on the potential parameters are such that˜B = 1 and˜A = 4A. In this

case corresponding eigenvalues are

ˆEn′

=

2B

|En|1/2= 4β{1 + 2n′+ [1 + 4Λ′(Λ′+ 1) +A

= L +1

2(N′− 3)

β2]} ,

Λ′

(39)

where B(= 2D0σ) and Enare the coupling parameter and the eigenenergy values

(Eq. 32), respectively, of the Kratzer potential.

The ensuing relationships among the dimensionalities and quantum numbers

of the two distinct systems considered here in this section are :

N′= 2N − 2 − 2λ , L = 2ℓ + λ

Clearly, the mapping parameter λ must be an integer if n′, L, n and ℓ are in-

tegers. It is worthwhile to discuss briefly the physics behind this transforma-

tion in the light of the comprehensive work of Kostelecky et al.. [22]. We note

that it is a general feature of this map that the spectrum of the N-dimensional

problem involving Kratzer potential is related to the half the spectrum of the

N′-dimensional problem involving Goldman-Krivchenkov potential for any even

integer N′. However, the quantities in Eq. (40) have parameter spaces that are

further restricted by the properties chosen for the map. For instance, suppose we

wish to map all states corresponding the N-dimensional Kratzer potential into

those corresponding Goldman-Krivchenkov potential. Since on physical grounds

n′= 2n − 2 + λ (40)

9

Page 10

we know that N′≥ 2, n′≥ 0, L ≥ 0, we must impose N ≥ 2 + λ, n ≥ 1 − λ/2,

ℓ ≥ −λ/2. This yields the bound −2ℓ ≤ λ ≤ N − 2. Further requiring n ≥ 1,

ℓ ≥ 0 restricts the bound to 0 ≤ λ ≤ N − 2. We conclude that all states of the

N-dimensional Kratzer problem can be mapped into the appropriate Goldman-

Krivchenkov problem, except for N = 1.

As an example, consider the three-dimensional Kratzer problem. Assuming

we wish to map all its states into those of its dual-the Goldman-Krivchenkov

potential, we must impose 0 ≤ λ ≤ 1. First, take λ = 0. Then, the s-orbitals in

Kratzer potential (n ≥ 1,ℓ = 0) are related to the (n′= 2n−2 ≥ 0,L = 0 ) states

of the four-dimensional Goldman-Krivchenkov problem. Similarly, the p-states

(n ≥ 2,ℓ = 1) correspond to the (n′= 2n − 2 ≥ 0,L = 0) same problem. Next,

suppose Λ = 1. The states corresponding the potential in Eq. (29) are then

mapped into the odd-integer states of the two-dimensional oscillator problem

of Eq. (38). The s-orbitals of Kratzer potential (n ≥ 1,ℓ = 0) map into the

(n′= 2n − 1 ≥ 1,L = 1) anharmonic oscillator states corresponding Goldman-

Krivchenkov potential, while the Kratzer p-orbitals (n ≥ 2,ℓ = 1) map into the

(n′= 2n − 1 ≥ 3,L = 1) oscillator states of Goldman-Krivchenkov problem. As

a rule, in both cases (λ = 0,1), the lowest-lying states of Goldman-Krivchenkov

potential are excluded , one by one, with each higher value of ℓ.

As a final remark, a student of introductory quantum mechanics often learns

that the Schr¨ odinger equation is exactly solvable (for all angular momenta) for

two central potentials in Eqs. (29) and (38), and for also their special cases

(A = 0) the Coulomb and harmonic oscillator problems. Less frequently, the

student made aware of the relation between these two problems, which are linked

by a simple change of the independent variable discussed in detail through the

paper. Under this transformation, energies and coupling constants trade places,

and orbital angular momenta are rescaled. Thus, we have in this section shown

that there is really only one quantum mechanical problem, not two involving the

Kratzer and Goldman-Krivchenkov potentials, which can be exactly solved for

all orbital angular momenta.

4Conclusion

The main aim of this work has been to establish a very general connection between

a class of singular potentials in higher dimensional space through the application

of a suitable transformation. Although much work had been done in the literature

on similar problems, an investigation as the one we have discussed in this paper

was missing to our knowledge. In addition, it is shown that the supersymmetric

quantum mechanics yields exact solutions for a single state only for the quasi-

exactly solvable potentials such as the ones given in Eqs. (1) and (2) with some

restrictions on the potential parameters in N-dimensional space, unlike the shape

invariant exactly solvable potentials. We have also shown how to obtain exact

10

Page 11

solutions to such problems in any dimension by applying an adequate transfor-

mation to previously known three-dimensional results. This simple and intuitive

method discussed through this paper is easy to be generalized. The application

of this method to other potentials involving non-central ones are in progress.

11

Page 12

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