Generalization of quasi-exactly solvable and isospectral potentials

—journal of September 2007

physics pp. 337–367

Generalization of quasi-exactly solvable and isospectral potentials

P K BERA¹, J DATTA¹, M M PANJA² and TAPAS SIL³

1Department of Physics, Dumkal College, Basantapur, Dumkal, Murshidabad 742 303, India

2Department of Mathematics, T.D.B. College, Raniganj, Burdwan 713 347, India

3Department of Physics, VIT University, Vellore 632 014, India E-mail: pbera321@sify.com; joydip.india@gmail.com

MS received 19 December 2006; revised 22 May 2007; accepted 8 June 2007

Abstract. A unified approach in the light of supersymmetric quantum mechanics (SSQM) has been suggested for generating multidimensional quasi-exactly solvable (QES) potentials. This method provides a convenient means to construct isospectral potentials of derived potentials.

Keywords. SUSY algebras; case study.

PACS Nos 11.30.Pb; 03.65.Fd; 03.65.Ge; 03.65.Sq

1. Introduction

Exact solutions of Schr¨odinger equations provide all important informations about the system concerned. But for physical systems, exactly solvable potentials are very few in number. Therefore, the quasi-exactly solvable (QES) potentials have received a lot of attention [1–3]. These QES models allow exact solutions only for a limited part of the discrete energies but not for the entire spectrum. Thus, these potentials fill up the gap between the exactly solvable and non-solvable potentials and help to understand many physical phenomena. Moreover, QES problem has its own inner mathematical beauty such as it can provide a good starting point for doing calculations perturbatively for complex systems.

On the other hand, supersymmetry inspired quantum mechanics proposed by Witten [4] is a very beautiful mathematical construct; it renders some very sugges- tive solutions to phenomenological questions such as the hierarchy problem. Super- symmetric quantum mechanics (SSQM) can provide an important testing ground for both physical and computational aspects of supersymmetry (SUSY) theories [4,5]. Moreover, a special attribute to the formal theory of SSQM is in its mathe- matical simplicity [6,7]. Therefore, it has been extensively used to analyse various physical processes in nuclear physics [8,9] and atomic physics [10,11].

Recently, the exact solutions for a single state for some singular potentials are obtained by using analytical methods [3] with some restrictions on the potential parameters. Singular potentials, specially Coulomb plus inverse power potentials and harmonic oscillator plus inverse power potentials have been extensively used in different branches of physics such as study of quark confinement problem in QCD [12] and false vacua in field theory [13], ion-atom scattering [14] and several interac- tions between the atoms [15] in atomic physics, interatomic interactions in molecu- lar physics [16], polaron formation and other problem in solid-state physics [17,18], magnetic resonances between massive and massless spin-¹₂ particle in low-energy physics [19] etc. In a recent work, G¨on¨oulet al [20] have shown that two kinds of potentials as mentioned above can be linked together inN-dimensional space. Non- singular (polynomial) potentials are also very much fascinating nowadays because of their mathematical beauty likeP T symmetry [21] and their application in differ- ent branches of physics [22]. Therefore, it will be very much interesting to develop a unified approach for constructing QES potentials of singular and non-singular type.

In SSQM, the shape-invariant potentials, the non-shape-invariant but exactly solvable potentials can be constructed and incorporated in the frame of the theorem of existence of the superpotential (W) [23]. Using this technique, we intend to derive a variety of generalized potentials in N-dimension by assuming different forms of the superpotentials for which Schr¨odinger equation yield exact solutions for a single state in each case. Then, we shall try to study the special cases of singular and non-singular potentials emerging from the generalized one for different choices of parameters. In this way, we shall search for a class of new QES potentials.

Furthermore, the invariance of the Hamiltonian under the addition of a suitable function to the SUSY operator can be exploited to construct a family of isospectral potentials associated with our generalized QES potentials.

In this article, several aspects of SUSY algebras related to our problem are re- viewed in§2 and applications of these algebras to construct generalized QES poten- tials and their family of isospectral potentials inN-dimensional space are discussed in§3. Finally, in§4, we will draw the conclusion.

2. SUSY algebras

Consider a particle of mass m moving in N-dimensional Euclidean space. The time-independent Schr¨odinger equation for any integral dimension is given by [24]

(in units of= 2m= 1)

HΦ = [−∇²_N +UN]Φ =EΦ. (1)

Here, the wave function Φ belongs to the energy eigenvalue E. The symbols ∇²_N andUN stand for the Laplacian operator and potential energy respectively inN- dimensional space. Investigation of physical process based on eq. (1) is a well- studied problem and many authors proceed by using the standard central potential U(r) in place ofUN. Hererrepresents theN-dimensional radius [_N

i x²_i]^1/2. Going over to the hyperspherical coordinate system withN−1 angular variable (θi) and one radial coordinate we can write

Φ =ψⁿ_l(r)

r^N−12 Yl^M(θi), (2)

whereY_l^M(θi) represents contributions from the hyperspherical harmonics that arise in higher dimensions. The eigenvalues and eigenfunctions for generalized angular momentum operators in hyperspherical coordinate are determined [25] using the results known from the factorization method [26]. However, from eqs (1) and (2) we have reduced radial Schr¨odinger equation for thelth partial wave on half-line as

Hψ_lⁿ(r) =

−d²

dr² +(l+^N₂⁻¹)(l−1 +^N₂⁻¹)

r² +V(r)

ψ_lⁿ(r)

=E_lⁿψⁿ_l(r). (3)

Here the superscript n refers to a quantum number, the interpretation of which depends on the choice ofV(r).

In SSQM, the Hamiltonian corresponding to the reduced radial Schr¨odinger equa- tion forlth partial wave can be written in the form

H=H1=−d²

dr² +U1(r), (4)

such that the binding energyE_1,l⁽⁰⁾ of the lowest bound state ofH1 is zero, i.e.

H1ψ_1,l⁽⁰⁾(r) = 0. (5)

The superscript (0) on reduced wave function ψ as well as E_1,l⁽⁰⁾ stands for the ground state wave function and energy while the subscript 1 merely indicates that the wave functionψ_1,l⁽⁰⁾(r) belongs toH1. Here, we shall use analogous notations for the partner Hamiltonians. The effective potential in the partner Hamiltonian H1 in eq. (4) consists of centrifugal as well as interaction term and can be written as

U1(r) = (l+^N−1₂ )(l−1 +^N₂⁻¹)

r² +V1(r). (6)

The underlying idea of SSQM is to factorizeH1 in the form

H1=O⁻O⁺, (7)

with the operators O^± =±d

dr+W(r). (8)

Here, the so-called superpotentialW(r) is related to the effective potential in eq.

(6) by

U1(r) =W²(r)−dW(r)

dr . (9)

The supersymmetric partnerH2 ofH1 is traditionally written as H2=O⁺O⁻=−d²

dr² +U2(r) (10)

with the concomitant supersymmetric partner potential [5,6] as U2(r) =W²(r) +dW(r)

dr . (11)

Equation (7) through eq. (5) automatically guarantees that the ground state ofH1

has zero energy and provides the ground state wave function ofH1 as ψ_1,l⁽⁰⁾(r) =N_1,l⁽⁰⁾exp

− _r

W(r)dr

, (12)

whereN_1,l⁽⁰⁾ is the normalization constant.

To construct a new class of potentials, two new operators are defined as C^± =±d

dr+F(r), (13)

withF(r) to be determined for the requirement that eq. (10) also holds good for these new operators [27,28]. So, one can write

H2=O⁺O⁻=C⁺C⁻. (14)

From this condition we obtain the Riccati differential equation dF(r)

dr +F²(r) =(l+^N−1₂ )(l+ 1 + ^N−1₂ )

r² +V(r). (15)

A particular solution of eq. (15) is W(r), such that the general solution can be written in the form

F(r) =W(r) +Z(r). (16)

With the help of eqs (6), (9) and (16), from eq. (15) we arrive at the equation dZ(r)

dr +Z²(r) + 2W(r)Z(r) = 0. (17) The solution of eq. (17) is given by

Z(r) = [ψ⁽⁰⁾_1,l(r)]² λ+_r

0[ψ⁽⁰⁾_1,l(r)]²dr, (18)

where λ is an arbitrary constant. In order to avoid problems with possible sin- gularity, we impose that λ >0 always for any type of problems with even or odd (2l+N−1). With the help of the technique of Alves and Filho [27], we get a new Hamiltonian

H=H1−2dZ(r)

dr =C⁻C⁺. (19)

This Hamiltonian can be regarded as a generalized version ofH1and is characterized by a new family of potentials

U(r) =U1(r)−2dZ(r)

dr . (20)

Again, the spectra ofH2 coincide with the spectra ofH1except for the missing ground state of the latter. By Mielnik [29] we now write

HC⁺=C⁺H1. (21)

From eq. (21) it is clear that eigenvectors Φ_2,l and C⁺Φ_2,l of H2 and H belong to the same eigenvalues. Therefore,Hand H2 will be isospectral provided we can show that the lowest energy level ofHlies at the ground state ofH1. To that end, we introduce a missing vectorφto the set{C⁺Φ_2,l;l= 1,2, . . .} such that

φ|C⁺Φ_2,l= 0, l= 1,2,3, . . . (22) This equation also implies that

C⁻φ|Φ_2,l= 0. (23)

Since Φ_2,l is an eigenfunction ofH2, we arrive at the condition

C⁻φ= 0 (24)

to determineφ(r) in the form φ(r) =ψ⁽⁰⁾_1,l(r) exp

−

Z(r)dr

. (25)

From eqs (24) and (25) we get

Hφ(r) = 0. (26)

Equation (26) shows thatφ(r), the eigenfunction ofH, belongs to the ground state eigenvalue ofH1.

3. Case study

We shall now use the aforementioned technique of supersymmetry-inspired quantum mechanics to generate several generalized potentials having the form of Coulomb plus singular and non-singular, harmonic oscillator plus singular and non-sigular, non-singular and finally singular plus non-singular potentials in N-dimensional space which are exactly solvable for a single state. In our construction process, eq. (9) has been used with a suitable choice ofW(r). Selection ofW(r) should be made in such a way that, the wave functionψ⁽⁰⁾_1,l(r) in eq. (12) is normalizable. The wave functionsψ_1,l⁽⁰⁾(r), of the derived potentials and their SUSY partners (U2) and also the family of isospectral potentials (U) have been derived. These information may turn out to be very instructive for the study of numerous aspects in various fields of physics. Such examples are in the study of loosely bound systems like halo nuclei, as well as, in scattering length and coupling parameter calculations [14].

3.1 Coulomb plus singular and non-singular potentials

Since the exponent ofein the Coulomb wave function contains a term linear inr, eq. (12) suggests us to choose the superpotentialW(r) to generate Coulomb plus singular or non-singular interactions as

W(r) =−l+^N−1₂

r +α+βdr^d−1, (27)

where the parametersα and β are to be determined by the demand of normaliz- ability ofψ⁽⁰⁾_1,l(r) in eq. (12). Using this superpotential, eq. (9) gives the effective potential functionU1(r) as

U1(r) =(l+^N−1₂ )(l−1 + ^N−1₂ )

r² −B

r

+Cr^2d−2−Dr^d−2+Er^d−1+A, (28)

which is exactly solvable at zero energy. Here, the coupling parameters areB = 2α(l+ ^N−1₂ ), C = β²d², D = βd(2l+N +d−2), E = 2αβd and the symbol A represents the change of reference point of the energy. Hence it is possible to describe a large number of Coulomb-dominated interactions by suitable choice of α, β, d, landN. For a given set of these parameters, the solution of the Schr¨odinger equation for the QES potential save the centrifugal term

V1(r) =−B

r +Cr^2d−2−Dr^d−2+Er^d−1 (29)

is

ψ_1,l⁽⁰⁾(r) =N_1,l⁽⁰⁾r^l+N−12 exp(−αr−βr^d), (30) with the ground state energy

E_1,l⁽⁰⁾=−A. (31)

Interestingly, the results for Coulomb interaction can be recovered from eqs (29)–

(31) by choosingβ= 0 andN = 3.

In a similar manner, it is staightforward to get the expressions of wave func- tions and energy derived by G¨on¨ul et al [20] by choosing d = −1 in our expres- sions eqs (29)–(31). In table 1, we present the wave function and energy for some QES Coulomb-dominated interactions which are exactly solvable for ground state.

From the expressions of wave functions presented in the table, it is clear that ψ⁽⁰⁾_1,l(r) satisfies the desired threshold (Ltr→0ψ_1,l⁽⁰⁾(r) = 0) and asymptotic behav- iour (Ltr→∞ψ_1,l⁽⁰⁾(r) = 0) for Re(α) > 0 and Re(β) > 0 and the normalization constantN_1,l⁽⁰⁾(1/ N_1,l⁽⁰⁾) are found to be finite. The representative values of these for somedare presented in Appendix A. It is interesting to note that except for the case ofd= 1, use of the analytic expressions for the coupling parametersE andD

Table 1. Ground state energies (E_1,l⁽⁰⁾) and reduced radial wave functions (ψ⁽⁰⁾_1,l(r)) for Coulomb-dominated singular and non-singular interactionV1(r) derived from eqs (29)–(31) with d = ±¹₂,±1,±³₂,−2,±3 and Re(α) > 0, Re(β)>0.

Parameter V1(r) ψ⁽⁰⁾_1,l(r) =N_1,l⁽⁰⁾r^l+N−12 × E_1,l⁽⁰⁾ d=¹₂ −^B−C_r +√^E

r−_r3/2^D exp(−αr−β√

r) −_4D^E22(2l+N−³₂)² d=−¹₂ −^B_r +_r^C₃ + ^E

r^3/2 −_r^D_5/2 exp(−αr−^√β_r) −_4D^E22(2l+N−⁵₂)²

d= 1 −^B_r −^D_r exp(−αr−βr) −(α+β)²

d=−1 −^B_r +_r^E₂−_r^D3 +_r^C₄ exp(−αr−^β_r) −_4D^E22(2l+N−3)² d=³₂ −^B_r +Cr+E√

r−^√D_r exp(−αr−βr^3/2) −_4D^E22(2l+N−¹₂)² d=−³₂ −^B_r +_r^C₅ −_r7/2^D + ^E

r^5/2 exp(−αr−_r3/2^β ) −_4D^E22(2l+N−⁷₂)² d=−2 −^B_r +_r^E₃−_r^D4 +_r^C₆ exp(−αr−_r^β2) −_4D^E22(2l+N−4)² d= 3 −^B_r +Cr⁴+Er² −Dr exp(−αr−βr³) −_4D^E22(2l+N+ 1)² d=−3 −^B_r +_r^C₈−_r^D5 +_r^E₄ exp(−αr−_r^β3) −_4D^E22(2l+N−5)²

in eq. (31) gives the exact energyE_1,l⁽⁰⁾ independent ofβ. Now, it is pertinent to in- vestigate whetherψ⁽⁰⁾_1,l(r) remains the eigenfunction belonging to theβ-independent eigenvalueE_1,l⁽⁰⁾(α) inspite of the violation of threshold or asymptotic behaviour of ψ⁽⁰⁾_1,l(r) for Re(β) < 0 and several choice of d. From our survey regarding this ambiguity, it reveals that (i) for β < 0 and d < 0, the depth of the interaction potentials V1(r) decreases as |β| increases (figure 1), (ii) for the same domain of parameters β and d, the wave function diverges to ∞ sharply near the origin as

|β| and|d| increase (figure 2), (iii) the behaviour of the wave function follows the inverted cup-shaped pattern of the ground state wave function in the interaction region with the energy given by eq. (31) (figure 3), (iv) forβ <0 andd > 0, the divergent part e^|β|rd ofψ_1,l⁽⁰⁾(r) asr → ∞ has been supressed by e^−αr(α >0) for d <1 and (v) figure 4 exhibits that althoughψ⁽⁰⁾_1,l(r) diverges in the asymptotic re- gion forβ <0 andd >1, the wave function possesses the single maximum without node in the interaction region for the energy given in eq. (31). Furthermore, the square integrability of the wave function for the above choice of parameters assures the convergence of improper integral appearing in the normalization constants and determination of expectation values of several physical observables. From these observations, we may conclude that despite the violation of SUSY for β <0 and d <0,ψ⁽⁰⁾_1,l(r) may play the role of ground state energy wave function for the in- teraction potentialV1(r) belonging to theβ-independent energy eigenvalueE_1,l⁽⁰⁾(α) in eq. (31) for the range−0.5< β <0,|d| ≤1.5.

The supersymmetric partner potential function U2(r) of the potential U1(r) is obtained from eq. (11) as

Figure 1. A plot of V1(r) in eq. (29) demonstrating the reduction of the depth of the potentials withβ= 0,−0.3,−0.5 for the exponentd=−0.5. Thex-axis denotes the variable r and the y-axis denotes the values ofV1(r).

Figure 2. The graph for ψ_1,l⁽⁰⁾(r) in eq. (30) ford=−0.5 andβ= 0,−0.3,−0.5 displaying the gradual change of the threshold behavior of the ground state wave function. The x-axis denotes the variablerand they-axis denotes the values ofψ_1,l⁽⁰⁾(r).

Figure 3. Figure displaying inverted cup shape of ground state wave function in the interaction region of the potential at the energy given by eq. (30) ford=−0.5, α= 1 andβ = −0.5 which supports the exis- tence of bound state despite the violation of threshold behaviour. Thex-axis denotes the variable r and the y-axis denotes the values ofE_1,l⁽⁰⁾, ψ_1,l⁽⁰⁾(r), V1(r).

Figure 4. The graphs ψ_1,l⁽⁰⁾(r) and V1(r) ford = 1.5, α= 1 and β =−0.1 demon- strate existence of the bound state in spite of the violation of asymptotic behaviour of the wave functions. Thex-axis denotes the variablerand they-axis denotes the values ofψ_1,l⁽⁰⁾(r), V1(r).

U2(r) =(l+^N₂⁻¹)(l+ 1 + ^N−1₂ ) r²

−B

r +Cr^2d−2−Dr^d−2+Er^d−1+A, (32) whereD=βd(2l+N−d).

Differentiating eq. (18) with respect tor and with the help of eq. (30), a new family of isospectral potentials of QES potentials can be obtained from eq. (20) as

U(r) =U1(r)−2r^2l+N−1exp(−2αr−2βr^d) I²(r)

×

(2l+N−1)

r −2α−2βdr^d−1

I(r)

−r^2l+N−1exp(−2αr−2βr^d)

, (33)

where the integration I(r) =λ+

_r

0 r^2l+N−1exp(−2αr−2βr^d)dr. (34) Expanding e^−2βrd to a series and with the help of integral representation [30]

_u

0 x^ν−1(u−x)^μ−1e^γxndx=B(μ, ν)u^μ+ν−1

×nFn

ν n,ν+ 1

n , ...,ν+n−1

n ;

μ+ν

n ,μ+ν+ 1

n , ...,μ+ν+n−1 n ;γuⁿ

,

Reμ >0, Reν >0, n= 2,3, ...; (35) we obtain

I(r) =λ+ ∞

0

(−2β)^k Γ(k+ 1)

Γ(2l+N+kd)

Γ(2l+N+kd+ 1)r^2l+N+kd

×₁F1(2l+N+kd; 2l+N+kd+ 1;−2αr), (36) where Γ(·) and nFn(·) are gamma function and confluent hypergeometric function respectively.

For the Coulomb potential (β= 0), we obtain the family of isospectral potentials from eqs (35) and (36) as

U^C(r) =U₁^C(r)−2r^2l+N−1 exp(−2αr) I_C²(r)

×

(2l+N−1)

r −2α

IC(r)−r^2l+N−1 exp(−2αr)

, (37)

where the integration

IC(r) =λ+ Γ(2l+N)

Γ(2l+N+ 1)r^2l+N1F1(2l+N; 2l+N+ 1;−2αr). (38)

3.2 Harmonic oscillator plus singular and non-singular potentials

The study of harmonic oscillator plus singular and non-singular potentials have been desirable to understand several physical phenomena like structural phase tran- sitions, polaron formation in solids etc. For this case, we consider a superpotential

W(r) =−l+^N−1₂

r + 2αr+βdr^d−1, (39)

where the parametersαandβ are also to be determined by the demand of normal- izability ofψ⁽⁰⁾_1,l(r) in eq. (12). Using this superpotential, we can write the effective potential functionU1(r) as

U1(r) = (l+^N−1₂ )(l−1 +^N₂⁻¹)

r² +Br²+Cr^2d−2−Dr^d−2+Er^d+A, (40) which is also exactly solvable at zero energy. Here, the coupling parameters are B = 4α², C =β²d², D=βd(2l+N +d−2), E= 4αβdand shift of the reference point of energy A is given by A = −2α(2l +N). As before, for a given set of parameters Re(α), Re(β), d, l andN, the ground state solution of the Schr¨odinger equation for the interaction

V1(r) =Br²+Cr^2d−2−Dr^d−2+Er^d (41) is

ψ_1,l⁽⁰⁾(r) =N_1,l⁽⁰⁾r^l+N−12 exp(−αr²−βr^d), (42) with the corresponding ground state energy

E_1,l⁽⁰⁾= E

2D(2l+N)(2l+N+d−2). (43)

It is easy to check that the choice β = 0 in (41)–(43) reproduces the harmonic oscillator potentialV1(r) = 4α²r² with the ground state energy

E_1,l^(0)H.O.= 2α(2l+N) (44)

and wave function as

ψ_1,l^(0)H.O.(r) =N_1,l⁽⁰⁾r^l+N−12 exp(−αr²). (45) Here, the parameterB can be written in terms of other parameters as

B= E²

4C; C, E >0 (46)

and ford=−1 we can recover the results of ref. [3]. For other choice ofdvalues, one can construct a number of harmonic oscillators plus different types of singular

Figure 5. ψ⁽⁰⁾_1,l(r) for d= 3, α = 1 andβ =−0.1,−0.5(−0.1). Thex-axis denotes the variablerand they-axis denotes the values ofψ⁽⁰⁾_1,l(r).

Table 2. Ground state energies (E_1,l⁽⁰⁾) and reduced radial wave functions (ψ⁽⁰⁾_1,l(r)) for harmonic oscillator plus singular and non-singular interaction V1(r) derived from eqs (41)–(43) with d = ±¹₂,±1,−2,3 and Re(α) > 0, Re(β)>0.

Parameter V1(r) = ψ⁽⁰⁾_1,l(r) =N_1,l⁽⁰⁾r^l+N−12 × E_1,l⁽⁰⁾ d=¹₂ Br²+E√

r+ ^C_r −_r3/2^D exp(−αr²−β√

r) _2D^E(2l+N)(2l+N−³₂) d=−¹₂ Br²+√^E

r+ _r^C₃ −_r5/2^D exp(−αr²−^√β_r) _2D^E(2l+N)(2l+N−⁵₂) d= 1 Br²+Er−^D_r exp(−αr²−βr) _2D^E(2l+N)(2l+N−1)−C d=−1 Br²+^E_r +−_r^D3 +_r^C₄ exp(−αr²−^β_r) _2D^E(2l+N)(2l+N−3) d=−2 Br²+_r^E₂−_r^D4 +_r^C₆ exp(−αr²−_r^β2) _2D^E(2l+N)(2l+N−4) d= 3 Br²+Er³−Dr+Cr⁴ exp(−αr²−βr³) _2D^E(2l+N)(2l+N+ 1)

and non-singular potentials from eq. (41). Analogous to our previous study, it is observed that for β < 0 and d > 0 the singular part e^|β|rd of ψ_1,l⁽⁰⁾(r) has been compensated by the strongly decaying harmonic oscillator part e^−αr2(α < 0) for 0 < d < 2. Consequently, the wave function behaves regular near the origin and in the asymptotic region for 0 < d < 2 and −1 < β < 0. However, for d > 2, the wave function behaves like the ground state in a wide region for finite part of r for −0.1 < β < 0 and increases to ∞ as r → ∞as shown in figure 5. But this feature gradually worsen as|β|increases. Therefore, the physically acceptable domain of parametersα, β andd for the harmonic plus singular and non-singular potentials should be considered asα >0. Table 2 consists of the QES potentials and their ground state wave functions and the ground state energy. The normalization constants of the ground state wave function for d = ±¹₂,±1,−2,3 are given in Appendix B.

The supersymmetric partner potential function U2(r) of the potential U1(r) is obtained as

U2(r) = (l+^N₂⁻¹)(l+ 1 +^N₂⁻¹)

r² +Br²+Cr^2d−2

−Dr^d−2+Er^d+A, (47) whereD=βd(2l+N−d).

Following the same mathematical artifice, we obtain a family of isospectral po- tential harmonic oscillator-dominated interactions as

U(r) =U1(r)−2r^2l+N−1exp(−2αr²−2βr^d) I²(r)

×

2l+N−1

r −4αr−2βdr^d−1

I(r)

−r^2l+N−1exp(−2αr²−2βr^d)

, (48)

where the value of the integration I(r) =λ+

∞ k=0

(−2β)^k Γ(k+ 1)

Γ(2l+N+kd)

Γ(2l+N+kd+ 1)r^2l+N+kd

×₁F1

l+N+kd

2 ;l+N+kd+ 2

2 ;−2αr²

. (49)

For harmonic oscillator (β= 0), the isospectral potential is U^H.O.(r) =U₁^H.O.(r)−2r^2l+N−1exp(−2αr²)

I_H.O.² (r)

×

(2l+N−1) r −4αr

IH.O.(r)−r^2l+N−1exp(−2αr²)

, (50) where the value of the integrationIH.O. is

IH.O.(r) =λ+ Γ(2l+N)

Γ(2l+N+ 1)r^2l+N1F1

l+N

2 ;l+N+ 2

2 ;−2αr²

. (51) 3.3 Non-singular potential

In this subsection, we shall consider the non-singular potentials mainly of the form of polynomials. These kinds of potentials have drawn attention to the study of P T-symmetry and pseudo-harmonicity of the Hamiltonian [21,22].

Type A:

Here, we consider a superpotential

Table 3. Ground state energies (E_1,l⁽⁰⁾) and reduced radial wave functions (ψ⁽⁰⁾_1,l(r)) for non-singular interaction V1(r) derived from eqs (54) and (55) withd=⁵₂,3,⁷₂,4,5 and Re(α)>0, Re(β)>0.

Parameter V1(r) = ψ_1,l⁽⁰⁾(r) =N_1,l⁽⁰⁾r^l+N−12 × E_1,l⁽⁰⁾ d=⁵₂ Ar⁸+Br^11/2+Cr³−D√

r exp(−αr⁵−βr^5/2) 0 d= 3 Ar¹⁰+Br⁷+Cr⁴−Dr exp(−αr⁶−βr³) 0 d=⁷₂ Ar¹²+Br^17/2+Cr⁵−Dr^3/2 exp(−αr⁷−βr^7/2) 0 d= 4 Ar¹⁴+Br¹⁰+Cr⁶−Dr² exp(−αr⁸−βr⁴) 0 d= 5 Ar¹⁸+Br¹³+Cr⁸−Dr³ exp(−αr¹⁰−βr⁵) 0

W(r) =−l+^N−1₂

r + 2αdr^2d−1+βdr^d−1, (52) where the parameters d, α and β are to be determined to match the interactions appropriately. Using this superpotential, we obtain the effective potential function U1(r) as

U1(r) = (l+^N₂⁻¹)(l−1 +^N₂⁻¹) r²

+Ar^4d−2+Br^3d−2+Cr^2d−2−Dr^d−2, (53) which is also exactly solvable at zero energy. Here, the coupling parameters are A= 4α²d², B= 4αβd², C=β²d²−2αd(2l+N+ 2d−2), D=βd(2l+N+d−2).

For a given set of parameters α, β, d, l and N, the solution representing ground state (E_1,l⁽⁰⁾ = 0) wave function for the Schr¨odinger equation with the interaction

V1(r) =Ar^4d−2+Br^3d−2+Cr^2d−2−Dr^d−2 (54) is

ψ_1,l⁽⁰⁾(r) =N_1,l⁽⁰⁾r^l+N−12 exp(−αr^2d−βr^d). (55) For the several choices of the exponent d, one can construct a number of non- singular potentials from eq. (54). The physically meaningful domain of the para- meters α(or β) for these interactions are found to be Re(α)>0 and Re(β)>0.

Table 3 consists of the representative QES potentials, ground state wave functions and the ground state energy for d = ⁵₂,3,⁷₂,4,5. As before, the normalization constants ofψ⁽⁰⁾_1,l(r) for the above choices ofdare presented in Appendix C.

The supersymmetric partner potential functionU2(r) of the potentialU1(r) is U2(r) =(l+^N₂⁻¹)(l+ 1 + ^N₂⁻¹)

r²

+Ar^4d−2+Br^3d−2+Cr^2d−2−Dr^d−2, (56) whereC=β²d²−2αd(2l+N−2d), D=βd(2l+N−d).

Following the same procedure as discussed in the case of Coulomb-dominated interaction, we obtain a new family of isospectral potentials for these QES non- singular potentials as

U(r) =U1(r)−2r^2l+N−1exp(−2αr^2d−2βr^d) I²(r)

×

2l+N−1

r −4αdr^2d−1−2βdr^d−1

I(r)

−r^2l+N−1exp(−2αr^2d−2βr^d)

, (57)

where the value of the integration I(r) =λ+

∞ k=0

(−2α)^k Γ(k+ 1)

Γ(2l+N+ 2kd)

Γ(2l+N+ 2kd+ 1)r^2l+N+2kd

×dFd

2l+N+ 2kd

d ,2l+N+ 2kd+ 1 d , ..., 2l+N+ 2kd+d−1

d ;2l+N+ 2kd+ 1

d ,2l+N+ 2kd+ 2

d ,

...,2l+N+ 2kd+d d ;−2βr^d

. (58)

Type B:

We consider a superpotential as W(r) =−l+^N−1₂

r +αdr^d−1+β(d−1)r^d−2. (59) Using this superpotential, we get the effective potential functionU1(r) as

U1(r) =(l+^N−1₂ )(l−1 + ^N−1₂ )

r² +Ar^2d−2

+Br^2d−3+Cr^2d−4−Dr^d−2−Er^d−3, (60) which is exactly solvable at zero energy. Here, the coupling parameters are A = α²d², B= 2αβd(d−1), C =β²(d−1)², D =αd(2l+N+d−2), andE =β(d− 1)(2l+N+d−3). So for a given set of parameters Re(α), Re(β), d, landN,ψ_1,l⁽⁰⁾(r) for the interaction

V1(r) =Ar^2d−2+Br^2d−3+Cr^2d−4−Dr^d−2−Er^d−3 (61) is

ψ_1,l⁽⁰⁾(r) =N_1,l⁽⁰⁾r^l+N−12 exp(−αr^d−βr^d−1). (62)

For the ground state wave function given in eq. (62) the normalization constants are found to be a combination of special functions whose representatives are pre- sented in Appendix C for d = 4 and 5. The supersymmetric partner potential functionU2(r) of the potentialU1(r) is

U2(r) =(l+^N−1₂ )(l+ 1 + ^N−1₂ ) r²

+Ar^2d−2+Br^2d−3+Cr^2d−4−Dr^d−2−Er^d−3, (63) whereD=αd(2l+N−d) andE=β(d−1)(2l+N−d+ 1).

The family of isospectral potentials for these QES potentials is found to be U(r) =U1(r)−2r^2l+N−1exp(−2αr^d−2βr^d−1)

I²(r)

×

2l+N−1

r −2αdr^d−1−2β(d−1)r^d−2

I(r)

−r^2l+N−1exp(−2αr^d−2βr^d−1)

, (64)

where the integration I(r) =λ+

∞ k=0

(−2β)^k Γ(k+ 1)

Γ(2l+N+k(d−1))

Γ(2l+N+k(d−1) + 1)r2l+N+k(d−1)

×dFd

2l+N+k(d−1)

d ,2l+N+k(d−1) + 1

d ,

...,2l+N+k(d−1) +d−1

d ;2l+N+k(d−1) + 1

d ,

2l+N+k(d−1) + 2

d , ...,2l+N+k(d−1) +d

d ;−2αr^d

. (65) Type C:

We next consider another choice of the superpotential W(r) =−l+^N−1₂

r +αdr^d−1+β(d−2)r^d−3. (66) Using this superpotential, we get the effective potential functionU1(r) as

U1(r) =(l+^N₂⁻¹)(l−1 + ^N−1₂ ) r²

+Ar^2d−2+Br^2d−4+Cr^2d−6−Dr^d−2−Er^d−4, (67) which is exactly solvable at zero energy. Here, the coupling parameters are A = α²d², B= 2αβd(d−2), C=β²(d−2)², D=αd(2l+N+d−2),andE=β(d−2)

(2l+N+d−4). For a given set of parameters Re(α), Re(β), d, landN, the ground state (E_1,l⁽⁰⁾ = 0) solution of the Schr¨odinger equation for the interaction

V1(r) =Ar^2d−2+Br^2d−4+Cr^2d−6−Dr^d−2−Er^d−4 (68) is

ψ_1,l⁽⁰⁾=N_1,l⁽⁰⁾r^l+N−12 exp(−αr^d−βr^d−2) (69) with the ground state energy. For the different d, Re(α)>0 and Re(β)>0, one can construct a number of different types of non-singular potentials from eq. (68).

The normalization constants for eq. (69) are given in Appendix C ford= 5 and 6.

The supersymmetric partner potential functionU2(r) of the potentialU1(r) is U2(r) = (l+^N₂⁻¹)(l+ 1 +^N₂⁻¹)

r²

+Ar^2d−2+Br^2d−4+Cr^2d−6−Dr^d−2−Er^d−4, (70) whereD=αd(2l+N−d) andE=β(d−2)(2l+N−d+ 2).

From eq. (20) we get a new family of isospectral potentials of these QES poten- tials as

U(r) =U1(r)−2r^2l+N−1exp(−2αr^d−2βr^d−2) I²(r)

×

2l+N−1

r −2αdr^d−1−2β(d−2)r^d−3

I(r)

−r^2l+N−1exp(−2αr^d−2βr^d−2)

, (71)

where the integration I(r) =λ+

∞ k=0

(−2β)^k Γ(k+ 1)

Γ(2l+N+k(d−2))

Γ(2l+N+k(d−2) + 1)r2l+N+k(d−2)

×dFd

2l+N+k(d−2)

d ,2l+N+k(d−2) + 1

d ,

...,2l+N+k(d−2) +d−1

d ;2l+N+k(d−2) + 1

d ,

2l+N+k(d−2) + 2

d , ...,2l+N+k(d−2) +d

d ;−2αr^d

. (72)

3.4 Singular plus non-singular potentials

Here, we shall construct a generalized potential consisting of both singular and non-singular parts. For this, we consider a superpotential

W(r) =−l+^N−1₂

r +αdr^d−1−βdr^−d−1. (73)

Using this superpotential, we get the effective potential functionU1(r) as U1(r) = (l+^N₂⁻¹)(l−1 +^N₂⁻¹)−E

r²

+Ar^2d−2−Br^d−2+ C

r^2d+2 + D

r^d+2, (74)

which is exactly solvable at zero energy. Here the coupling parameters are A = α²d², B=αd(2l+N+d−2), C =β²d², D=βd(2l+N−d−2), E= 2αβd². Hence for a given set of parameters Re(α), Re(β), d, l andN, the ground state (E_1,l⁽⁰⁾= 0) solution of the Schr¨odinger equation for the interaction

V1(r) =Ar^2d−2−Br^d−2+ C

r^2d+2 + D r^d+2 −E

r² (75)

is

ψ_1,l⁽⁰⁾(r) =N_1,l⁽⁰⁾r^l+N−12 exp(−αr^d−βr^−d). (76) For different values of d, one can construct different types of singular and non- singular potentials from eq. (75). Here, we also avoid the unphysical choices of the parameters. The physically meaningful domain of the parameters α (andβ), for these interactions are found to be Re(α)>0 and Re(β)>0. Table 4 consists of the QES potentials, their ground state wave functions and the ground state energy.

The normalization constants of the ground state wave function for different values ofdare given in Appendix D.

The supersymmetric partner potential functionU2(r) of the potentialU1(r) is U2(r) =(l+^N−1₂ )(l+ 1 + ^N−1₂ ) +E

r²

+Ar^2d−2+Br^d−2+ C

r^2d+2 + D

r^d+2, (77)

whereB=−αd(2l+N−d), D=βd(2l+N+d). We get a new family of isospectral potentials for these QES potentials as

U(r) =U1(r)−2r^2l+N−1exp(−2αr^d−2βr^−d) I²(r)

×

2l+N−1

r −2αdr^d−1+ 2β dr^−d−1

×I(r)−r^2l+N−1exp(−2αr^d−2βr^−d)

, (78)

where the integration