Rationally extended shape invariant potentials in arbitrary D-dimensions associated with exceptional $X_m$ polynomials

Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of X_m Laguerre or X_m Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.


Introduction
In recent years the discovery of exceptional orthogonal polynomials (EOPs) (also known as X 1 Laguerre and X 1 Jacobi polynomials) [1,2] has increased the list of exactly solvable potentials.The EOPs are the solutions of second-order Sturm-Liouvilli eigenvalue problem with rational coefficients.Unlike the usual orthogonal polynomials, the EOPs starts with degree n ≥ 1 and still form a complete orthonormal set with respect to a positive definite innerproduct defined over a compact interval.After the discovery of these two polynomials Quesne et.al.reported three shape invariant potentials whose solutions are in terms of X 1 Laguerre polynomial (extended radial oscillator potentials) and X 1 Jacobi polynomials (extended trigonometric scarf and generalized Pöschl Teller (GPT) potentials) [3,4].Subsequently Odake and Sasaki generalize these and obtain the solutions in terms of exceptional X m orthogonal polynomials [5].The properties of these X m exceptional orthogonal polynomials have been studied in detail in Ref. [6,7,8,9].Subsequently, the extension of other exactly solvable shape invariant potentials have also been done [10,11,12,13] by using different approaches such as supersymmetry Quantum mechanics (SUSYQM) [17,18], point canonical transformation (PCT) [21], Darboux-Crum transformation (DCT) [22] etc.The scattering amplitude of some of the newly found exactly solvable potentials in terms of X m EOPs are studied in Ref. [14,15,16].The bound state solutions of these extended (deformed) potentials are in terms of EOPs or some type of new polynomials (y n ) ( which can be expressed in terms of combination of usual Laguerre or Jacobi orthogonal polynomials).
The bound state spectrum of all these extended potentials are investigated in a fixed dimension (D = 1 or 3).Recently the extension of some exactly solvable potentials have been made in arbitrary D dimensions whose solutions are in terms of X 1 EOPs [26].The obvious question then if one can extend this discussion and obtain potentials whose solutions are in terms of X m EOPs in arbitrary dimensions.The purpose of this paper is to answer this question.In particular, in this paper we apply the PCT approach, which consists of a coordinate transformation and a functional transformation, that allows generation of normalized exact analytic bound state solutions of the Schrödinger equation, starting from an analytically solvable conventional potential.Here we consider two analytically solved conventional potentials (they are isotropic oscillator and GPT potential) [3,28] corresponding to which D-dimensional rationally extended potentials are obtained whose solutions are in terms of X m exceptional Laguerre or Jacobi polynomials.This paper is organized as follow: In Sec. 2, details about the point canonical transformation (PCT) method for arbitrary D-dimensions is given.In Sec. 3, We have written the differential equation corresponding to X m EOPs and discussed some important properties for EOPs.Arbitrary D-dimensional rationally extended exactly solvable potentials whose solutions are in terms of X m Laguerre or X m Jacobi EOPs are obtained in Sec. 4. The approximate solutions corresponding to X m Jacobi case are also discussed in this section.In Sec. 5, new shape invariant potentials for the rationally extended X m Laguerre and X m Jacobi polynomials are obtained in arbitrary D dimensions.In particular, for D = 2 and D = 4, the shape invariant partner potentials for the extended radial oscillator are obtained explicitly.Sec. 6, is reserved for results and discussions.in arbitrary D-dimensional Euclidean space [19,20] given by ( = 2m = 1) To solve this equation we apply PCT approach and assume the solution of the form where f (r) and g(r) are two undetermined functions and F (g(r)) will be later identified as one of the orthogonal polynomials which satisfies a second-order differential equation Here a prime denote derivative with respect to g(r).
Using Eq. ( 2) in Eq.( 1) and comparing the results with Eq. ( 3), we get and (5) where N is the integration constant and plays the role of the normalization constant of the wavefunctions and {g(r), r} is the Schwartzian derivative symbol [29], {g, r} defined as Here the prime denotes derivative with respect to r.From ( 2) and (4), the normalizable wavefunction is given by where The radial wavefunction ψ(r) = χ(r) 2 ) has to satisfy the boundary condition χ(r) = 0 to be more precise, it must at least vanish as fast as r (D−1)/2 as r goes to zero in order to rule out singular solutions [30].For Eq. ( 5) to be satisfied, one needs to find some function g(r) ensuring the presence of a constant term on its right hand side to compensate E n on its left hand one, while giving rise to a potential V (r) with well behaved wavefunctions.

Exceptional X m orthogonal polynomials
For completeness, we now give the differential equations corresponding to the X m EOPs and summarize the important properties for these two polynomials.

Exceptional X m Laguerre orthogonal polynomials
For an integer m ≥ 0, n ≥ m and k > m, the X m Laguerre orthogonal polynomial L(α) n,m (g(r)) satisfy the differential equation [23] L′′ (α) n,m (g(r)) + n,m (g(r)) The L 2 norms of the X m Laguerre polynomials are given by where is the weight factor for the X m Laguerre polynomials.
In terms of classical Laguerre polynomials the X m Laguerre polynomials can be written as For m = 0, the above definitions reduces to their classical counterparts i.e.
and for m = 1 this satisfy Eq. (80) of Ref. [1].The other properties related to the X m Laguerre polynomials are discussed in detain in Ref. [23].

Exceptional X m Jacobi orthogonal polynomials
For an integer m ≥ 1 and α, β > −1 the exceptional X m Jacobi orthogonal polynomials P (α,β) n,m (g(r)) [24,27] satisfies the differential equation The L 2 norms of the X m Jacobi polynomials are given by is the weight factor for the X m Jacobi polynomials.The above L 2 norms of the X m Jacobi polynomials holds, when the denominator of the above weight factor is non-zero for −1 ≤ g ≤ 1.To ensure this, the following two conditions must be satisfied simultaneously: where sgn(g) is the signum function.In terms of classical Jacobi polynomials P (α,β) n (g), the X m Jacobi polynomials can be written as For m = 0, the above definitions reduces to their classical counterparts i.e.
and for m = 1 this satisfy Eq. (56) of Ref. [1].The other properties related to the X m Jacobi polynomials are discussed in detain in Ref. [24] 4 Extended potentials in D-dimensions

Potentials associated with X m exceptional Laguerre polynomial
In this section we consider the extension of the usual radial oscillator potential.For this potential let us define the function F (g) as an X m (m ≥ 1) exceptional Laguerre polynomial L(α) n,m (g), where n = 0, 1, 2, 3, ....., and α > 0, the associated second order differential Eq. ( 3) is equivalent to X m Laguerre differential equation (9) where the functions Q(g) and R(g) are Using Q(g) and R(g) in Eq. ( 5), we get To get E n as explained in the above section, here we assume (g ′ (r)) 2 g(r) = C 1 (a constant not equal to zero), and for the radial oscillator potential this constant C 1 can be obtained by setting Putting g(r) in the above Eq.( 23) and define the quantum number n → n + m, we get The wavefunction can be obtained by putting Q(g) and g(r) in Eq.( 8) and is given by where N n,m is the normalization constant given by To get the correct centrifugal barrier term in D-dimensional Euclidean space, we have to identify the coefficient of 1 r 2 in Eq. ( 26) to be equal to ℓ(ℓ + D − 2), which fixes the value of α as and identifying the constant C 1 = 2ω , the energy eigenvalues (25), extended potential (26) and the corresponding wavefunction (27) in any arbitrary D-dimensions are and n+m,m ( respectively.Where ) is conventional radial oscillator potential in arbitrary D-dimensional space [31].Note that the full eigenfunction ψ as given by Eq. ( 7) with χ as given by Eq. (32).
For a check on our calculations, we now discuss few special cases of the results obtained in equations ( 31) and (32).
Case (a): m = 0 For m = 0, from Eq. ( 31) and (32) we get the well known usual radial oscillator potential in D-dimensions [31], and the corresponding wavefunctions which can be written in terms of usual Laguerre polynomials For D = 3 above expressions reduces to the well known 3-D harmonic oscillator potential.

Case (b): m = 1
For m = 1, the obtained potential (35) is the rationally extended D dimensional oscillator potential studied earlier in Ref. [26] The corresponding wavefunctions in terms of exceptional X 1 Laguerre orthogonal polynomial can be written as For D = 3, the above expressions matches exactly with the expressions given in Ref. [3].
Case (c): m = 2 For m = 2, In this case the extended potential and the corresponding wavefunctions in terms of X 2 Laguerre orthogonal polynomials are given by and 2 ).(38)

Potentials associated with X m exceptional Jacobi polynomial
Let us consider the case where the second order differential equation ( 3) coincides with that satisfied by X m Jacobi polynomial P (α,β) n,m , where n = 1, 2, 3, .....; m ≥ 1; α, β > −1 and α = β.Thus the function F (g) in Eq. ( 3) is equivalent to P (α,β) n,m (g) and the other two functions Q(g) and R(g) are given by Eq. ( 15) Using the above equations in Eq. ( 5) and after doing some straightforward calculations for s-wave (ℓ = 0), we get and the wavefunction (8) becomes 2 ) where the effective potential, V ef f,m (r) is given by and the normalization constant It is interesting here to note that when this extended potential is purely non-power law, the potential given by Eq.( 42) has an extra term (D−1)(D−3) 4r 2 which behaves as constant background attractive inverse square potential in any arbitrary dimensions except for D = 1 or 3.For power law cases (e.g.radial oscillator potential), this background potential give the correct barrier potential in arbitrary dimensions (as shown in Eq. ( 31)).On using Eq. ( 6) in Eq.( 40) and assume a term g ′2 (r) 1−g 2 (r) = C 2 (a real constant not equal to zero), we get a constant term on right hand side which gives the energy eigenvalue E n .There are several possibilities of g(r) which produces this constant C 2 .If we consider g(r) = cosh r ; 0 ≤ r ≤ ∞ and define the parameters α 2 and the quantum number n → n + m; m ≥ 1, Eq. (40) and Eq.(41) gives, and the wave function is the conventional generalized Pöschl Teller (GPT) potential.Note that the full eigenfunction is given by Eq. ( 7) with χ as given by Eq. ( 46).
Here we see that the energy eigenvalues of conventional potentials are same as the rationally extended D-dimensional potentials (i.e they are isospectral).
It is interesting to note that compared to D = 3, the only change in the potential in D-dimensions is the extra centrifugal barrier term (D−1)(D−3) 4r2 , and χ(r) is unaltered while only ψ(r) is slightly different due to r (D−1)/2 .It is also worth pointing out that even in three dimensions, the GPT potential (47) can be analytically solved only in the case of S-wave, i.e. l = 0.
We now show that approximate solution of the GPT potential problem for arbitrary l can, however, be obtained in D dimensions.

Approximate solutions for arbitrary ℓ
In this section, we solve the D-dimensional Schödringer equation (1) with arbitrary ℓ and obtain the effective potential (as given in Eq. ( 40)) with an extra ℓ dependent term i.e., So, in order to get the appropriate centrifugal barrier terms in the above effective potential, we have to apply some approximation.Following [25], we consider the approximation Thus effectively, one has approximated a problem for the l'th partial wave to that of l = 0 but with different set of parameters compared to the usual l = 0 case.In that case, the effective potential (42) becomes Now we define the parameters α and β in terms of modified parameters 1 B ′ and where E 0 is factorization energy2 .For unbroken SUSY, these partner potentials satisfy a shape invariant property where a 1 is a set of parameters, a 2 is a function of a 1 (say a 2 = f (a 1 )) and the remainder R(a 1 ) is independent of x.
The eigenstates of these partner potentials are related by Where A, A † and superpotential W (x) are defined as The factorized Hamiltonians in terms of A and A † or in terms of partner potentials V (±) are given by

Extended radial oscillator potentials
We now show that the extended radial oscillator potentials that we have obtained in D dimensions in Sec. 4 provide us with yet another example of shape invariant potentials with translation.
For the radial oscillator case the ground state wave function χ (−) 0,m (r) is given by Eq. (32) i.e. where ) and φ m (r) ∝ L Here we see that the ground state wave function of the extended radial oscillator potential in higher dimensions whose solutions are in terms of EOPs differs from that of the usual potential by an extra term φ m (r) and the corresponding superpotential where Using W (r), we get V (−) m (r)(= W (r) 2 − W ′ (r)) same as in Eq. ( 31) and V (+) m (r)(= W (r) 2 + W ′ (r)) is given by From the above equations ( 31) and ( 73), the potential V (+) m (r) can be obtained directly by replacing l −→ l + 1 in V (−) m (r) and satisfy Eq. ( 65).This means these two partner potentials are shape invariant potentials (with translation).Thus we see that the same oscillator potential V (r) = and For D = 4 and

Extended Pöschl-Teller potentials
Let us first discuss the extended GPT (with l = 0) as discussed in Sec.4.2.In this case, the ground state wave function χ Where Using W (r), the partner potential V The potential V (+) ef f,0 (r) can also be obtained simply by replacing A ′ → A ′ − 1 in V (−) ef f,0 (r) and satisfy Eq.(65).For D = 3 and ℓ = 0 the parameters A ′ → A; B ′ → B and thus these potentials corresponds to the conventional shape invariant Pöschl Teller potentials given in ref. [18].