Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions

In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schrödinger-like DE. Our proposal is based on an auxiliary function g(x) which determines the transformation needed to find exactly-solvable potentials associated to a known DE. To show the usefulness of the proposed approach, we consider explicitly their application to the hypergeometric DE with the aim to find quantum potentials with hypergeometric wavefunctions. As a result, different potentials are obtained depending on the choice of the auxiliary function; the generalized Scarf, Posh-Teller, Eckart and Rosen-Morse trigonometric and hyperbolic potentials, are derived by selecting g(x) as constant and proportional to the P(x) hypergeometric coefficient. Similarly, the choices ( ) ( ) 2 ~ g x P x x and ( ) ( ) 2 ~ g x x P x give rise to a class of exactly-solvable generalized multiparameter exponential-type potentials, which contain as particular cases the Hulthén, Manning-Rosen and Woods-Saxon models, among others. Our proposition is general and can be used with other important DE within the frame of applied matematics and physics.


Introduction
Exactly and quasi-exactly solvable potential models are important in practically any field of theoretical quantum chemistry and physics for two principal reasons: first, they are useful to understand the behavior of quantum systems and second, can be used as a basis to study problems that can only be treated using perturbative and nonperturbative procedures.In spite of the above, the exactly solvable Schrödinger equations are rather scanty and in their research different analytical or operational approaches have been used.Also, the well known exactly-solvable Scarf, Eckart, Rosen-Morse I and II, Poschl-Teller I and II as well as Hulthén, Manning-Rosen and Woods-Saxon potentials, all they have, as common feature, hypergeometric wavefunctions.Similarly, the harmonic oscillator, Morse, Coulomb or Kratzer potential models have confluent hypergeometric solutions.Consequently, it becomes clear that the exact solution for the Schrödinger equation is reduced to the study of hypergeometric and/or confluent hypergeometric Differential Equations (DE).At this regard, many efforts have been conducted to find the intermapping between different solvable potentials [1] [2] with the aim to give a unified treatment of partner potentials [3].For example, in the case of potentials with the hypergeometric wavefunctions, the hexagonal diagram proposed by Cooper et al. [4] is very useful to show how all the shape invariant potentials are inter-related.Also, it has been proposed a pre-potential approach to study of Eckart-type potentials [5] and a five-parameter exponential-type potential to unify the treatment of exactly solvable trigonometric potential models [6]- [8].Furthermore, to find exactly solvable Schrödinger equations different methods based on Supersymetric Quantum Mechanics (SUSY-QM) [9], Asymptotic Iteration Method (AIM) [10] and on the transformation of a Schrödinger equation into a hypergeometric-type DE by the Nikiforov-Uvarov (NU) [11] approach, have been used.In this work, we present a proposal that can be considered inverse to the NU method.However, instead of transforming a DE into a Schrödinger equation we consider the transformation of a general homogeneous linear second order DE to their canonical form.Obviously, the general DE has as particular cases the hypergeometric and confluent hypergeometric DE.For that reason, this work aims at finding solvable potentials with hypergeometric wavefunctions leaving the treatment of models with confluent hypergeometric solutions elsewhere [12].Thus, the proposed approach to transform a general DE into a Schrödinger-like equation is given in next section by means of the canonical transformation method given in the Appendix.The application of the present proposal is given in Section 3, where we consider the hypergeometric DE by means of an auxiliary function ( ) g x , defined in Section 2, that indicates the required transformation.That is, as will be shown, each possibility of ( ) g x leads to different generalized potentials which are reduced to well known particular cases.Finally, in Section 4 we presents the concluding remarks emphasizing that our proposition is general for which can be directly applied to other important DE.

Transformation of a General DE into a Schrödinger-Like Equation
According to the proposition given in Equation (A5), the generalized canonical transformation of Equation (A1) becomes that can be rewritten as where that will be referred as auxiliary function hereafter.Consequently, from the above, the variable r is given by in such a way that Equation (A5) is feasible on condition to have the inverse function Furthermore, the general transform given in Equation (A6) can be rewritten as where we have used the auxiliary function and the fact that Consequently, by substituting Equation (7) into Equation ( 2), the generalized canonical form of Equation (A1) will be ( At this point, it should be noticed that above equation can be identified with a Schrödinger-like equation where the ( ) wavefunction can be obtained from Equation (7) on condition that coefficients ( ) ( ) ( ) , , , , being { }

Application to the Hypergeometric DE
To show the usefulness of the approach given in above section, in the search of exactly-solvable Schrödinger equations, let us apply the above results to the hypergeometric DE ( ) ( ) ( ) ( ) ( ) whose solution is [13] ( ) ( ) ( ) ( ) ( ) where is the Pochhammer symbol and parameters a and b are constant with 0, 1, 2, 3, .c ≠ − − −  That is, in this case Equation (12) matches with Equation (A1) provided that where we have used the explicit form of function ( ) So, in order to have at least one constant term associated to the eigenvalue ( ) E a b of Equation (11), one option is to propose the following cases: ( ) ( ) . All these options are considered explicitly in what follows by using ( )

Generalized Potentials from g(x) Constant
Let us consider the identity ( ) ( ) to rewrite Equation ( 5) as with the purpose to use the integral [14] That is, the above integral and the choice of ( ) g x as constant given by ( ) allows to evaluate the integral of Equation ( 16) as Consequently, in this case the corresponding transformation is ( ) leading to the particular cases + and − specified by ( ) Thus, according to Equation (3), which means and Thus, as mentioned before, the potential function ( ) ( ) leading to the corresponding generalized potentials with eigenvalues ( ) ( ) and eigenfunctions Also, in order to have physically acceptable wavefunctions, the original parameters a, b and c are redefined as ( )  , for which the potentials of Equation ( 26) become as well as their respective wavefunctions ( ) with energy spectra ( ) At this point, is important to notice that particular cases and are identified with the well known trigonometric respectively, where Jacobi polynomials.As a consequence the following question arises: From where comes the Posch-Teller potential?The answer to that question follows from the negative of ( ) g x given in Equation ( 18) but now along with the integral [14] As can be proved, the plus (+) case leads to given in Equation (33).So, we are going to consider only the minus (−) case.That is, now Equation ( 16) gives rise to ( ) which indicates that the corresponding transformation is ( ) Then, according to Equation (3), in this case ( ) W r will be ( ) ( ) ( ) leading, from Equation (11), to the potential with eigenvalue ( ) and eigenfunction ( ) Finally, using the conditions for physically acceptable wavefunctions one obtains that leads to the potential with eigenfunctions ( ) and energy spectra ( ) that coincides with the well known Posch-Teller potential.

Generalized V(r) from g(x) Proportional to P(x)
Similarly to the cases considered in above section, the choice of ( ) ( ) ( ) can be worked by means of the integrals In fact, the use of Equation (49) lets write Equation ( 16) as leading to the corresponding transformation ( ) So, according to Equation (3) Thus, the potential function V(r) and the eigenvalue E can be identified from Equation (11) as to obtain the corresponding generalized potentials As before, with the aim to have physically acceptable solutions it becomes necessary to redefine the original parameters a, b and c as follow This fact, leads to ( ) with their corresponding wavefunctions and energy spectra ( ) ( ) where corresponds to the Rosen-Morse II hyperbolic potential and ( ) to the exactly solvable Eckart potential.Also, we want to point out that ( ) respectively.Similarly to the cases analyzed in section III.1), the question is now; How can obtain the Rosen-Morse II trigonometric potential?To answer that question, we are going to use the same choice of ( ) but now proportional to the negative ( ) that together with Equation (49) in Equation ( 16) permit us to obtain Thus, with the aid of the identity the corresponding transformation is Consequently, according to Equation ( Similarly, to have physically acceptable wavefunctions the original parameters a, b and c are defined as ( )  and p a new real parameter.Consequently, the potentials are corrected as ( ) with energy spectra ( ) ( ) ) Specifically, the ( ) with their respective wavefunctions with the same energy spectra.At this point, we want to notice that potential ( ) V r − corresponds, as expected, to the exactly solvable Rosen-Morse II trigonometric potential and ( ) V r + seems to be a new exactly solvable potential. .

A class of Multiparameter
According to the third option for ( ) g x in this case one have ( ) ( ) 2 0 and 1. 1 In fact, by choosing the minus sign in Equation ( 5) and the down integral limit as a new parameter q, one obtains an additional set of new parameters.In consequence, from Equation (3),

( )
W r is given by ( ) ( ) ( ) ( ) leading, from Equation (11), to Thus, by defining ( ) and, from Equation ( 7), the eigenfunction In what follow, the parameters a,b and c will be calculated by considering the boundary conditions of the system in order to have physically acceptable wavefunctions.For example, by combining Equation (89) and Equation (90) one obtains ( ) Thus, to have a node in s r is necessary to apply the condition ( ) On the other hand, by using Equation (94) in Equation (89) we obtain and Also, since the hypergeometric function of Equation ( 93) is an infinite series, the condition , 0,1, 2,3, leads to a polynomial of n degree in the variable ( ) 93) which gives the number of states for the system These assumptions on the original parameters {a, b, c} assure that boundary conditions are fulfilled, leading to a physically acceptable wavefunction for the Schrödinger equation under consideration.Additionally, the energy spectra for the potential ( ) V r is obtained by using Equation (99) in Equation (92).That is ( ) with the corresponding wavefunctions It should be pointed out that in this case the potential V(r) has a minimum value since in this case the argument of the logarithm function, given in Equation (105), is always positive.Consequently, Equations ( 104)-(106) ensures that potential V(r) is attractive with a infinite wall in s r .In short, the potential given in Equation ( 91) is general and contains as particular cases exactly-solvable potentials for specific values of A, B, and C. For example, the choice of which is identified with the standard Hulthén potential with eigenvalues [15] [16] ( ) ( ) Hulthén potential already given by Morales et al., [17].Also, the selection 0 , = and 1 q = gives rise to the Manning-Rosen potential [18] ( ) Another important exactly-solvable exponential potential that have hypergeometric wavefunctions is the Woods-Saxon potential which has been worked using the Numerov method for the standard model [19] or by means of the Nikiforov-Uvarov procedure for the generalized case [20].According to our results, one can show  Thus, with these new parameters, the energy spectra and wavefunctions of the generalized Wood-Saxon potential are respectively ( ) Similarly to the above case, in this new situation one have of parameters that come from Equation (A1) and Equation (5) respectively.In short, it is worth noting that ( )x F r =is involved in the construction of potential ( )(7) through the function ( ) W r .Also, according to Equation (4), there will be different transformations ( ) F r depending on the selection of the auxiliary function which means the existence of various exactly-solvable potentials having the same wavefunctions of the former DE.
=As mentioned before, depending on the choice of ( ) F r , or their corresponding auxiliary function( ) ( ) g F r ,our proposal leads to different exactly-solvable potentials.Consequently, the different options for can be obtained by Equation(3) given by Another possibility arises when us to obtain the supersymmetric or generalized using Equation (114) and the negative of Equation (89) we obtain r