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In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-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 g(x)~P(x)/x
^{2} and g(x)~x
^{2}/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.

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 [

According to the proposition given in Equation (A5), the generalized canonical transformation of Equation (A1) becomes

that can be rewritten as

where

and

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

such 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

being

In short, it is worth noting that

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 [

where

As mentioned before, depending on the choice of

where we have used the explicit form of function

Let us consider the identity

to rewrite Equation (5) as

with the purpose to use the integral [

That is, the above integral and the choice of

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

with

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

and

respectively, where

As can be proved, the plus (+) case leads to

which indicates that the corresponding transformation is

Then, according to Equation (3), in this case

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.

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

with eigenvalue

and wavefunction

As before, with the aim to have physically acceptable solutions it becomes necessary to redefine the original parameters a, b and c as follow

with

with their corresponding wavefunctions

and energy spectra

where

At this point, is important to notice that potential

corresponds to the Rosen-Morse II hyperbolic potential and

to the exactly solvable Eckart potential. Also, we want to point out that

and

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

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 (3)

such that

leading to

with eigenvalue

and eigenfunction

Similarly, to have physically acceptable wavefunctions the original parameters a, b and c are defined as

where

with energy spectra

and eigenfunctions

where

Specifically, the

and

with their respective wavefunctions

and

with the same energy spectra. At this point, we want to notice that potential

a). Option with

According to the third option for

In fact, by choosing the minus sign in Equation (5) and the down integral limit as a new parameter q, one obtains

from which

being

leading, from Equation (11), to

Thus, by defining

and

it becomes possible to identify the potential

the eigenvalue

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

with

Thus, to have a node in

and

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

leads to a polynomial of n degree in the variable

that

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

with the corresponding wavefunctions

It should be pointed out that in this case the potential V(r) has a minimum value

in

on condition that

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

values of A, B, and C. For example, the choice of

which is identified with the standard Hulthén potential with eigenvalues [

and eigenfunctions

In a similar way, the selection

Hulthén potential already given by Morales et al., [

with energy spectra

and wavefunctions

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 [

However, due to the fact that the approach considered until now in this work is consistent with

with

and

as well as

Thus, by using Equation (114) and the negative of Equation (89) we obtain

and

with the condition

and

b). Option with

Similarly to the above case, in this new situation one have

that leads to the transformation

where

that, from Equation (3), lets write

In this new situation, by considering the identities

and

as well as

one find the multiparameter exponential-type potential

with eigenvalue

and eigenfunction

At first glance, the potential of Equation (129) can be considered as a new one and however it should be noticed that it can be written as

That is, the above

approximation schemes proposed to

other particular radial potentials can be derived from our proposal as, for example, the recently considered Manning-Rosen potential [

In this paper, we present a method to obtain the general canonical form of second order differential equations on the field of theoretical physics. The procedure is similar to the method proposed by Levai [

This work was partially supported by the projects UAM-A-CBI-2232004 and 009. JGR thanks to the Instituto Politécnico Nacional for the financial support given through the COFAA-IPN project SIP-200150019. JGM acknowledges to the IPN-ESFM, for the hospitality during his PhD studies in Science and Technology.

J.Morales,J.García-Martínez,J.García-Ravelo,J. J.Peña, (2015) Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions. Journal of Applied Mathematics and Physics,03,1454-1471. doi: 10.4236/jamp.2015.311173

The general second order DE

is transformed into their standard canonical form [

by means of the transformation

i.e.

where apostrophes indicates derivative with respect to the argument.

In order to have a generalized canonical transformation of Equation A1, we consider the variable change

to obtain the general transformation [

Consequently, one have

being

where

It will be noticed that, for the particular case