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We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of

Supersymmetric quantum mechanics (SUSYQM) is a generalization of the factorization method commonly used for the harmonic oscillator. The factorization technique begun by Darboux [

The current form of SUSYQM appeared in 1981 [

In the next section, we will describe the general formalism of SUSYQM and in

Throughout this paper, we use units such that

Thus, the product

Operators

The groundstate energy of this Hamiltonian is then given by

Thus, the non-vanishing of the groundstate energy implies that either

The Hamiltonians

Since _{±}

where

Consider a system described by the superpotential

Although we assume that our Hamiltonians are hermitian, hermiticity is not necessary to generate real eigenvalues. Replacing the sufficient but not necessary condition of hermiticity with the weaker condition of PT symmetry, has led to the discovery of new potentials with real energy eigenvalues [

The remainder of this manuscript is devoted to the study of the shape invariance condition and its solutions.

If the superpotential

where

From Equation (7) it follows that for a shape invariant system, the partner potentials

In terms of operators _{±}

As we will see in

From Equation (9), we see that Hamiltonians

Again from Equation (9), we see that since

Thus, for a system with a given shape invariant superpotential, the eigenvalues and eigenfunctions can be determined analytically. This result makes it very important to find all such potentials. In the past, researchers had found a list of additive shape invariant potentials [

We will now show that the symmetry behind the shape invariance is essential in building the algebraic structures known as

The starting point of our construct is the shape invariance condition given by Equation (9), which we rewrite as

where

The left hand side of Equation (12) resembles a commutation relation. This suggests that we use

where

Observe now that the right hand side of Equation (14) matches the left hand side of Equation (12) provided that we make the following mappings

Since we know that

Let us look at some examples to illustrate this procedure.

• Translation:

If the change of parameters is a translation, then the function

We have

• Scaling:

For shape invariant potentials characterized by a scaling change of parameters, the corresponding function

Indeed

• Cyclic:

Cyclic potentials form a series of shape invariant potentials that repeats after a cycle of

where

• Other choices of parameters follow from more complicated choices for

we obtain the change of parameters:

Now, let us get back to the building of the potential algebra. In terms of operators

Thus, the commutation relation of operators

where

Putting together the above results, we arrive at the following

Lemma:

As an example let us build the potential algebra corresponding to the Pöschl-Teller II potential. The potential

is generated by the superpotential

The shape invariance is now evident if we observe that

Now we can identify the main objects of our model and build the corresponding algebra:

1. The parameters of the model are

2. The change of parameter

3. From the concrete shape invariance condition (32) of this potential we get

4. Defining

satisfying the commutation relations

Once we know the potential algebra for a given potential, we can use its representations to obtain the energy spectrum for the Hamiltonian. Using Equations (25) and (26), we observe that

From the reciprocal of the mapping Equation (13), we obtain

with the function

It can be explicitly checked that

where we have used

Keeping in mind that

Next, we will determine the allowed values of

Iterating this procedure we can generate a general formula for

Substituting

The profile of

Generic behaviors of

One obtains the finite dimensional representations of

For example, let us consider the scaling change of parameters

which is a deformation of the standard

For this case, from Equations (42) and (36) one gets

Note that for scaling problems [

The spectrum of the Hamiltonian

Therefore, the eigenenergies are

in agreement with the known results [

Since we have demonstrated the value of shape-invariant superpotentials, the question now becomes how to find such superpotentials. This question is equivalent to asking how to solve the difference-differential Equation (9) to find the list of desired superpotentials

We begin our discussion of known shape invariant systems by considering only superpotentials that do not depend explicitly on

Previous work [

The complete family of

Name | Superpotential |
---|---|

Harmonic Oscillator | |

Coulomb | |

-D oscillator | |

Morse | |

Rosen-Morse I | |

Rosen-Morse II | |

Eckart | |

Scarf I | |

Scarf II | |

Gen. Pöschl-Teller | |

Because of additive shape invariance, the dependence of

Thus, all conventional additive shape invariant superpotentials are solutions of the above set of non-linear partial differential equations [

and

In doing so, we derive a new proof that the superpotentials shown in

The general solution to Equation (51) is

Therefore, to generate all shape invariant superpotentials, we need to determine all possible combinations of

To determine

where dots and primes represent derivatives taken with respect to

Inserting the form of the general solution (52) into (53) yields

where

• Case 1:

• Case 2:

• Case 3: Neither

For each of these cases we can determine the form of

Inserting the form of the general solution (52) into (55) yields

Now, we will analyze each of the three cases in detail.

Let

We now find

or equivalently,

where

Since

The solution depends on the value of the constants

• Case 1A:

• Case 1B:

• Case 1C:

In this case, let

or equivalently,

Integrating it once, we get

This equation can be simplified by setting

The solutions for

• Case 2A:

Setting

• Case 2B:

• Case 2C:

In this case, since

Integrating,

Thus, again we have

Note that this is the same differential equation as (60) and will therefore give the same solutions for

This equation is again simplified by setting

Since

For different values of

• Case 3A:

Its solution is

• Case 3B:

where we have set

• Case 3C:

Thus, we have generated all the superpotentials of

In the previous section we generated the complete list of additive shape-invariant superpotentials that do not depend explicitly on

We now extend our formalism to include “extended" superpotentials that contain

We will now substitute Equation (68) into the shape invariance condition given in Equation (7), for which we compute

and

Since

Similarly,

We substitute these into Equation (7) and stipulate that the equation hold for any value of

For

This equation is identical to Equation (50) for

Additionally, Equation (70) provides a constraint for the possible energy spectra of the extended potentials. from Equation (70), the function

Thus, the function

Thus far, each of the known extended potentials contains a solution from

1. Further extended superpotentials may be found based on the conventional superpotentials. In this case, the potentials derived from the extended superpotential will be isospectral with the potentials derived from the corresponding conventional superpotential;

2. While

While supersymmetric quantum mechanics began as a simplified model to account for dynamical symmetry breaking, the application of this formalism to quantum mechanics has become an important field in its own right. In this manuscript we have reviewed research on supersymmetric quantum mechanics with a particular emphasis on the property of shape invariance. As we have shown, shape invariance is a sufficient condition for exact solvability of quantum mechanical problems;

However, in its traditional form, the shape invariance condition Equation (7) is a difference-differential equation and is difficult to solve. It has recently been established that for additive shape-invariant superpotentials that do not explicitly depend on

Since 2008, new sets of additive shape invariant potentials have been discovered [

It may also be possible to extend this method to other forms of shape invariance such as multiplicative or cyclic. For these, the potentials are generally not available in terms of known functions, except in very special cases (

This research was supported by an award from Research Corporation for Science Advancement

Note the constant

We assume that as

In the last line we have used the fact that

These constraints are:

We have used

In some cases these are additive constants and subtracted away. If we provide a common floor to all potentials, demanding that their groundstate energies be zero, we will find that all known solvable potentials pick up a

Normalizability of the groundstate

By substituting