Consider a system described by the superpotential , defined over the domain . Corresponding partner potentials are given by . This is a rather complicated potential, and rarely analyzed in quantum mechanics courses. However, for , the potential reduces to the infinite square well, with the bottom of the potential at and zero groundstate energy. We know that the corresponding eigenstates are given by and eigenvalues . Thus, using the familiarity with the relatively simpler potential , we are able to derive all of the eigenvalues and eigenfunctions of . Then the inter-relations expressed through Equation (5) enable us to determine eigenvalues and eigenfunctions of .

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 [14,15,16]. Considerable work has been done on the study of SI potentials with PT symmetry. In reference [17], the author examined the shape invariant hyperbolic Rosen-Morse potential as a case of exact solvability with PT invariance. In reference [18], the Scarf II potential was shown as an example of spontaneous PT-symmetry-breaking. The case of a square well with discrete PT symmetry but with intervals of non-hermiticity was shown in reference [19] to produce real eigenvalues. Authors of [20] constructed the spectrum of a square well of imaginary strength, to obtain the hierarchy of SI potentials, while authors of [21] constructed a set of solvable rational potentials, and related the PT symmetry condition to the condition that they be free of singularities. However, in this paper we limit ourselves to the case of hermitian Hamiltonians, i.e., real .

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

From Equation (9), we see that Hamiltonians and differ only by the constant . We already know that . Let us determine the first excited state of ; i.e., . Hence, using [33], we find that the energy of the first excited state of and of the groundstate of both are given by . Similarly, to determine , we use the isospectrality condition (5) to relate it to . But by the shape invariance condition (9), = . Following the method used for determining , we find , and hence, . Extending this argument to higher excited states of , we get

(10)

Again from Equation (9), we see that since and only differ by the constant , they must have common eigenfunctions. Hence, from Equation (6), we have . Then the isospectrality condition (5), requires . Iterating this procedure, we obtain

(11)

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 [10,11], mostly by trial and error [34,35]. In Section 5, we will discuss how to find solutions of Equation (9) for the additive case. Before that, however, in the next section we will show why the shape invariance condition leads to solvability.

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

(12)

where , and is a function that models the change of parameter . For example, if the change of parameter is a translation, .

The left hand side of Equation (12) resembles a commutation relation. This suggests that we use and to build the generators of the potential algebra. To transform the above shape invariance condition into an exact commutator, we replace operators and by

(13)

where is a constant parameter, is an auxiliary variable, and . The function will be appropriately chosen to emulate the relationship between parameters and . Thus, to generate , we multiplied the operator from right by and replaced the parameter by the differential operator . If we now compute the commutator between operators and , we find [36]

(14)

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

(15)

Since we know that , these mappings require that the function satisfy

(16)

Let us look at some examples to illustrate this procedure.

• Translation:

If the change of parameters is a translation, then the function that models it is the identity function

(17)

We have , which gives the desired change of parameters.

• Scaling:

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

(18)

Indeed where we denoted .

• Cyclic:

Cyclic potentials form a series of shape invariant potentials that repeats after a cycle of iterations. These potentials appear also in connection with the dressing chain formalism [37]. To satisfy the cyclic parameter change, the function should obey . The projective map with specific constraints [38] on the parameters , and satisfies such a condition [27]. The function satisfying Equation (16) in this case is given by

(19)

where are solutions of the equation and is an arbitrary periodic function of with period .

• Other choices of parameters follow from more complicated choices for . For example taking

(20)

we obtain the change of parameters: .

Now, let us get back to the building of the potential algebra. In terms of operators and , the shape invariance condition (12) becomes:

(21)

Thus, the commutation relation of operators and generates an operator that has no -dependence. If we now define a third operator in terms of the operator , the shape invariance condition becomes simply one of the commutation relations of the potential algebra. In particular, if we define

(22)

where is an arbitrary constant, the three commutators among these generators are given by [39]

(23)

Putting together the above results, we arrive at the following

Lemma: To any shape invariant system characterized by

(24)

for which we can find a function such that for arbitrary parameters and , we can associate an algebra [40]generated by

(25)

(26)

(27)

satisfying the commutation relations

(28)

where

(29)

The function in Equation (29) is given by the shape invariance condition (24), while the function satisfies the compatibility equation: , where models the change of parameter of Equation (24).

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

(30)

is generated by the superpotential , through , where . Its supersymmetric partner is given by

(31)

The shape invariance is now evident if we observe that . Therefore, in terms of and operators, the shape invariance (24) for the Pöschl-Teller II potential reads

(32)

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

1. The parameters of the model are and , where is an arbitrary positive constant greater than so that is a positive quantity;

2. The change of parameter is thus given by . This is a translational change of parameter with the translation parameter . Translation implies that the function satisfying (16) is the identity function ;

3. From the concrete shape invariance condition (32) of this potential we get . Then, the function is given by if we choose the arbitrary constant ;

4. Defining and as prescribed by Equations (25) and (27), we obtain the differential realization of the algebra’s generators:

(33)

satisfying the commutation relations , and . Thus, shape invariance of this system implies that the system has a potential algebra given by [?, 41, 42].

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

(34)

From the reciprocal of the mapping Equation (13), we obtain . Consequently, the spectrum of the operator gives the spectrum of the Hamiltonian. To find the concrete values for the energy, we need to know the action of individual operators and respectively on a set of eigenvectors of the operator . In this general case, , and commute with the Casimir operator given by

(35)

with the function (defined up to an additive constant) such that

(36)

It can be explicitly checked that does indeed commute with , and [41]. In a basis in which and are diagonal, play the role of raising and lowering operators, respectively. Operating on an arbitrary eigenstate we have

(37)

where we have used .

Keeping in mind that , and observing that , we see that in order to find the energies of the system one must find the coefficients . If we apply the third commutation relation of Equation (23) to , we obtain using Equation (36)

(38)

Next, we will determine the allowed values of and the corresponding values . Let us say corresponds to the lowest state in a given representation of the algebra. This implies that , which means . From Equation (38) we get

(39)

Iterating this procedure we can generate a general formula for

(40)

Substituting leads to

(41)

The profile of determines the dimension of the representation. For example, let us consider the two cases presented in Figure 1.

One obtains the finite dimensional representations of Figure 1a, by starting from a point on the vs. graph corresponding to , and moving in integer steps parallel to the -axis to the point corresponding to . At the end points, , and we get a finite representation. If is decreasing monotonically, as in Figure 1b, there exists only one end point at . Starting from the value of can be increased in integer steps up to infinity. In this case we have an infinite dimensional representation. As in the finite case, labels the representation. The difference is that here takes continuous values. Similar arguments apply for a monotonically increasing function . Having the representation of the algebra associated with a characteristic model, we obtain, using Equation (41), the complete spectrum of the system.

For example, let us consider the scaling change of parameters . Consider the simple choice , where is a constant. This choice generates self-similar potentials studied in references [44,26,25]. In this case, combining Equation (18) with Equation (28) yields:

(42)

which is a deformation of the standard Lie algebra.

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

(43)

Note that for scaling problems [25], one requires , which leads to . From the monotonically decreasing profile of the function , it follows that the unitary representations of this algebra are infinite dimensional. If we label the lowest weight state of the operator by , then . Without loss of generality we can choose the coefficients to be real. Then one obtains from (38) for an arbitrary

(44)

The spectrum of the Hamiltonian is given by

(45)

Therefore, the eigenenergies are

(46)

in agreement with the known results [25].

We begin our discussion of known shape invariant systems by considering only superpotentials that do not depend explicitly on , which we call “conventional” superpotentials. In Table 1 we list the known “conventional" superpotentials that meet this criterion.

Previous work [45,46,47] has proven that this list of conventional shape-invariant superpotentials is complete. We now show a new proof of this completeness which has the advantage of being significantly more straightforward and elegant than it predecessors.

Because of additive shape invariance, the dependence of on and is through the linear combination ; therefore, the derivatives of with respect to and are related by: . Since Equation (7) must hold for an arbitrary value of , if we assume that does not depend explicitly on , we can expand in powers of , and the coefficient of each power must separately vanish. Expanding the right hand side in powers of yields

(47)

(48)

(49)

Thus, all conventional additive shape invariant superpotentials are solutions of the above set of non-linear partial differential equations [46,47]. Although this represents an infinite set, note that if equations of and are satisfied, all others automatically follow. Therefore, we proceed to find all possible solutions to the two partial differential equations:

(50)

and

(51)

In doing so, we derive a new proof that the superpotentials shown in Table 1 are the only possible solutions.

The general solution to Equation (51) is

(52)

Therefore, to generate all shape invariant superpotentials, we need to determine all possible combinations of , , and that satisfiy Equation (50). We will ignore the case when both , and are constants, as this corresponds to a flat potential with no -dependence. We will also ignore the case in which , and are linearly dependent on each other; i.e., . In this case, . If we redefine , this case becomes equivalent to a superpotential with a shifted parameter and constant which will be considered shortly. We can therefore assume that and are linearly independent of each other without loss of generality. Note that from here onward, we will use lower case Greek letters to denote constants that are independent of both and .

To determine , we first focus on determining . To do so, we take two derivatives of (50) with respect to . This leads to the following differential equation:

where dots and primes represent derivatives taken with respect to and respectively. Since , this simplifies to:

(53)

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

(54)

where is a function of , and is independent of . Since and are linearly independent, we find that there are only three possible ways for the left-hand-side of Equation (54) to be independent of :

• Case 1: is a constant and ;

• Case 2: is a constant and ;

• Case 3: Neither nor are not constants, but and .

For each of these cases we can determine the form of . Then we can determine and for these three cases. This we do by taking two derivatives of (50), this time one with respect to and another with respect to . This yields:

(55)

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

(56)

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

In the previous section we generated the complete list of additive shape-invariant superpotentials that do not depend explicitly on . However, a new class of superpotentials was discovered by Quesne [50,51]. It has been shown [46] that these “extended" superpotentials obey the shape invariance condition in the form of Equation (7) only when is allowed to depend explicitly on . While this dependence is frequently ignored by the conventional notation that sets , we will show that this constraint results in important consequences for the energy spectrum of the resulting potentials. In each case, the new potential is isospectral with a potential that arises from one of the “conventional" superpotentials listed in Table 1. Authors of [52,53,54] have added an infinite number of potentials that belong in this class, and extended shape invariant potentials continue to be objects of intense research [55,56].

We now extend our formalism to include “extended" superpotentials that contain explicitly. To do so, we expand the superpotentials in powers of . Hence, we define

(68)

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

and

Since , Expanding in powers of

Similarly,

We substitute these into Equation (7) and stipulate that the equation hold for any value of . After some significant algebraic manupulation we find that the following equation must be true separately for each positive integer value of :

(69)

For , we obtain

(70)

This equation is identical to Equation (50) for -independent ’s. We have already found a set of solutions for Equation (70) that includes all known -independent superpotentials. The extended cases [50,51] are solutions to (69) as well, as shown in [46]. Note that Equation (69) provides a consistency condition for all -dependent potentials; however, these are not easy to solve to determine new potentials.

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

(71)

(72)

Thus, the function , and hence the energy of the system, is given entirely in terms of the -independent part of the superpotential. Hence, the eigenvalues are not affected by the -dependent extension of the superpotential.

Thus far, each of the known extended potentials contains a solution from Table 1 as the -independent term of the superpotential . Therefore, each of the expanded potentials is isospectral with its corresponding conventional potential. Future possibilities for finding new shape-invariant superpotentials fall into one of two categories:

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 is required to satisfy Equation (70), which is equivalent to Equation (50) for -independent ’s, is not required to satisfy Equation (48). Rather, the additional constraints for an extended are supplied by Equation (69). It therefore may be possible to find an -dependent superpotential whose -independent term is not equivalent to a conventional superpotential. Intriguingly, it therefore may still be possible to discover shape-invariant systems with new energy spectra.