## Abstract

We present the general ideas on supersymmetric quantum mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges, which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found, there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved.

### Keywords

- supersymmetric quantum mechanics
- quasi-exactly solvable potentials

## 1. Introduction

We present the general ideas on supersymmetric quantum mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges,

In this work, we try three main schemes; the first one consists on finding the eigenvalue Schrödinger equation in one dimension using the matrix representation via the appropriate factorization with ladder-like operators and finding the one parameter isospectral equation for this one. In this scheme, the wave function is written as a supermultiplet. Continuing with the Schrödinger model, we extend SUSY to include two-parameter factorizations, which include the SUSY factorization as particular case. As examples, we include the case of the harmonic oscillator and the Pöschl-Teller potentials. Also, we include the steps for the two-dimensional case and apply it to particular cases. The second scheme uses the differential representation in Grassmann numbers, where the wave function can be written as an n-dimensional vector or as an expansion in Grassmann variables multiplied by bosonic functions. We apply the scheme in two bosonic variables a particular cosmological model and compare the corresponding solutions found. The third scheme tries on extensions to the SUSY factorization and to the case of quasi-exactly solvable potentials; we present a particular case which does not form part of the class of potentials found using Lie algebras.

To establish the different approaches presented here, we will briefly describe the different main formalisms applied to supersymmetric quantum mechanics, techniques that are now widely used in a rich spectrum of physical problems, covering such diverse fields as particle physics, quantum field theory, quantum gravity, quantum cosmology, and statistical mechanics, to mention some of them:

In one dimension, SUSY-QM may be considered an equivalent formulation of the Darboux transformation method, which is well-known in mathematics from the original paper of Darboux [4], the book by Ince [5], and the book by Matweev and Salle [6], where the method is widely used in the context of the soliton theory. An essential ingredient of the method is the particular choice of a transformation operator in the form of a differential operator which intertwines two Hamiltonian and relates their eigenfunctions. When this approach is applied to quantum theory, it allows to generate a huge family of exactly solvable local potential starting with a given exactly solvable local potential [7]. This technique is also known in the literature as isospectral formalism [7, 8, 9, 10].

Those defined by means of the use of supersymmetry as a square root [11, 12, 13, 14], in which the Grassmann variables are auxiliary variables and are not identified as the supersymmetric partners of the bosonic variables. In this formalism, a differential representation is used for the Grassmann variables. Also the supercharges for the n-dimensional case read as

where

There are two forms where the equations in 1D are satisfied: in the literature we find either the matrix representation or the differential operator scheme. However for more than one dimensions, there exist many applications to cosmological models, where the differential representation for the Grassmann variables is widely applied [14, 15, 16, 17, 18]. There are few works in more dimensions in the first scheme [19]; we present in this work the main ideas to build the 2D case, where the supercharge operators become

## 2. Factorization method in one dimension: matrix approach

We begin by introducing the main ideas for the one-dimensional quantum harmonic oscillator. The corresponding Hamiltonian is written in operator form as

where

This Hamiltonian can be written in terms of the anti-commutation relation between these operators as

The symmetric nature of

Now, we build the operators

and in analogy to (5), we define the corresponding new Hamiltonian as

The antisymmetric nature of

These operators

with

Now, consider both Hamiltonians as a composite system, that is, we consider the superposition of two oscillators, one being bosonic and one fermionic, with energy

When we demand that both frequencies are the same,

In this way, we have the super-Hamiltonian

where I is a

From Eqs. (18) and (19), we can see that

The question is, what are the generators for this SUSY Hamiltonian? The answer is, considering that the degeneration is the result of the simultaneous destruction (creation) of quantum boson and the creation (destruction) of quantum fermion, the corresponding generators for this symmetry must be written as

implying that

and satisfying the following relations

We can generalize this procedure for a certain function W(q), and at this point, we can define two new operators

In order to obtain the general solutions, we can use an arbitrary potential in Eq. (3), that is,

The Hamiltonians

where the potential term V_{+}(q) is related to the superpotential function W(q) via the Riccati equation

(modulo constant

In a general way, let us now find the general form of the function W. The quantum equation (17) applied to stationary wave function

where

Then, this equation is the same as the original one, Eq. (21), that is, W is related to an initial solution of the bosonic Hamiltonian. What happens to the isopotential

the question is, what is

The family of potentials

Finally

is the isospectral solution of the Schrödinger-like equation for the new family potential (23), with the condition

This

### 2.1 Two-dimensional case

We use Witten’s idea [20] to find the supersymmetric supercharge operators

where the Schrödinger-like equation can be obtained as the bosonic sector of this super-Hamiltonian in the superspace, i.e., when all fermionic fields are set equal to zero (classical limit).

In two dimensions, the supercharges are defined by the tensorial products

with

where

where the super-Hamiltonian, (14), can be written as

where

and

The Riccati equation (20) is written in 2D as

and, using separation variables, we get

In general, we find that each potential

and we can find the isopotential as

Following the same steps as in the 1D case, we find that the solutions (22) are the same in this case. So, the general solution for

where

In the same manner, we have that

with

On the other hand, using the Riccati equation, we can build a generalization for the isopotential, using the new potential

For the other coordinate, we have

The general solutions for

where the

### 2.2 Application to cosmological Taub model

The Wheeler-DeWitt equation for the cosmological Taub model is given by

where

where the parameter

where K (or I) is the modified Bessel function of imaginary order and the function L is defined as

Using Eqs. (38) and (39), we obtain the isopotential for this model

Using Eq. (40) we can obtain general solutions for the functions

## 3. Differential approach: Grassmann variables

The supersymmetric scheme has the particularity of being very restrictive, because there are many constraint equations applied to the wave function. So, in this work and in others, we found that there exist a tendency for supersymmetric vacua to remain close to their semiclassical limits, because the exact solutions found are also the lowest-order WKB-like approximations and do not correspond to the full quantum solutions found previously for particular models [14, 15, 16, 17, 18].

Maintaining the structure of Eqs. (13)–(16), taking the differential representation for the fermionic operator

where

These rules are satisfied when we use a differential representation for these

where

where

The superspace for three-dimensional model becomes

where the indices

It is well-known that the physical states are determined by the applications of the supercharges

where we use the usual representation for the momentum

From (54)–(55), we obtain the relation

On the other hand, the first equation in (53) gives

The free term equation is written as

with the solution to

Also, Eqs. (57) and (58) are written as

whose solution is

### 3.1 The unnormalized probability density

To obtain the wave function probability density

and the integral over the Grassmann numbers is

In 2D, the main contributions to the term

and using that

By demanding that

## 4. Beyond SUSY factorization

Although most of the SUSY partners of 1D Schrödinger problems have been found [1], there are still some unveiled aspects of the factorization procedure. We have shown this for the simple harmonic oscillator in previous works [2, 3] and will proceed here in the same way for the problem of the modified Pöschl-Teller potential. The factorization operators depend on two supersymmetric type parameters, which when the operator product is inverted, allow us to define a new SL operator, which includes the original QM problem.

The Hamiltonian of a particle in a modified Pöschl-Teller potential is [1, 22]

where

The eigenvalue problem may be solved using the Infeld and Hull’s (IH) factorizations [23],

where the IH raising/lowering operators are given by

and where

Beginning with the zeroth-order eigenfunctions, the eigenfunctions can be found by successive applications of the raising operator, which only increases the value of the upper index. That is,

We repeatedly apply the creation operator

### 4.1 Two-parameter factorization of the Pöschl-Teller Hamiltonian

Following our previous work [2, 3], we define two non-mutually adjoint first-order operators,

where

By multiplying the first equation by

This Riccati equation was found in [24]; it has the solution

The constant

The general solution to the pair of coupled equations (69) is

and

where

### 4.2 Reversing the operator product: new Sturm-Liouville operator

Now we invert the first-order operators’ product, keeping in mind Eq. (64),

Then we can define a new Sturm-Liouville (SL) eigenvalue problem

with the weight function

This new SL operator is isospectral to the original PT problem. The zeroth-order eigenfunction is easily found by solving

### 4.3 Regions in the two-parameter space

We may recover the original QM problem when

where

where the partner SUSY potentials are given by

The zeroth-order eigenfunction is defined by

## 5. Quasi-exactly solvable potentials

In exactly solvable problems, the whole spectrum is found analytically, but the vast majority of problems have to be solved numerically. A new possibility arises with the class of QES potentials, where a subset of the spectrum may be found analytically [25, 26, 27]. QES potentials have been studied using the Lie algebraic method [25]: Manning [28], Razavy [29], and Ushveridze [30] potentials belong to this class (see also [31]). These are double-well potentials, which received much attention due to their applications in theoretical and experimental problems. Furthermore, hyperbolic-type potentials are found in many physical applications, like the Rosen-Morse potential [32], Dirac-type hyperbolic potentials [33], bidimensional quantum dot [34], Scarf-type entangled states [35], etc. QES potentials’ classification has been given by Turbiner [25] and Ushveridze [30].

Here we show that the Lie algebraic procedure may impose strict restrictions on the solutions: we shall construct here analytical solutions for the Razavy-type potential

### 5.1 A Razavy-type QES potential

Let us consider Schrödinger’s problem for the Razavy-type potential

For simplicity, we set

Here the potential function is the hyperbolic Razavy potential

In the case of the Razavy potential, the solutions obtained by Finkel et al. are

with the parameters

with

### 5.2 Symmetric solutions for V x = V 0 sinh 4 x

To find the even solutions to Eq. (81) with

and to ensure that

We shall look for rank

The highest power of

For

Solving these, we find that

### 5.3 Antisymmetric solutions

In order to find antisymmetric solutions to Eq. (86), we set

This CHE can be solved in power series:

Here,

We find four eigenvalues,

## 6. The potential function V x = V 0 sinh 4 x − k sinh 2 x

Now we apply our analysis to the problem with

We now find that

meaning that for

For the case with

Now, to find the antisymmetric eigenfunctions, we set

For

Note that in this case

### 6.1 The case with k = − 1

As was seen in Section VI, the ground-state energy diverges as

### 6.2 Unclassified QES potentials

Finally, we would like to emphasize that there should be other potential functions which may not be classified form the Lie algebraic method [25].

Indeed, let us consider Schrödinger’s problem with the potential function

For this problem, the ground-state eigenfunction and eigenvalue are given by

while this particular problem does not belong to the class of potentials found using the Lie algebraic method. Similar potentials may be found which do not belong to that class, leaving space for further developments.

## Acknowledgments

This work was partially supported by CONACYT 179881 grants and PROMEP grants UGTO-CA-3. This work is part of the collaboration within the Instituto Avanzado de Cosmología. E. Condori-Pozo is supported by a CONACYT graduate fellowship.