Abstract

We construct two-dimensional integrable and superintegrable systems in terms of the master function formalism and relate them to Mielnik’s and Marquette’s construction in supersymmetric quantum mechanics. For two different cases of the master functions, we obtain two different two-dimensional superintegrable systems with higher order integrals of motion.

1. Introduction

It is known from classical and quantum mechanics that a system with N degrees of freedom is called completely integrable if it allows N functionally independent constants of the motion [1]. From the mathematical and physical point of view, these systems play a fundamental role in description of physical systems due to their many interesting properties. A system is superintegrable if one could obtain more than N constants of the motion and if there exist constants of the motion, the system is maximally superintegrable or just superintegrable [2–5]. Recently the study of superintegrable systems has been considered for different potentials and many researches have been studied for calculating the spectrum of these systems by different methods. In [6, 7], the spectrum of these systems has been calculated by an algebraic method using the realization of some Lie groups.

For a two-dimensional quantum integrable system with Hamiltonian , there is always one operator like which commutes with Hamiltonian of the system, that is, . For a quantum superintegrable system, one should define another operator such as which commutates with the Hamiltonian of system, that is, , but . In other words, for a two-dimensional superintegrable system, there are two integrals of the motion in addition to the Hamiltonian. The superintegrability with the second- and third-order integrals was the object of a series of articles [8–11]. The systems studied have second- and third-order integrals. Although superintegrability and supersymmetric quantum mechanics (SUSYQM) are two separated fields, many quantum systems, such as the harmonic oscillator, the Hydrogen atom, and the Smorodinsky-Winternitz potential, have both supersymmetry and superintegrable conditions [12–16]. These articles show that superintegrability is accurately connected with supersymmetry. For example, in [17], Marquette used the results obtained by Mielnik [18] and generated new superintegrable systems. Mielnik has shown that the factorization of second-order operators is not essentially unique. He has considered the Hamiltonian of the harmonic oscillator in one dimension as the simplest case:where it can be factorized by two types of the first-order operators of creation and annihilation as follows:For two superpartner Hamiltonians and where and , he has demanded that and obtained the inverted product as a certain new Hamiltonian:where is a function obtained from the general solution of Riccati equation considering . The creation and annihilation operators of the third order for are described by expressions , , where and are the creation and annihilation operators for . Marquette [17] has taken the Hamiltonian in the -axis and its superpartner given by (3) in the -axis. Hence, he has obtained a two-dimensional superintegrable system as , which can be separated in Cartesian coordinates with creation and annihilation operators , , , and . Also, he has shown that the Hamiltonian possesses the following integrals of motion:where these integrals are of order 2, 3, and 4 for shape invariant potentials [17].

On the other hand, in [19, 20], the authors have shown that the second-order differential equations and their associated differential equations in mathematical physics have the shape invariant property of supersymmetry quantum mechanics. They have shown that by using a polynomial of a degree not exceeding two, called the master function, the associated differential equations can be factorized into the product of rising and lowering operators. The master function formalism has been used in relativistic quantum mechanics for solving the Dirac equation [21, 22].

As Mielnik’s and Marquette’s method for generating superintegrable systems can be applied to other systems obtained in the context of supersymmetric quantum mechanics, hence, in this paper, we show that the supersymmetry method for obtaining the integrable and superintegrable systems can be related to master function formalism. In fact, we use the master function approach for 1-dimensional shape invariant potentials and generate 2-dimensional integrable systems. Also for a particular class of shape invariant systems, we generate 2-dimensional superintegrable systems. This class contains the harmonic oscillator, the singular harmonic oscillator, and their supersymmetric isospectral deformations.

The paper is presented as follows: in Section 2, we review how one can generate integrals of motion for two-dimensional superintegrable system from the creation and annihilation operators. In Section 3, we consider a particular quantum system for applying the Mielnik’s and Marquette’s method and obtain a superintegrable potential separable in Cartesian coordinates. In Section 4, we briefly review the master function formalism and then in Section 5, we use this approach to obtain integrable systems and particular cases of the superintegrable systems that satisfy the oscillator-like (Heisenberg) algebra with higher order integrals of motion in terms of the master function and weight function. In Section 6, we give two examples to show how this method works in constructing oscillator-like two-dimensional superintegrable systems. Paper ends with a brief conclusion in Section 7.

2. Two-Dimensional Superintegrable System and Its Integrals of Motion

According to [17, 23, 24], for a two-dimensional Hamiltonian separable in Cartesian coordinates aswhere the creation and annihilation operators (polynomial in momenta) , , , and satisfy the following equations:one can show that the operators and commute with the Hamiltonian : that is,ifAlso the following sums of and commute with the Hamiltonianthat is, and are the integrals of motion. The order of these integrals of motion depends on the order of the creation and annihilation operators. On the other hand, the Hamiltonian possesses a second-order integral as , such that the integral is the commutator of and . Thus, the Hamiltonian is a superintegrable system and , , and are its integrals of motion.

3. Mielnik-Marquette Method and Superintegrable Model Obtained from Shifted Oscillator Hamiltonian

In this section, for reviewing the Mielnik-Marquette method, we consider shifted oscillator Hamiltonian asWe introduce the following first-order operators:where the supersymmetric partner Hamiltonians are calculated asIt is obvious that and have the shape invariant properties. Now, according to (2), we define the new operators and such thatThe above equation gives the following Riccati equation:where a particular solution isNow, if we considerthen we can obtain the following first-order linear inhomogeneous equation:where . After solving the above equation, we getwhere is the constant of integration. Using the function , we obtainwhere its creation and annihilation operators are given by the following expressions:According to Marquette method, we take the -axis for Hamiltonian and the -axis for its superpartner and we have the following two-dimensional superintegrable system:This Hamiltonian possesses the integral of motion given by (4), which are of order 2, 3, and 4.

4. The Master Function Formalism

According to [19, 20], the general form of the differential equation in master function approach is written aswhere as master function is at most a second-order polynomial and is the nonnegative weight function in interval . By differentiating (22) times and then multiplying it by , we get the following associated second-order differential equation in terms of the master function and weight function:whereChanging the variable and defining the new function , one can obtain the Schrodinger equation aswhere the most general shape invariant potential isand the energy spectrum is asAccording to [19, 20] the first-order deferential operators are written aswhere the superpotential is expressed in terms of the master function and weigh function asThe Hamiltonian and called the superpartner Hamiltonians are written aswhere and are called the partner potentials in the concept of supersymmetry in nonrelativistic quantum mechanics. Furthermore, if the partner potentials have the same shape and differ only in parameters, then potentials and are called the shape invariant potentials that satisfy where is independent of any dynamical variable and is a function of . Potentials which satisfy in this condition are exactly solvable, although shape invariance is not the most general integrability or superintegrability condition.

5. Integrable and Superintegrable Systems Obtained from the Master Function Formalism

In this section, we try to relate the Mielnik-Marquette method to the master function approach. Hence, we define the following new operators:where as the new superpotential must be related to the general form of the master function superpotential . Their product yields to Hamiltonians asNow if we demand then we can obtain the following Riccati equation in terms of master function:where a particular solution is . The general solution can be obtained likewhich yieldsWe consider the transformation and obtain a first-order linear inhomogeneous differential equation aswhere the general solution iswhere is constant. Hence, Using the function given by (38), the superpartner Hamiltonian is given bywhich is the general form of Hamiltonian in terms of master function. Now if we catch and (the Hamiltonian is thus given in terms of the variable vertical to ), then we obtain a new two-dimensional integrable Hamiltonian asTherefore, we have obtained the general form of the 2-dimensional integrable Hamiltonian in terms of master function which can be separated in radial coordinates. This separation of variable implies the existence of a second-order integral as . Hence, is an integrable system. Now, for generating superintegrable systems, we can obtain the creation and annihilation operators for from aswhere and are given in (28) and (32). It is necessary to mention that these ladder operators satisfy (6) only for a particular class of shape invariant systems and in general, the 2-dimensional system , obtained from a given master function, is not a superintegrable system. In other words, we cannot obtain 2-dimensional superintegrable system for all of the shape invariant cases given in [19, 20]. This particular class contains the harmonic oscillator, the singular harmonic oscillator, and their supersymmetric isospectral deformations.

Hence, if relation (6) exists, according to (9) we can obtain the integrals of motion for Hamiltonian (41) asIn the next section, we apply this formalism for some particular cases of shape invariant potentials in terms of master function.

6. Examples of Two-Dimensional Superintegrable Systems as a Result of Master Function Approach

In this section, we would apply the master function formalism of the previous section for two examples and show how these results allow us to obtain 2-dimensional superintegrable systems with higher order integrals.

Example 1. Let ; then according to [19], that , and the interval is . Using (29), we obtain the superpotential asAccording to (27), the energy spectrum is asand also the ladder operators given by (28) related to (44) satisfy a Heisenberg algebras (6). Now, substituting expression in (38) yields the following relation in terms of the error function:and soSubstituting this expression in (40) and (41) yields the family of superpartner and a two-dimensional superintegrable Hamiltonian , respectively, aswhereIt is seen that 2-dimensional superintegrable Hamiltonian (48) is the same as (21). We can find the general form of the operators and in terms of the master function for this oscillator-like potential as follows:Thus, we have obtained a 2-dimensional superintegrable system with integrals given by (43) aswhereThese integrals are of order 2, 3, and 4.