Integrable and superintegrable systems with higher order integrals of motion: master function formalism

In this article, we construct two-dimensional integrable and superintegrable sys- tems in terms of the master function formalism and relate them to Mielnik;s and Marquette;s construction in supersymmetric quantum mechanics. For two diferent cases of the master functions, we obtain two diferent two-dimensional superintegrable systems with higher order integrals of motion.


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 2N-1 constants of the motion, the system is maximally superintegrable or just superintegrable provided that the commutator of operators as constants of the motion be zero with Hamiltonian of the system [2][3][4][5][6][7][8]. Recently the study of superintegrable systems has been considered for different potentials and many researches have been studied for calculating of the spectrum of these systems by different methods. In Refs. [9,10], 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 H, there is always one operator like A 1 which can be commutated with Hamiltonian of the system i.e. [H, A 1 ] = 0.
For a quantum superintegrable system, one should define another operator such as A 2 which commutates with the Hamiltonian of system i.e. [H, A 2 ] = 0, but [A 1 , A 2 ] = 0. In other words, for a two-dimensional superintegrable system, there are two integrals of the motion (A 1 , A 2 ) in addition to the Hamiltonian. The superintegrability with the second and third order integrals was the object of a series of articles [11][12][13][14][15][16][17]. 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-Winternits potential , have both supersymmetry and superintegrable conditions [18][19][20][21][22][23]. These articles show that superintegrability is accurately connected with supersymmetry. For example, in Ref. [24], Marquette used the results obtained by Mielnik [25] 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 can be factorized by two-type of the first order operators of creation and annihilation as follows: For two superpartner Hamiltonians H 1 and H 2 where a + a − = H − 1 2 = H 1 and a − a + = H + 1 2 = H 2 , he has demanded that H 2 = b − b + and obtained the inverted product b + b − as a certain new Hamiltonian: where ϕ(x) is a function obtained from the general solution of Riccati equation considering . The creation and annihilation operators of the third order for H ′ are described by expressions s with creation and annihilation operators a + (x), a − (x), s + (y) and s − (y). Also, he has shown that the Hamiltonian H s possesses the following integrals of motion where these integrals are of order 2, 3 and 4 for shape invariant potentials [24].
On the other hand, in Refs. [26,27], 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 [28,29].
As Mielnik , s-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-demensional shape invariant potentials and generate 2-dimensional integrable systems. Also for a particular class of shape invariant systems, we generate 2dimensional supperintegrable systems. This class contains the harmonic oscillator, the singular harmonic oscillator and their suppersymmetric 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-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 in-

tegrals of motion
According to Refs. [24,30,31], for a two-dimensional Hamiltonian separable in Cartesian coordinates as: where the creation and annihilation operators (polynomial in momenta) and A − (y) satisfy the following equations one can show that the operators if Also the following sums of f 1 and f 2 commute with the Hamiltonian that is, I 1 and I 2 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 H possesses a second order integral as K = H x − H y , such that the integral I 2 is the commutator of I 1 and K. Thus the Hamiltonian H is a superintegrable system and H, I 1 and K are its integrals of motion.

Mielnik-Marquette method and superintegrable model obtained from shifted oscillator Hamiltonian
In this section, for reviewing of the Mielnik-Marquette method, we consider shifted oscillator Hamiltonian as We introduce the following first order operators where the supersymmetric partner Hamiltonians are calculated as It is obvious that H 1 and H 2 have the shape invariant properties. Now, according to Eq.(2), we define the new operators b − and b + such that The above equation gives the following Riccati equation where a particular solution is Now, if we consider then we can obtain the following first order linear inhomogeneous equation where z = 1 ϕ(x) . After solving the above equation, we get where C is the constant of integration. Using the function ϕ(x), we obtain where its creation and annihilation operators are given by following expressions According to Marquette method, we take the x axis for Hamiltonian H 1 and the y axis for its superpartner H ′ and we have the following two-dimensional superintegrable system This Hamiltonian possesses the integral of motion given by Eq.(4), which are of order 2, 3 and 4.

The Master function formalism
According to Refs. [26,27], the general form of the differential equation in master function approach is written as: where A(x) as master function is at most a second order polynomial and w(x) is the nonnegative weight function in interval (a, b). By differentiating Eq. (22) m times and then multiplying it by (−1) m A m 2 (x), we get the following associated second-order differential equation in terms of the master function and weight function where Changing the variable dx dr = A(x), and defining the new function Ψ m n (r) = A where the most general shape invariant potential is: and the energy spectrum E(n, m) is as: According to Refs. [26,27] the first-order deferential operators are written as: where the superpotential W m (x(r)) is expressed in terms of the master function A(x) and weigh function w(x) as: The Hamiltonian H 1 and H 2 called the superpartner Hamiltonians are written as where v 1 (r) and v 2 (r) 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 v 1 (r) and v 2 (r) are called the shape invariant potentials that satisfy in where R(a 1 ) is independent of any dynamical variable and a 1 is a function of a 0 . Potentials which satisfy in this condition are exactly solvable, although shape invariance is not the most general integrability or superintegrability condition.

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 ω(r) as the new superpotential must be related to the general form of the master function superpotential W m (x(r)). Their product yields to Hamiltonians as: Now if we demand A − A + = B − B + then we can obtain the following Riccati equation in terms of master function: where a particular solution is ω(r) = W m (r). The general solution can be obtained like : which yields: We consider the transformation f (r) = 1 λ(r) and obtain a first order linear inhomogeneous differential equation as: which the general solution is : where C is constant. Hence: Using the function f (r) given by (38), the superpartner Hamiltonian is given by: which is the general form of Hamiltonian in terms of master function. Now if we catch H r = H 2 and H r ′ = H ′ (the Hamiltonian H ′ is thus given in terms of the variable r ′ vertical to r) then we obtain a new two-dimensional integrable Hamiltonian as: Therefore we have obtained the general form of the 2-dimensional integrable Hamiltonian in terms of master function in which can be separated in radial coordinates. This separation of variable implies the existence of a second order integral as K = H r − H r ′ . Hence, H s is a integrable system. Now, for generating superintegrable systems, we can obtain the creation and annihilation operators for H ′ from H 2 as where A ± and B ± were given in Eqs. (28),(32). As these ladder operators satisfy the relation given by (6) only for a particular class of shape invariant systems so in general form, the 2-dimensional system H s , obtained from a given master function, is not a superintegrable system. In fact, this class contains the harmonic oscillator, the singular harmonic oscillator and their suppersymmetric iso-spectral deformations.
Hence if it exists, according to Eq. (9) we can obtain the integrals of motion for Hamiltonian (41) as In 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 A(x) = 1, then according to Ref. [26] , w(x) = e − β 2 x 2 that x = r − 2α β , β > 0 and the interval is (−∞, +∞). Using Eq. (29), we obtain the superpotential as: According to Eq. (27), the energy spectrum is as also the ladder operators given by equation (28) and so Substituting this expression in Eqs. (40),(41), yield the family of superpartner H ′ and a two-dimensional superintegrable Hamiltonian respectively as: where and We can find the general form of the operators S + and S − in terms of the master function for this oscillator-like potentials as follows: Thus we have obtained a 2-dimensional superintegrable system with integrals given by Eq.(43) as: where the Whittaker function M µ,ν (z) is the solution of the following differential equation: It can be also defined in terms of the confluent hypergeometric function as: where A ± (r) is given in Eq. (28)and from Eq.(42), we have the creation and annihilation operators of the Hamiltonian H ′ as: where B ± (r) is given by (32)and (39). We can also find the integrals of motion of the Hamiltonian H s from Eq.(43) as: that are of the order 2, 7 and 8.

Conclusion
In this article, we have shown how the supersymmetric quantum mechanics gives a procedure for constructing two-dimensional integrable and superintegrable systems with higher order integrals of motion. We have used the results obtained by Mielnik in the concept of SUSYQM and related it to master function formalism for constructing two-dimensional integrable and superintegrable systems. From this procedure, we have generated the superintegrable systems for two different cases of master functions A(x) = 1 and A(x) = x, and have shown that the higher integrals of motion are in order 2, 3, 4 and 2, 7, 8 respectively.