Symmetry, Integrability and Geometry: Methods and Applications Complex SUSY Transformations and the Painlevé IV Equation ⋆

In this paper we will explicitly work out the complex first-order SUSY transformation for the harmonic oscillator in order to obtain both real and complex new exactly-solvable potentials. Furthermore, we will show that this systems lead us to exact complex solutions of the Painlev\'e IV equation with complex parameters. We present some concrete examples of such solutions.


Introduction
Supersymmetry was introduced in quantum field theory in order to relate the bosonic and fermionic sectors into one unified superfield [37]. This theoretical framework is described by some deformations of Lie algebras and their commutation relations. Although there is no experimental evidence of supersymmetry in nature yet, its concepts have aroused new ideas in other fields of science.
The supersymmetric quantum mechanics (SUSY QM) is one of these new ideas: while first developed by Witten [39], it was soon related to the Darboux transformation and to the factorization method, which was thought to be completed in 1951 by Infeld and Hull [25] but it was developed further as a paradigm to obtain new exactly-solvable by Mielnik [30] and others [6,7,17,31,35] and deepen into the algebraic structure of quantum systems. Even more, for those potentials that are not exactly-solvable we still have a powerful new tool to develop approximation methods, giving rise to a whole new branch of study called spectral design [5], since one can obtain an exactly-solvable potential with a prescribed spectrum, i.e. with some given energy levels.
In this paper we will use the scheme of SUSY QM to obtain complex partner potentials. As far as we know, they were first developed in [4], where complex potentials with real energy spectra were obtained using complex transformation functions associated with real factorization energies and later in [19] with complex factorization energies. On the other hand, in this work we will study complex potentials with complex energy spectra.
The theoretical framework of SUSY QM have connections with other branches of study, for example with non-classical orthogonal polynomials and the special functions [34], which are solutions of linear differential equations. In this paper, we are interested in some other functions that play an analogous role as special functions, but now for non-linear differential equations. Specifically, we are talking about the Painlevé equations. Some specialists (for example [15,26]) consider that solutions of the Painlevé equations will be the future members of special functions This paper is a contribution to the Special Issue "Superintegrability, Exact Solvability, and Special Functions". The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html in the twenty-first century. It turns out that one of these equations, the Painlevé IV equation (P IV ) (see [24]), is closely related to potentials obtained by SUSY QM from the harmonic oscillator. Note that P IV also appears in other areas of physics, for example in fluid mechanics [38], non-linear optics [21], and quantum gravity [22]. It also arises in mathematics, e.g., in the symmetric reduction of several partial differential equations like Boussinesq equation [14], dispersive wave equation [32], non-linear cubic Schrödinger equation [12], and auto-dual Yang-Mills field equations [1].
The SUSY QM technique has been used before to obtain solutions of P IV ; however, they are either real or complex but associated with a differential equation having real parameters [9,10,11]. In this work we will expand the solution set to obtain complex solutions, associated now with complex parameters of P IV .
To accomplish this, the structure of the paper is the following: in Section 2 we will work out the complex scheme of the first-order SUSY QM. In Section 3 we will describe the second-order polynomial Heisenberg algebras and its relationship with P IV . Then, in Section 4 we will use the first-order SUSY partners of the harmonic oscillator in order to connect with complex solutions to P IV . These solutions shall be studied in Section 5. Finally, we will present our conclusions in Section 6.

Complex SUSY QM
Let us begin with a given Hamiltonian H which has been completely solved, i.e., all of its eigenvalues E k and eigenfunctions ψ k , k = 0, 1, . . . are known: where ∂ = d/dx and we are using natural units such that = m = 1. Now, we use the factorization method which is equivalent to SUSY QM but much simpler in this case. We propose that H is factorizable as In the standard real factorization method there is an extra condition A + ≡ (A − ) † . In this paper we do not use this constrain, but rather we simply ask that where β(x) is a complex function to be found. This choice represents a more general factorization than the usual real one [33]. Working out the operations in (3), using the definitions in (1) and (4), we obtain one condition for β(x) On the other hand, if we consider a similar factorization but in a reversed order and introduce a new HamiltonianH, defined byH = −∂ 2 +Ṽ and it turns out that Besides, from equations (3) and (5) it is straightforward to show that [4,33] HA which are the well known intertwining relationships with A + , A − being the intertwining operators. From equations (2) and (6) we can obtain the eigenvalues and eigenfunctions of the new HamiltonianH as follows Therefore, the eigenfunctionsψ k ofH associated with the eigenvalues E k becomẽ , and u(x) is an eigenfunction of H (in general a non-physical one) associated with a complex eigenvalue , i.e.
Furthermore, the eigenstatesψ k are not automatically normalized as in the real SUSY QM since now and in this case (A + ) † A + = (H − ). Nevertheless, since they are normalizable we can introduce a normalizing constant C n , chosen for simplicity as C n ∈ R + , so that ψ n (x) = C n A + ψ n (x), ψ n |ψ n = 1.
Finally, there is a wavefunctioñ that is also eigenfunction ofH If it is normalizable, it turns out thatṼ (x) is a complex potential which HamiltonianH has the following spectrum and ∈ C although in particular could be real.

Polynomial Heisenberg algebras and P IV
A polynomial Heisenberg algebra (PHA) is a particular deformation of the Heisenberg-Weyl algebra [18,36]. Although there is a general definition of a m-th order PHA, in this work we will only be interested in the second-order case. A second-order PHA is defined by where P and Q are second-and third-order polynomials, respectively. This definition means that L ± act like ladder operators for the system described by the Hamiltonian H [13,20]. As usual, L + will be a creation operator and L − an annihilation one, i.e., In this case, both L ± will be third-order differential ladder operators. Now, we propose a closed-chain of three SUSY transformations [2,3,16,23,28,36] so that L ± can be expressed as In general, (L − ) † = L + , except in the case where all f i ∈ R. As a matter of fact, in this article we are interested precisely in the case where f i ∈ R. Each L i fulfills two intertwining relationships of kind where i = 1, 2, 3. In Fig. 1 we present a diagram of the SUSY transformations. If we equate the two different factorizations associated with (9) which lead to the same Hamiltonians we get In addition, the closure condition is given by By making the corresponding operator products we get the following system [2,29,36] Eliminating f 2 2 from (10a) and (10b) we get and from here we substitute f 2 1 from (10c) to obtain which, after integration becomes Now, substituting (11) into (10a) Let us define now a useful new function as g ≡ f 3 − x, from which we get Similarly, by plugging (11) into (10b) and using g we obtain Now that we have f 1 , f 2 , f 3 in terms of g, we replace them in (10c) in order to obtain which is the Painlevé IV equation (P IV ) Since, in general f ∈ C then g ∈ C. In addition, i ∈ C which implies that a, b ∈ C and so g is a complex solution to P IV associated with the complex parameters a, b.

First-order SUSY partners of the harmonic oscillator
In this section we will show that the complex first-order SUSY transformation applied to the harmonic oscillator leads to a system with third-order ladder operators which is naturally ruled by a second-order PHA. Let us begin with the harmonic oscillator potential As in Section 2, we propose two operators A + and A − that fulfill equations (4) and (6) so that with the condition that β(x) should solve the Riccati equation The previous scheme can be written in terms of a solution of the initial stationary Schrödinger equation u by substituting β = (ln u) in (12) in order to obtain whose general solution, for any ∈ C, is given by With this formalism, the first-order SUSY partner potentialṼ of the harmonic oscillator is given bỹ The previously known results for the real case [27] are obtained by taking ∈ R, κ = 0 and expressing λ as On the other hand, for ∈ C the transformation function u(x) is complex and so isṼ (x). In Fig. 2 we present some examples of complex SUSY partner potentials of the harmonic oscillator generated for ∈ C and compare them with the initial potential. Let us note that these new potentials have the same real spectra as the harmonic oscillator, except that they have one extra energy level, located at the complex value . This kind of spectrum is represented in the diagram of Fig. 3. Note that this can be interpreted as the superposition of the two ladders shown in equation (7).

Complex solutions to Painlevé IV equation
In this Section we will see that for each first-order SUSY partner of the harmonic oscillator we will have three different complex solutions of P IV .
First of all, let us note that the ladder operators associated withH are given by or, explicitly, This system has third-order differential ladder operators, therefore it is ruled by a second-order PHA and so we can apply our analysis of Section 3 by identifying the ladder operators L ± given in equations (14) with those of equations (8). This identification leads to Recall that the function g = f 3 − x fulfills the Painlevé IV equation, then one solution to P IV is Let ψ E i , i = 1, 2, 3, be the states annihilated by L − , where E i represents the factorization energy for the corresponding extremal state. In particular, let ψ E 3 be annihilated by L − 3 , and from equation (8) we can see that it is also annihilated by L − . By solving L − 3 ψ E 3 = 0 we get from which it can be shown that the corresponding solution to P IV reads For our complex first-order SUSY partner potentialṼ (x), generated by using the complex seed solution u(x) of equation (13), the three extremal states (up to a numerical factor) and their corresponding energies consistent with equations (15) are given by: Nevertheless, the labelling given in equations (16) is not essential, so that by making cyclic permutations, associated with the three extremal states of equations (16) we get three solutions to P IV : with i = 1, 2, 3. The corresponding parameters a, b of P IV are given by: where we have added the subscript corresponding to the extremal state used. In Fig. 4 we have presented one example for each of the three families of solutions. From equations (17) we can see that a i is linear in for all three cases. So, instead of studying the parametric relationship of a, b in terms of let us analyze b i = b i (a i ), namely, Then, we can choose a i ∈ C but b i will be fixed by its corresponding relationship with a i . In Fig. 5 we show the domain for b 3 from equations (18). There are similar plots for b 1 but for b 2 remember that b 2 = −2 ∀ a 2 ∈ C.

Conclusions
In this work we have applied the first-order SUSY QM to the harmonic oscillator potential to obtain systems with third-order ladder operators which are ruled by second-order PHA. From this system we have obtained complex solutions g 1 , g 2 , and g 3 of the Painlevé IV equation.
It is worth to notice that complex solutions have not been generated in this context, nevertheless, the work of Bassom et al. [8,Section 3.3] is general and can be applied to a P IV with complex parameters. An extension of the parameter space to the complex plane is helpful at least when P IV appears in the reduction of other non-linear differential equations. Moreover, it can be used in the future to study complex extensions of those physical systems described by P IV . We first discussed the complex first-order SUSY transformation for general systems. We defined then the PHA and established explicitly the relationship of the second-order case with the Painlevé IV equation. We also studied the complex SUSY partner potentials of the harmonic oscillator with an analysis of the energy spectra of the new potentials. Then, we obtained complex solutions of the Painlevé IV equation and we analized the complex parameter space where the solutions can appear. We concluded that there are three different families of complex solutions for each complex first-order SUSY transformation.
In the future we would like to address the higher-order SUSY transformation in order to extend the space of complex solutions of the Painlevé IV equation and to study in more detail the properties of these solutions.