Painlevé IV Transcendents Generated
from the Complex Oscillator
David J. Fernández
Abstract Supersymmetry transformations are used to generate exactly solvable
potentials departing from the complex oscillator. The corresponding Hamiltonians
are shown to be ruled by polynomial Heisenberg algebras. A process for reducing
the degree of these algebras to 2 is used to connect such systems with the
Painlevé IV equation, thus leading to a simple algorithm for generating Painlevé
IV transcendents.
Keywords Painlevé transcendents · Complex oscillator · Supersymmetric
quantum mechanics
1 Introduction
The recent scientific advances make it important to study the links that could
exist between supersymmetric quantum mechanics (SUSY QM) and nonlinear
differential equations [1]. Indeed, there is a well known connection between SUSY
partners of the free particle and solutions of the KdV equation [2–4]. Similarly, it has
been shown that there is a link between systems ruled by second-degree polynomial
Heisenberg algebras and Painlevé IV (PIV) equation [5–12]. This connection helped
to design further an algorithm for generating solutions to the PIV equation, called
Painlevé IV transcendents in the literature [13, 14]. The simplest systems that can be
used to supply explicit expressions for PIV transcendents are the harmonic oscillator
and its SUSY partners [15, 16]. It would be important to know if the so-called
complex oscillator [17], which arises from making complex the oscillator frequency
and includes the harmonic oscillator as a limit, does the same. This subject is going
to be explored in this article. In order to do this, in Sect. 2 we will make a quick
D. J. Fernández ()
Département de physique, Université de Montréal, Montréal, QC, Canada
e-mail: david@fis.cinvestav.mx
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
http://www.fis.cinvestav.mx/~david
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_4
47
from the Complex Oscillator
David J. Fernández
Abstract Supersymmetry transformations are used to generate exactly solvable
potentials departing from the complex oscillator. The corresponding Hamiltonians
are shown to be ruled by polynomial Heisenberg algebras. A process for reducing
the degree of these algebras to 2 is used to connect such systems with the
Painlevé IV equation, thus leading to a simple algorithm for generating Painlevé
IV transcendents.
Keywords Painlevé transcendents · Complex oscillator · Supersymmetric
quantum mechanics
1 Introduction
The recent scientific advances make it important to study the links that could
exist between supersymmetric quantum mechanics (SUSY QM) and nonlinear
differential equations [1]. Indeed, there is a well known connection between SUSY
partners of the free particle and solutions of the KdV equation [2–4]. Similarly, it has
been shown that there is a link between systems ruled by second-degree polynomial
Heisenberg algebras and Painlevé IV (PIV) equation [5–12]. This connection helped
to design further an algorithm for generating solutions to the PIV equation, called
Painlevé IV transcendents in the literature [13, 14]. The simplest systems that can be
used to supply explicit expressions for PIV transcendents are the harmonic oscillator
and its SUSY partners [15, 16]. It would be important to know if the so-called
complex oscillator [17], which arises from making complex the oscillator frequency
and includes the harmonic oscillator as a limit, does the same. This subject is going
to be explored in this article. In order to do this, in Sect. 2 we will make a quick
D. J. Fernández ()
Département de physique, Université de Montréal, Montréal, QC, Canada
e-mail: david@fis.cinvestav.mx
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
http://www.fis.cinvestav.mx/~david
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_4
47