Complex oscillator and Painlev\'e IV equation

TL;DR

This paper uses supersymmetric quantum mechanics to generate new solvable potentials from the complex oscillator and links polynomial Heisenberg algebras to solutions of the Painlevé IV equation.

Contribution

It introduces a method to derive new Painlevé IV solutions via supersymmetric transformations of the complex oscillator.

Findings

New exactly solvable potentials from the complex oscillator

Connection established between polynomial Heisenberg algebras and Painlevé IV solutions

Reduction of algebra order yields novel Painlevé IV solutions

Abstract

Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. In this article the first- and second- order supersymmetric transformations will be used to obtain new exactly solvable potentials departing from the complex oscillator. The corresponding Hamiltonians turn out to be ruled by polynomial Heisenberg algebras. By applying a mechanism to reduce to second the order of these algebras, the connection with the Painlev\'{e} IV equation is achieved, thus giving place to new solutions for the Painlev\'{e} IV equation.