Supersymmetric Quantum Mechanics and Painlev\'e IV Equation
David Bermudez, David J. Fern\'andez C
TL;DR
This paper links supersymmetric quantum mechanics to the Painlevé IV equation, providing a new method to generate solutions and classifying them into hierarchies, thus advancing understanding of quantum systems with special symmetries.
Contribution
It introduces a simple technique for generating Painlevé IV solutions via higher-order supersymmetric partners of the harmonic oscillator.
Findings
Identifies subsets of supersymmetric partners with third-order ladder operators
Provides a classification of Painlevé IV solutions into three hierarchies
Establishes a method to generate solutions using supersymmetric quantum mechanics
Abstract
As it has been proven, the determination of general one-dimensional Schr\"odinger Hamiltonians having third-order differential ladder operators requires to solve the Painlev\'e IV equation. In this work, it will be shown that some specific subsets of the higher-order supersymmetric partners of the harmonic oscillator possess third-order differential ladder operators. This allows us to introduce a simple technique for generating solutions of the Painlev\'e IV equation. Finally, we classify these solutions into three relevant hierarchies.
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