Supersymmetric Quantum Mechanics and Painlevé IV Equation

David Bermudez, David J. Fernández C

Published 2010-12-01, updated 2011-12-12Version 4

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.