Supersymmetric Quantum Mechanics and Painlevé IV Equation - Bermudez, David; Fern'andez C., David J. (Symmetry, Integrability and Geometry: Methods and Applications, 2011)
As it has been proven, the determination of general one-dimensional Schrödinger Hamiltonians having third-order differential ladder operators requires to solve the Painlevé 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 Painlevé IV equation. Finally, we classify these solutions into three relevant hierarchies.