Supersymmetric Quantum Mechanics and Painlevé IV Equation

Details Supersymmetric Quantum Mechanics and Painlevé IV Equation Supersymmetric Quantum Mechanics and Painlevé IV Equation

2010-12-01

arxiv.org/abs/1012.0290v4

SIGMA 7:025,2011

Physics

Mathematical Physics

Proceedings of the Workshop 'Supersymmetric Quantum Mechanics and Spectral Design' (July 18-30, 2010, Benasque, Spain)

Scientific paper

10.3842/SIGMA.2011.025

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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