Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 025, 14 pages      arXiv:1012.0290
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Supersymmetric Quantum Mechanics and Painlevé IV Equation

David Bermúdez and David J. Fernández C.
Departamento de Física, Cinvestav, AP 14-740, 07000 México DF, Mexico

Received November 30, 2010, in final form March 04, 2011; Published online March 08, 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.

Key words: supersymmetric quantum mechanics; Painlevé equations.

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