Supersymmetric Quantum Mechanics and Painlevé IV Equation

Abstract

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.

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

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• Supersymmetric partners of the harmonic oscillator with an infinite potential barrier
• Solutions to the Painlevé V equation through supersymmetric quantum mechanics
• SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations
• Complex SUSY Transformations and the Painlevé IV Equation
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials
• On Integrable Ermakov-Painlevé IV Systems
• Solution hierarchies for the Painleve IV equation
• Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators
• Harmonic Oscillator SUSY Partners and Evolution Loops
• On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials
• Supersymmetric Quantum Mechanics, Engineered Hierarchies of Integrable Potentials, and the Generalised Laguerre Polynomials
• Complete classification of rational solutions of $A_{2n}$-Painlevé systems
• Singular Isotonic Oscillator, Supersymmetry and Superintegrability
• Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent