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Supersymmetric quantum mechanics and Painlevé IV equation. (English) Zbl 1217.81098

Summary: 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.

MSC:

81Q60 Supersymmetry and quantum mechanics
35G20 Nonlinear higher-order PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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