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

SIGMA 8 (2012), 069, 10 pages      arXiv:1208.1782
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Complex SUSY Transformations and the Painlevé IV Equation

David Bermúdez
Departamento de Física, Cinvestav, AP 14-740, 07000 México DF, Mexico

Received July 29, 2012, in final form September 28, 2012; Published online October 11, 2012

In this paper we will explicitly work out the complex first-order SUSY transformation for the harmonic oscillator in order to obtain both real and complex new exactly-solvable potentials. Furthermore, we will show that this systems lead us to exact complex solutions of the Painlevé IV equation with complex parameters. We present some concrete examples of such solutions.

Key words: supersymmetric quantum mechanics; Painlevé equations; differential equations; quantum harmonic oscillator; polynomial Heisenberg algebras.

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