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


SIGMA 8 (2012), 063, 14 pages      arXiv:1209.4151      http://dx.doi.org/10.3842/SIGMA.2012.063
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Singular Isotonic Oscillator, Supersymmetry and Superintegrability

Ian Marquette
School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

Received July 20, 2012, in final form September 14, 2012; Published online September 19, 2012

Abstract
In the case of a one-dimensional nonsingular Hamiltonian H and a singular supersymmetric partner Ha, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied these results to four 2D superintegrable systems with second and third-order integrals of motion obtained by Gravel for which polynomial algebras approach does not allow to obtain the energy spectrum of square integrable wavefunctions. We obtain solutions involving parabolic cylinder functions.

Key words: supersymmetric quantum mechanics; superintegrability; isotonic oscillator; polynomial algebra; special functions.

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