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Factorization solution of a family of quantum nonlinear oscillators. (English) Zbl 1177.81034

Summary: In a recent paper, J. F. Cariñena, A. M. Perelomov, M. F. Rañada and M. Santander [J. Phys. A, Math. Theor. 41, No. 8, Article ID 085301 (2008; Zbl 1138.81380)] analyzed a non-polynomial one-dimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators. In particular, they proved that it is Schrödinger soluble, and explicitly obtained the wavefunctions and energies of the bound states.

In this paper, we show that these results can be obtained much more simply by noting that this potential is a supersymmetric partner potential of the harmonic oscillator. We then use this observation to generate an infinite set of potentials which can exactly be solved in a similar manner.

81Q05Closed and approximate solutions to quantum-mechanical equations
35Q55NLS-like (nonlinear Schrödinger) equations