An exactly solvable three-dimensional nonlinear quantum oscillator. (English) Zbl 1287.81048
Authors’ abstract: Exact analytical, closed-form solutions, expressed in terms of special functions, are presented for the case of a three-dimensional nonlinear quantum oscillator with a position dependent mass. This system is the generalization of the corresponding one-dimensional system, which has been the focus of recent attention. In contrast to other approaches, we are able to obtain solutions in terms of special functions, without a reliance upon a Rodrigues-type of formula. The wave functions of the quantum oscillator have the familiar spherical harmonic solutions for the angular part. For the \(s\)-states of the system, the radial equation accepts solutions that have been recently found for the one-dimensional nonlinear quantum oscillator, given in terms of associated Legendre functions, along with a constant shift in the energy eigenvalues. Radial solutions are obtained for all angular momentum states, along with the complete energy spectrum of the bound states.
Reviewer: Aurelian Gheondea (Bucureşti)
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81U15 | Exactly and quasi-solvable systems arising in quantum theory |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
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