Abstract
We present two maximally superintegrable Hamiltonian systems Hλ and Hη that are defined, respectively, on an N-dimensional spherically symmetric generalization of the Darboux surface of type III and on an N-dimensional Taub-NUT space. Afterwards, we show that the quantization of Hλ and Hη leads, respectively, to exactly solvable deformations (with parameters λ and η) of the two basic quantum mechanical systems: the harmonic oscillator and the Coulomb problem. In both cases the quantization is performed in such a way that the maximal superintegrability of the classical Hamiltonian is fully preserved. In particular, we prove that this strong condition is fulfilled by applying the so-called conformal Laplace-Beltrami quantization prescription, where the conformal Laplacian operator contains the usual Laplace- Beltrami operator on the underlying manifold plus a term proportional to its scalar curvature (which in both cases has non-constant value). In this way, the eigenvalue problems for the quantum counterparts of Hλ and Hη can be rigorously solved, and it is found that their discrete spectrum is just a smooth deformation (in terms of the parameters λ and η) of the oscillator and Coulomb spectrum, respectively. Moreover, it turns out that the maximal degeneracy of both systems is preserved under deformation. Finally, new further multiparametric generalizations of both systems that preserve their superintegrability are envisaged.
Highlights
It is well known that if we consider a natural classical Hamiltonian system on the N -dimensional (N D) Euclidean spaceH = T (p) + U(q), (1)the harmonic oscillator potential U (q) = ω2q2 and the Coulomb potential U(q) = −k/|q| define two maximally superintegrable (MS) systems, since both systems are endowed with (2N −1) functionally independent and globally defined integrals of the motion
When the quantization of these systems is performed it is found that such superintegrability implies that their spectrum exhibits maximal degeneracy due to a superabundance of quantum integrals of the motion
In this paper we review two spherically symmetric deformations of the oscillator and Coulomb systems that define two new MS systems [5, 6]
Summary
It is well known that if we consider a natural classical Hamiltonian system on the N -dimensional (N D) Euclidean space. In this paper we review two spherically symmetric deformations of the oscillator and Coulomb systems that define two new MS systems [5, 6] As a consequence, their quantization [7, 8, 9] is shown to present maximal degeneracy in the spectra. Where μ can be regarded as a (generic) deformation parameter in such a manner that we will be no longer working on the flat Euclidean space, but on a suitable curved space with metric and kinetic energy given by. This fact will provide additional interesting geometric features to the systems we will deal with. New results are sketched in the last section by presenting the only possible multiparametric spherically symmetric generalizations of the above systems which are MS with quadratic integrals of motion, that is, the most generic deformations that can be endowed, respectively, with a generalized Demkov–Fradkin tensor and with a Runge–Lenz N -vector
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