Exactly solvable deformations of the oscillator and Coulomb systems and their generalization

30th International Colloquium on Group Theoretical Methods in Physics

Ballesteros, A.^1 ; Enciso, A.^2 ; Herranz, F.J.^1 ; Ragnisco, O.^3 ; Riglioni, D.^4

Departamento de Fisica, Universidad de Burgos, E-09001 Burgos, Spain^1

Instituto de Ciencias Matematicas, CSIC, Nicolas Cabrera 13-15, Madrid

E-28049, Spain^2

Dipartimento di Matematica e Fisica, Universita di Roma Tre and Istituto Nazionale, Fisica Nucleare Sezione di Roma Tre, Via Vasca Navale 84, Roma

I-00146, Italy^3

Centre de Recherches Mathematiques, Universite de Montreal, H3T 1J4 2920 Chemin de la tour, Montreal, Canada^4

关键词: Eigenvalue problem;    Hamiltonian systems;    Harmonic oscillators;    Laplace-Beltrami operator;    Laplacian operator;    Quantum counterpart;    Quantum-mechanical system;    Superintegrability;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012014/pdf
DOI  :  10.1088/1742-6596/597/1/012014

来源: IOP

【 摘 要 】

We present two maximally superintegrable Hamiltonian systems H 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 H 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 H 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.