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

TL;DR

This paper introduces exactly solvable, maximally superintegrable deformations of the harmonic oscillator and Coulomb systems on curved spaces, preserving their spectral degeneracy and integrability through a specific quantization method.

Contribution

It presents new superintegrable Hamiltonian systems on curved spaces and their quantization, maintaining superintegrability and spectral properties in deformed quantum systems.

Findings

Eigenvalue problems are exactly solvable for the deformed systems.

Deformed spectra are smooth continuations of the original oscillator and Coulomb spectra.

Maximal degeneracy of the systems is preserved under deformation.

Abstract

We present two maximally superintegrable Hamiltonian systems ${\cal H}_\lambda$ and ${\cal H}_\eta$ 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 ${\cal H}_\lambda$ and ${\cal H}_\eta$ leads, respectively, to exactly solvable deformations (with parameters $\lambda$ and $\eta$) 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 ${\cal H}_\lambda$ and ${\cal H}_\eta$ can be rigorously solved, and it is found that their discrete spectrum is just a smooth deformation (in terms of the parameters $\lambda$ and $\eta$) 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.