An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric

TL;DR

This paper introduces an exactly solvable quantum deformation of the Coulomb problem linked to the Taub-NUT space, preserving superintegrability and degeneracy, with a spectrum explicitly computed for positive parameters.

Contribution

It proposes a quantization method using the conformal Laplacian to preserve superintegrability in a deformed Coulomb system related to Taub-NUT geometry, providing exact solutions.

Findings

Spectrum computed in closed form for positive parameters.

Maximal degeneracy of the flat system is preserved.

New exactly solvable quantum deformation of Coulomb problem.

Abstract

In this paper we quantize the $N$-dimensional classical Hamiltonian system $H= \frac{|q|}{2(\eta + |q|)} p^2-\frac{k}{\eta +|q|}$, that can be regarded as a deformation of the Coulomb problem with coupling constant $k$, that it is smoothly recovered in the limit $\eta\to 0$. Moreover, the kinetic energy term in $H$ is just the one corresponding to an $N$-dimensional Taub-NUT space, a fact that makes this system relevant from a geometric viewpoint. Since the Hamiltonian $H$ is known to be maximally superintegrable, we propose a quantization prescription that preserves such superintegrability in the quantum mechanical setting. We show that, to this end, one must choose as the kinetic part of the Hamiltonian the conformal Laplacian of the underlying Riemannian manifold, which combines the usual Laplace-Beltrami operator on the Taub-NUT manifold and a multiple of its scalar curvature. As a consequence, we obtain a novel exactly solvable deformation of the quantum Coulomb problem, whose spectrum is computed in closed form for positive values of $\eta$ and $k$, and showing that the well-known maximal degeneracy of the flat system is preserved in the deformed case. Several interesting algebraic and physical features of this new exactly solvable quantum system are analysed, and the quantization problem for negative values of $\eta$ and/or $k$ is also sketched.