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

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.