A maximally superintegrable deformation of the N-dimensional quantum Kepler-Coulomb system

Abstract

The $N$-dimensional quantum Hamiltonian $\hat{H}=-\frac{{\hslash }^2|\mathbf{q}|}{2(\eta +|\mathbf{q}|)}{\nabla }^2-\frac{k}{\eta +|\mathbf{q}|}$ is shown to be exactly solvable for any real positive value of the parameter $\eta$. Algebraically, this Hamiltonian system can be regarded as a new maximally superintegrable $\eta$-deformation of the $N$-dimensional Kepler-Coulomb Hamiltonian while, from a geometric viewpoint, this superintegrable Hamiltonian can be interpreted as a system on an $N$-dimensional Riemannian space with nonconstant curvature. The eigenvalues and eigenfunctions of the model are explicitly obtained, and the spectrum presents a hydrogen-like shape for positive values of the deformation parameter $\eta$ and of the coupling constant $k$.

10 pages, 1 figure. Based on the contribution presented at "XXIst International Conference on Integrable Systems and Quantum symmetries (ISQS-21)", June 12-16, 2013, Prague, Czech Republic

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Cited by in corpus (2)

• An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric
• The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion