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

Journal Article · Mon Dec 15 04:00:00 EST 2014 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/J.AOP.2014.09.013· OSTI ID:22403483

Ballesteros, Ángel ^[1]; Enciso, Alberto ^[2]; Ragnisco, Orlando ^[3]; Riglioni, Danilo ^[4]

---

1. Departamento de Física, Universidad de Burgos, E-09001 Burgos (Spain)
2. Instituto de Ciencias Matemáticas, CSIC, Nicolás Cabrera 13-15, E-28049 Madrid (Spain)
3. Dipartimento di Matematica e Fisica, Università di Roma Tre and Istituto Nazionale di Fisica Nucleare sezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma (Italy)
4. Centre de Recherches Mathématiques, Université de Montreal, H3T 1J4 2920 Chemin de la tour, Montreal (Canada)

In this paper we quantize the N-dimensional classical Hamiltonian system H=(|q|)/(2(η+|q|)) p{sup 2}−k/(η+|q|) , that can be regarded as a deformation of the Coulomb problem with coupling constant k, that it is smoothly recovered in the limit η→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 η 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 analyzed, and the quantization problem for negative values of η and/or k is also sketched.

• OSTI ID: 22403483
• Journal Information: Annals of Physics (New York), Journal Name: Annals of Physics (New York) Vol. 351; ISSN 0003-4916; ISSN APNYA6
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
COULOMB FIELD
COUPLING CONSTANTS
DEFORMATION
EXACT SOLUTIONS
HAMILTONIANS
KINETIC ENERGY
LAPLACIAN
MANY-DIMENSIONAL CALCULATIONS
METRICS
QUANTIZATION
QUANTUM MECHANICS
QUANTUM SYSTEMS