Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 097, 10 pages      arXiv:1010.0641      https://doi.org/10.3842/SIGMA.2010.097
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

A Family of Exactly Solvable Radial Quantum Systems on Space of Non-Constant Curvature with Accidental Degeneracy in the Spectrum

Orlando Ragnisco and Danilo Riglioni
Dipartimento di Fisica Universitá Roma Tre and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, I-00146 Roma, Italy

Received October 05, 2010, in final form December 07, 2010; Published online December 15, 2010

Abstract
A novel family of exactly solvable quantum systems on curved space is presented. The family is the quantum version of the classical Perlick family, which comprises all maximally superintegrable 3-dimensional Hamiltonian systems with spherical symmetry. The high number of symmetries (both geometrical and dynamical) exhibited by the classical systems has a counterpart in the accidental degeneracy in the spectrum of the quantum systems. This family of quantum problem is completely solved with the techniques of the SUSYQM (supersymmetric quantum mechanics). We also analyze in detail the ordering problem arising in the quantization of the kinetic term of the classical Hamiltonian, stressing the link existing between two physically meaningful quantizations: the geometrical quantization and the position dependent mass quantization.

Key words: superintegrable quantum systems; curved spaces; PDM and LB quantisation.

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