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

TL;DR

This paper introduces a new family of exactly solvable quantum systems on curved spaces, extending classical superintegrable systems, and analyzes their spectral degeneracies and quantization methods using supersymmetric quantum mechanics.

Contribution

It presents the quantum analog of the classical Perlick family, providing exact solutions and exploring the connection between different quantization schemes.

Findings

Complete solution of the quantum systems using SUSYQM techniques

Identification of accidental degeneracy in the spectrum

Analysis of ordering ambiguities in quantization

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.