Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability

TL;DR

This paper rigorously solves the quantum nonlinear oscillator on an N-dimensional space with nonconstant curvature, preserving superintegrability through three quantization methods, and explores their relationships and implications for position-dependent mass quantum systems.

Contribution

It introduces the first example of a maximally superintegrable quantum system on a curved space with nonconstant curvature, analyzing three quantization approaches and their interrelations.

Findings

Full spectrum and eigenfunctions are rigorously determined.

Three quantization prescriptions are developed to preserve superintegrability.

Relationships among the quantization methods are explicitly described.

Abstract

The full spectrum and eigenfunctions of the quantum version of a nonlinear oscillator defined on an N-dimensional space with nonconstant curvature are rigorously found. Since the underlying curved space generates a position-dependent kinetic energy, three different quantization prescriptions are worked out by imposing that the maximal superintegrability of the system has to be preserved after quantization. The relationships among these three Schroedinger problems are described in detail through appropriate similarity transformations. These three approaches are used to illustrate different features of the quantization problem on N-dimensional curved spaces or, alternatively, of position-dependent mass quantum Hamiltonians. This quantum oscillator is, to the best of our knowledge, the first example of a maximally superintegrable quantum system on an N-dimensional space with nonconstant curvature.