Quantum Oscillator on $\DC P^n$ in a constant magnetic field

TL;DR

This paper constructs and analyzes a quantum oscillator on complex projective and Lobachewski spaces under a magnetic field, revealing that the magnetic field does not alter the energy spectrum or superintegrability, and extends results via transformations to Coulomb-like systems.

Contribution

It introduces a quantum oscillator model on complex projective and Lobachewski spaces with magnetic fields, showing spectral invariance and superintegrability preservation, and connects these to Coulomb-like systems through transformations.

Findings

Magnetic field does not change the energy spectrum.

Superintegrability is preserved for N>1.

System remains exactly solvable for N=1.

Abstract

We construct the quantum oscillator interacting with a constant magnetic field on complex projective spaces $\DC P^N$, as well as on their non-compact counterparts, i. e. the $N-$dimensional Lobachewski spaces ${\cal L}_N$. We find the spectrum of this system and the complete basis of wavefunctions. Surprisingly, the inclusion of a magnetic field does not yield any qualitative change in the energy spectrum. For $N>1$ the magnetic field does not break the superintegrability of the system, whereas for N=1 it preserves the exact solvability of the system. We extend this results to the cones constructed over $\DC P^N$ and ${\cal L}_N$, and perform the (Kustaanheimo-Stiefel) transformation of these systems to the three-dimensional Coulomb-like systems.