(Super)Oscillator on CP(N) and Constant Magnetic Field

TL;DR

This paper explores the properties of a maximally integrable oscillator on complex projective spaces CP(N), examining effects of magnetic fields, supersymmetry, and transformations, revealing qualitative differences between the N=1 and N>1 cases.

Contribution

It introduces a new integrable oscillator model on CP(N), analyzes its behavior under magnetic fields, and constructs supersymmetric extensions and transformations, expanding understanding of geometric quantum systems.

Findings

Properties differ qualitatively for N=1 and N>1.

Magnetic fields preserve supersymmetry for N>1.

Constructed generalized MIC-Kepler problem via Kustaanheimo-Stiefel transformation.

Abstract

We define the "maximally integrable" isotropic oscillator on CP(N) and discuss its various properties, in particular, the behaviour of the system with respect to a constant magnetic field. We show that the properties of the oscillator on CP(N) qualitatively differ in the N>1 and N=1 cases. In the former case we construct the ``axially symmetric'' system which is locally equivalent to the oscillator. We perform the Kustaanheimo-Stiefel transformation of the oscillator on CP(2) and construct some generalized MIC-Kepler problem. We also define a N=2 superextension of the oscillator on CP(N) and show that for N>1 the inclusion of a constant magnetic field preserves the supersymmetry of the system.