Quantum oscillator on complex projective space (Lobachewski space) in constant magnetic field and the issue of generic boundary conditions

TL;DR

This paper develops a framework for quantum oscillators on complex projective and Lobachewski spaces with magnetic fields, introducing generic boundary conditions via self-adjoint extensions, leading to a family of energy spectra that preserve symmetries.

Contribution

It introduces a one-parameter family of self-adjoint extensions for quantum oscillators on complex projective and Lobachewski spaces, allowing for generic boundary conditions and symmetry-preserving spectra.

Findings

Derived a family of energy spectra parametrized by boundary conditions.

Included known oscillator results as special cases within the family.

Extended the self-adjoint extension method to conic oscillators with magnetic fields.

Abstract

We perform a 1-parameter family of self-adjoint extensions characterized by the parameter $\omega_0$. This allows us to get generic boundary conditions for the quantum oscillator on $N$ dimensional complex projective space($\mathbb{C}P^N$) and on its non-compact version i.e., Lobachewski space($\mathcal L_N$) in presence of constant magnetic field. As a result, we get a family of energy spectrums for the oscillator. In our formulation the already known result of this oscillator is also belong to the family. We have also obtained energy spectrum which preserve all the symmetry (full hidden symmetry and rotational symmetry) of the oscillator. The method of self-adjoint extensions have been discussed for conic oscillator in presence of constant magnetic field also.