On the spectrum of an oscillator in a magnetic field

TL;DR

This paper analyzes the spectral properties of a charged particle in a harmonic potential under a magnetic field, revealing a transition from bounded to unbounded spectra and the role of algebraic methods in understanding this quadratic Hamiltonian.

Contribution

It introduces an algebraic approach to study the spectral transition in a quadratic Hamiltonian with magnetic field influence.

Findings

Spectrum transitions from bounded to unbounded as parameter varies.

At the transition point, eigenvalues have infinite multiplicity.

Algebraic methods effectively analyze the Hamiltonian's spectral properties.

Abstract

We consider the Hamiltonian for a charged particle in a harmonic potential in the presence of a magnetic field. The most symmetric case depends on one parameter, the variation of which leads from a spectrum bounded from below to an unbounded spectrum. At the transition point the spectrum is bounded from below but each eigenvalue has infinite multiplicity. The algebraic method proves to be a remarkable tool for the analysis of this quadratic Hamiltonian.