Two Particles with Zero-Range Interaction in a Magnetic Field

TL;DR

This paper analyzes the energy spectrum of two charged particles with zero-range interaction in a magnetic field, revealing universal bound states and resonance behavior as the field varies.

Contribution

It introduces a transcendental equation for the energy levels and explores non-perturbative effects, including the universal bound state in the strong-field limit.

Findings

Existence of a universal bound state in strong magnetic fields.

Resonance states transition into the continuum at critical field strengths.

A hyperbolic approximation for the number of bound levels as a function of field strength.

Abstract

Energy levels are investigated for two charged particles possessing an attractive, momentum-independent, zero-range interaction in a uniform magnetic field. A transcendental equation governs the spectrum, which is characterized by a collective Landau-level quantum number incorporating both center-of-mass and relative degrees of freedom. Results are obtained for a system of one charged and one neutral particle, with the interaction chosen to produce a bound state in vanishing magnetic field. Beyond deriving the weak-field expansion of the energy levels, we focus on non-perturbative aspects. In the strong-field limit, or equivalently for a system in the unitary limit, a single bound level with universal binding energy exists. By contrast, excited states are resonances that disappear into the continuum as the magnetic field is raised beyond critical values. A hyperbola is derived that approximates the number of bound levels as a function of the field strength remarkably well.