On the Maximal Excess Charge of the Chandrasekhar-Coulomb Hamiltonian in Two Dimensions

TL;DR

This paper proves that in a two-dimensional relativistic atomic model, the maximum number of electrons is bounded by twice the nuclear charge, extending the result to magnetic fields and using properties of the ultra-relativistic kinetic energy.

Contribution

It establishes a bound on the maximal excess charge for 2D relativistic Coulomb Hamiltonians, a result not previously known for this setting.

Findings

Maximal electron number does not exceed twice the nuclear charge in 2D relativistic models.

The positivity of a specific operator is proven for the ultra-relativistic kinetic energy.

The result extends to systems with external magnetic fields.

Abstract

We show that for the straightforward quantized relativistic Coulomb Hamiltonian of a two-dimensional atom -- or the corresponding magnetic quantum dot -- the maximal number of electrons does not exceed twice the nuclear charge. It result is then generalized to the presence of external magnetic fields and atomic Hamiltonians. This is based on the positivity of $$|\bx| T(\bp) + T(\bp) |\bx| $$ which -- in two dimensions -- is false for the non-relativistic case $T(\bp) = \bp^2$, but is proven in this paper for $T(\bp) = |\bp|$, i.e., the ultra-relativistic kinetic energy.