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

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.