Stability of the one electron atom Schr\"odinger model with magnetic field in two dimensions

Abstract

We study the stability of the one electron atom Schr\"odinger model with self-generated magnetic field in two dimensions. The magnetic energy is taken of the general form $K\int_{\mathbb{R}^2} |B|^p$ and we study the stability of the model as a function of the power $p$ and the coupling constant $K$. We show that for $p>3/2$, the model is always stable, and for $p<3/2$, the model is always unstable. In the critical case $p=3/2$, there is a critical stability constant $K_c$, that we characterize in terms of zero modes of the Dirac-Weyl operator. The value of $K_c$ is approximated using analytic and numerical methods.