Resonances at the Threshold for Pauli Operators in Dimension Two

Abstract

It is well-known that, due to the interaction between the spin and the magnetic field, the two-dimensional Pauli operator has an eigenvalue $0$ at the threshold of its essential spectrum. We show that when perturbed by an effectively positive perturbation, $V$, coupled with a small parameter $\varepsilon$, these eigenvalues become resonances. Moreover, we derive explicit expressions for the leading terms of their imaginary parts in the limit $\varepsilon\searrow 0$. These show, in particular, that the dependence of the imaginary part of the resonances on $\varepsilon$ is determined by the flux of the magnetic field. The cases of non-degenerate and degenerate zero eigenvalue are treated separately. We also discuss applications of our main results to particles with anomalous magnetic moments.