Criterion for nonaccumulation of complex eigenvalues near zero for Pauli operators

Let $V=\eta W$ satisfy Assumption (C1), with $Eta iC\setminusR e^{ik\frac{\pi}{2}}$, $k\inZ$, and $W\in\mathfrak{B}_h(\mathbb{R}^3)$ of definite sign. Here $P_3(b,V)$ denotes the three-dimensional Pauli operator with magnetic field $b$ and potential $V$, and $\Re(V)$ is the real part of the potential.

Nonaccumulation criterion. There is no accumulation of complex eigenvalues of $P_3(b,V)$ near zero if and only if $\Re(V)>0$.

The statement proposes a generalization of the preceding result for the specific perturbations considered in the paper. The source presents more general magnetic fields and perturbations of the magnetic field itself as open problems.