  We consider the semi-classical Dirichlet Pauli operator in bounded connected
domains in the plane. Rather optimal results have been obtained in previous
papers by Ekholm-Kova\v{r}\'ik-Portmann and Helffer-Sundqvist for the
asymptotics of the ground state energy in the semi-classical limit when the
magnetic field has constant sign. In this paper, we focus on the case when the
magnetic field changes sign. We show, in particular, that the ground state
energy of this Pauli operator will be exponentially small as the semi-classical
parameter tends to zero and give lower bounds and upper bounds for this decay
rate. Concrete examples of magnetic fields changing sign on the unit disc are
discussed. Various natural conjectures are disproved and this leaves the
research of an optimal result in the general case still open.
