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

On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator III: Magnetic fields that change sign