Dia- and paramagnetism for nonhomogeneous magnetic fields

Diamagnetism of the magnetic Schrödinger operator and paramagnetism of the Pauli operator are rigorously proven for nonhomogeneous magnetic fields in the large field, in the large temperature and in the semiclassical asymptotic regimes. New counterexamples are presented which show that neither dia-nor paramagnetism is true in a robust sense (without asymptotics). In particular, we demonstrate that the recent diamagnetic comparison result by Loss and Thaller [M. Loss and B. Thaller, Commun. Math. Phys. (submitted)] is essentially the best one can hope for.