The talk presents recent results on magnetic Hamiltonians for two-well potentials. We show that tunneling may be completely suppressed by the magnetic field. Moreover, an even two-well potential may give rise to an odd ground state eigenfunction. In natural topologies on two-well potentials, complete suppression of tunneling occurs in codimension 1, while odd ground states occur for an open set of potentials.

Joint work with Jacob Shapiro and Michael Weinstein.