Quantum tunneling and its absence in deep wells and strong magnetic fields

We present results on quantum tunneling between deep potential wells, in the presence of a strong constant magnetic field. We construct a family of double-well potentials containing examples for which the low-energy eigenvalue splitting vanishes, and hence quantum tunneling is eliminated. Further, by deforming within this family, the magnetic ground state can be made to transition from symmetric to antisymmetric. However, for typical double wells in a certain regime, tunneling is not suppressed, and we provide a lower bound for the eigenvalue splitting.