Concentration of bloch eigenstates in the presence of gauge at the semi-classical limit

We prove a concentration result of a Bloch eigenstate in a periodic channel under a constant gauge. In the semi-classical limit h → 0 these eigenstates concentrate near a maximizer of the scalar potential of the associated Schrödinger operator, provided the constant gauge converges to a critical value from above. This is in contrast with the ground states which concentrate for any gauge in this limit near a minimizer of the scalar potential.