Abstract:

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 Schr¨odinger 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.

Key words: Schr¨odinger equation, gauge, semi classical limit, Bloch eigenstate