On the Lipschitz continuity of spectral bands of Harper-like and magnetic Schroedinger operators

We show for a large class of discrete Harper-like and continuous magnetic Schrodinger operators that their band edges are Lipschitz continuous with respect to the intensity of the external constant magnetic field. We generalize a result obtained by J. Bellissard in 1994, and give examples in favor of a recent conjecture of G. Nenciu.