We consider the magnetic Schrödinger operator H in dimension d with metric, magnetic potential, and electric potential. All coefficients are supposed to be periodic with respect to some lattice. The Floquet theory shows that the spectrum of H has a band structure. Ruling out the possibility for some bands to degenerate into points is a more subtle problem. The absence of degenerate bands is equivalent to absolute continuity of the spectrum. For the periodic operator Schrödinger operator with electric potential only the absence of degenerate bands was proved by L. Thomas in 1973. Thomas suggested an original method of analytic continuation in the complex quasimomentum. In recent years the problem of absolute continuity of the spectrum of H has been intensively studied. We propose a survey of the recent results and unsolved problems in this area.
Heinz Siedentop