Positive Measure Spectrum for Schrödinger Operators with Periodic Magnetic Fields

We study Schrödinger operators with periodic magnetic field in R2, in the case of irrational magnetic flux. Positive measure Cantor spectrum is generically expected in the presence of an electric potential. We show that, even without electric potential, the spectrum has positive measure if the magnetic field is a perturbation of a constant one. Introduction Magnetic Schrödinger operators have been studied in Solid State Physics, especially in connection with the Quantum Hall effect, as well as on their own right. In a regular crystal ‘physics’ is periodic, i.e., the electric potential caused by the background field of the ions is a periodic function. Magnetic fields internal as well as external ones are periodic as well, the latter ones typically being constant. Alas, as is well known, magnetic fields enter the Schrödinger operator through a vector potential, so that the resulting operator is not necessarily periodic. Indeed, it is so only in the simple and well-understood case of ‘zero flux’, where one has absolutely continuous spectrum and band-structure (Birman & Suslina, 1998; Sobolev, 1999). Here, the magnetic flux (in units of flux quanta) is defined by