Spectral theory of Schrödinger operators with periodic magnetic field

Starting with Schrödinger operators with periodic electric fields I will briefly review ordinary (commutative) Floquet/Bloch theory and its implications, and then proceed to describe the complications that arise in the presence of a periodic magnetic field. In some cases known as rational magnetic flux, we can deal with these complications by putting Bloch theory in a more geometric context, i.e. studying sections of vector bundles instead of functions. Doing so allows us to draw conclusions about the measure-theoretic nature of the spectrum of the magnetic Schrödinger operator. In the more complicated case of irrational flux commutative Floquet/Bloch theory does not apply. I will describe how to overcome this by setting up a new non-commutative form of Bloch theory which extends the theory from commutative to non-commutative groups and C*-algebras. It allows to read off spectral properties from properties of the C*-algebras that are generated by the (magnetic) symmetries.