Schrödinger operators with periodic magnetic fields

Schrödinger operators with periodic electric and magnetic fields play a central role in solid state physics for understanding the low temperature behaviour of periodic crystals. Mathematically, periodic electric fields are best dealt with using Floquet/Bloch theory: The space of solutions of the periodic equation can be decomposed into subspaces consisting of quasiperiodic solutions, and the spectrum of the periodic Schrödinger operator is the union of the spectra in these subspaces.

I will briefly review this theory and its implications, and then proceed to describe the complications that arise under the presence of a magnetic field (such as gauge properties, magnetic translations). 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 -- for example, the absence of singular continuous spectrum. In the more complicated case of irrational flux there is no such decomposition of the space of solutions into subspaces. I will describe how to overcome this by setting up a completely new 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.