The list of known examples of zero modes supports the idea that they have a geometric (rather than analytic) origin (). It is possible to obtain a local description of the structure of the set of zero mode producing magnetic potentials when working within certain classes of potentials. The general picture is of a submanifold whose co-dimension relates to the degeneracy of the corresponding zero modes (); it is tempting to view these submanifolds as the level sets of some (as yet to be determined) functional. Other ongoing problems relate to a full understanding of the zero-mode producing fields which posses an extra symmetry (similar to fields of constant direction) and applications to level crossing results (for such things as the Chern-Simons action).
An operator corresponding to a uniform magnetic field and periodic electric field is not itself periodic (since the magnetic potential appears as coefficients in the operator). The group of magnetic translations provides an alternative discrete symmetry group in this case, although it is only under the additional assumption of flux rationality that the usual analysis for periodic operators can be carried through to produce a band-gap picture for the spectrum. I have been interested in the number of spectral gaps in the case of the 3 dimensional Schrödinger operator (the so called Bethe-Sommerfeld conjecture is the statement that this should be finite; ) and the way in which generic periodic electric fields cause the Landau levels of the 2 dimensional Schrödinger operator to smear into bands; in particular, the band-gap picture in 2 dimensions shows a fractal structure, with the emergence of objects related to the Hofstadter butterfly. The 1 dimensional harmonic oscillator with a quasi-periodic perturbation arises naturally in this context; the eigenvalue asymptotics for such problems show interesting non-standard (specifically non-power) behaviour ().
Another area of interest related to periodic problems concerns the density of states for the periodic Schrödinger operator; in a physical sense the density of states can be viewed as the number of electron states of a given energy per unit volume. Full large energy asymptotics for this object appear to be computable and give detailed information about the original periodic potential.