Areas of Research Interest

Zero modes

Zero modes are simply the zero energy eigenfunctions of a given operator - in my case the Pauli operator on 3 dimensional Euclidean space or on the 3 sphere. Apart from their intrinsic interest, problems concerning zero modes have significant physical applications in areas such as the stability of matter and the non-perturbative behaviour of the 3 dimensional Fermionic determinant in quantum electrodynamics. The first examples of zero modes were only found in 1986 and there's still plenty of work to be done to obtain a complete picture of which magnetic fields produce zero modes.

The list of known examples of zero modes supports the idea that they have a geometric (rather than analytic) origin ([5]). 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 ([8]); 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).

Periodic problems

Operators such as the Schrödinger and Pauli operators corresponding to periodic electric and magnetic fields arise naturally in the study of the electronic properties of crystals. Such operators are also of independent mathematical interest, not least because they lie in the interesting gap between operators with short range potentials (which typically have bound states below a certain level and continuous scattering states above) and confining potentials (with pure point spectrum; e.g., the harmonic oscillator). The existence of a discrete symmetry group for periodic operators (the translations preserving a lattice) can be used to show that the spectrum of these operators has a band-gap structure. Basic questions concerning the spectrum consider such things as the existence of eigenvalues, the number of gaps and the density of electron states in the bands.

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; [9]) 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 ([10]).

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.

Unique continuation theorems

A longer term project is the study of problems of unique continuation at infinity. At one level this is a question of determining the possible decay rates for solutions of partial differential equations on $\mathbb{R}^n$ , but the problem can be re-expressed in terms of ordinary differential equations with operator coefficients or local unique continuation theorems for operators with possibly singular coefficients. The problem also has important links to the existence of eigenvalues for periodic operators; in particular the hope is to develop new approaches to eigenvalue questions for periodic operators that don't rely so heavily on the discrete symmetry groups and can then be used to get some spectral information for quasi-periodic operators and operators with random potentials.