# 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 ,
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.