Salmon Lecture Theatre
School of Mathematics, Hamilton Building
Trinity College Dublin
Time: 4 pm
February 23 Daniel Elton (Heriot Watt U., Scotland)
Zero energy bound states of the Pauli operator in three dimensions
Aside from their intrinsic mathematical
interest, zero energy $L^2$ eigenfunctions
of the Pauli operator on $\mathbb{R}^3$ (or \emph{zero modes}) have
important applications in several areas of mathematical physics (such
as the stability of matter and the non-perturbative behaviour of the
Fermionic determinant in QED). Unlike their two dimensional
counterparts, the well known Aharonov-Casher zero modes, results
relating to three dimensional zero modes are relatively limited and
recent --- the first examples were only
found around 15 years ago. To date most results relate to
specific classes of examples, generally constructed by pulling back
two dimensional magnetic fields via conformal submersions. General
results are still surprisingly limited; an example
of such results is that within the class of magnetic fields decaying
as $o(\lvert x\rvert^{-2})$ at $\infty$, the set of zero mode
producing fields is generically a sub-manifold of co-dimension~1.