Zero modes of the Weyl-Dirac operator and their strong field asymptotics

Daniel Elton (Lancaster)

Tuesday 12th May, 2009 16:00-17:00 214

Abstract

For a given magnetic potential $A$ one can define the Weyl-Dirac operator $\sigma.(-i\nabla-A)$ on $\mathbb{R}3$. The question of whether or not this operator posses a zero-energy $L2$ eigenfunction, or \emph{zero mode}, is quite subtle. An expanding collection of results now exists on the existence and asymptotic properties of zero modes, as well as on the local structure of the set of zero mode producing potentials. The strong field (or, equivalently, semi-classical) regime can be studied by considering the asymptotic occurrence of zero modes for scaled potentials $tA$ ($A$ fixed, $t\to\infty$). General $O(t3)$ bounds on the number of zero modes can be obtained. These bounds can be refined to $O(t2)$ asymptotics for a special class of potentials $A$ that are constructed from potentials on $\mathbb{R}2$; a key step involves localising the Aharonov-Casher theorem to obtain good estimates for the number of approximate zero modes'' for two-dimensional Pauli operators.