Completeness of the localised Landau orbits

Abstract

It is shown that the set of localised Landau orbits Psi R=(2 pi a02)-1/2 exp(-((r-R)2-2i(xY-yX))/4a02) (where the magnetic length a0=(h(cross)/eB)1/2), is overcomplete in the lowest Landau level if R is continuous. In the discrete case when R=sa+nb ranges over lattice sites as s and n vary over all integers, the set ( Psi sn) is still overcomplete in this subspace if Omega 2 pi a02, where Omega = mod a*b mod is the area of a unit cell. For Omega =2 pi a02, the set remains complete if a single state is removed but becomes incomplete if any two or more states are absent.