Possible coexistence of localized and extended states in the quantum Hall effect

We study the system of a single electron in the plane, subject to a strong transverse magnetic field and an impurity potential, within a subspace scrP pertaining to a single Landau level. It is shown that dilute impurities build a potential V which is active only in a subspace P of scrP. The complementary subspace scrP-P, the null eigenspace of V, is an infinite-dimensional space of extended states. In other words, there is more than just one extended state available for conductance.