Characterization of bulk states in one-edge quantum Hall systems
We study magnetic quantum Hall systems in a half-plane with Dirichlet boundary conditions along the edge. Much work has been done on the analysis of the currents associated with states whose energy is located between Landau levels. These edge states are localized near the boundary and they carry a non-zero current. In this article, we study the behavior of states with energy close to a Landau level that are referred to as bulk states in the physics literature. The magnetic Schr\"odinger operator is invariant with respect to translations in the direction of the edge and is a direct integral of operators indexed by a real wave number. We analyse the fiber operators and prove new asymptotics on the band functions and their first derivative as the wave number goes to infinity. We apply these results to prove that the current carried by a bulk state is small compared to the current carried by an edge state. We also prove that the bulk states are exponentially small near the edge.