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 carry a non-zero current that remains well-localized in a neighborhood of the boundary. In this article, we study the behavior of states with energies close to a Landau level. Such states are referred to as bulk states in the physics literature. Since the magnetic Schrödinger operator is invariant with respect to translations along the edge, it is a direct integral of operators indexed by a real wave number. We analyse these 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 small near the edge.