Localized states in strong magnetic field: resonant scattering and the Dicke effect

We study the energy spectrum of a system of localized states coupled to a 2D electron gas in strong magnetic field. If the energy levels of localized states are close to the electron energy in the plane, the system exhibits a kind of collective behavior analogous to the Dicke effect in optics. The latter manifests itself in ``trapping'' of electronic states by localized states. At the same time, the electronic density of states develops a gap near the resonance. The gap and the trapping of states appear to be complementary and reflect an intimate relation between the resonant scattering and the Dicke effect. We reveal this relation by presenting the exact solution of the problem for the lowest Landau level. In particular, we show that in the absence of disorder the system undergoes a phase transition at some critical concentration of localized states.