Counting Function of Magnetic Resonances for Exterior Problems

We study the asymptotic distribution of the resonances near the Landau levels ({Lambda_q =(2q+1)b}), ({q in mathbb{N}}), of the Dirichlet (resp. Neumann, resp. Robin) realization in the exterior of a compact domain of ({mathbb{R}^3}) of the 3D Schrödinger operator with constant magnetic field of scalar intensity ({b > 0}). We investigate the corresponding resonance counting function and obtain the main asymptotic term. In particular, we prove the accumulation of resonances at the Landau levels and the existence of resonance-free sectors. In some cases, it provides the discreteness of the set of embedded eigenvalues near the Landau levels.