COUNTING FUNCTION OF MAGNETIC RESONANCES FOR EXTERIOR PROBLEMS

24 pagesWe 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 Schr\"odinger operator with constant magnetic field of scalar intensity $b>0$. We investigate the corresponding resonance counting function and obtain the main asymptotic term. %giving a precise asymptotic formula of the rate accumulation of the resonances near a given Landau level $\Lambda_q$.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