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Semiclassical Szego limit of resonance clusters for the hydrogen atom Stark Hamiltonian

We study the weighted averages of resonance clusters for the hydrogen atom with a Stark electric field in the weak field limit. We prove a semiclassical Szego-type theorem for resonance clusters showing that the limiting distribution of the resonance shifts concentrates on the classical energy surface corresponding to a rescaled eigenvalue of the hydrogen atom Hamiltonian. This result extends Szego-type results on eigenvalue clusters to resonance clusters. There are two new features in this work: first, the study of resonance clusters requires the use of non self-adjoint operators, and second, the Stark perturbation is unbounded so control of the perturbation is achieved using localization properties of coherent states corresponding to hydrogen atom eigenvalues.