  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.
