Affiliations: Department of Mathematics, University of Kentucky, Lexington, KY, USA. E-mail: hislop@ms.uky.edu | Universidad Nacional Autónoma de México, Instituto de Matemáticas, Unidad Cuernavaca, Mexico. E-mail: villegas@matcuer.unam.mx
Note: [] Corresponding author: Peter D. Hislop, Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA. E-mail: hislop@ms.uky.edu.
Abstract: 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 Szegö-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 Szegö-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.
Keywords: hydrogen atom Stark effect, resonance clusters, semiclassical Szegö limit
