TY - JOUR
T1 - Semiclassical Szegö limit of resonance clusters for the hydrogen atom Stark Hamiltonian
AU - Hislop, Peter D.
AU - Villegas-Blas, Carlos
PY - 2012
Y1 - 2012
N2 - 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.
AB - 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.
KW - hydrogen atom Stark effect
KW - resonance clusters
KW - semiclassical Szegö limit
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U2 - 10.3233/ASY-2011-1087
DO - 10.3233/ASY-2011-1087
M3 - Article
AN - SCOPUS:84865461066
SN - 0921-7134
VL - 79
SP - 17
EP - 44
JO - Asymptotic Analysis
JF - Asymptotic Analysis
IS - 1-2
ER -