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Sum rules and other properties involving resonance projection operatorsA sum rule is derived for the auxiliary eigenvalues of an equation whose eigenspectrum pertains to projection operators which describe electron scattering from multielectron atoms and ions. The sum rule's right-hand side depends on an integral involving the target system eigenfunctions. The sum rule is checked for several approximations of the two-electron target. It is shown that target functions which have a unit eigenvalue in their auxiliary eigenspectrum do not give rise to well-defined projection operators except through a limiting process. For Hylleraas target approximations, the auxiliary equations are shown to contain an infinite spectrum. However, using a Rayleigh-Ritz variational principle, it is shown that a comparatively simple aproximation can exhaust the sum rule to better than five significant figures. The auxiliary Hylleraas equation is greatly simplified by conversion to a square root equation containing the same eigenfunction spectrum and from which the required eigenvalues are trivially recovered by squaring.
Document ID
Document Type
Reprint (Version printed in journal)
Berk, A.
(NASA Goddard Space Flight Center Greenbelt, MD, United States)
Temkin, A.
(NASA Goddard Space Flight Center Greenbelt, MD, United States)
Date Acquired
August 12, 2013
Publication Date
December 1, 1985
Publication Information
Publication: Physical Review A - General Physics, 3rd Series
Volume: 32
ISSN: 0556-2791
Subject Category
Atomic And Molecular Physics
Accession Number
Distribution Limits

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