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Bound states and scattered states for contraction semigroups (1985)

Goldstein, Jerome A.

Springer

Acta applicandae mathematicae 4 (1985), S. 93-98

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ISSN: 1572-9036

Keywords: Primary ; 47D05 ; secondary ; 34G10 ; 35P25 ; 47D45 ; RAGE theorem ; contraction semigroups ; asymptotics ; bound state ; scattered state ; mean ergodic theorem ; Wiener's theorem ; almost convergence

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract LetA generate a strongly continuous contraction semigroup {T(t)} on a Hilbert space and letL be a bounded operator. IfL(ζI−A)−1 is compact, then the Cesàro limit of ‖LT(t)f‖2 (ast→∞) is computed for all vectorsf. This limit is interpreted in terms of bound and scattered states in the context of quantum mechanical and classical wave propagation problems.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF02293492

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On Löwdin orthogonalization (1980)

Aiken, John G. ; Erdos, John A. ; Goldstein, Jerome A.

New York, NY : Wiley-Blackwell

International Journal of Quantum Chemistry 18 (1980), S. 1101-1108

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ISSN: 0020-7608

Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Chemistry and Pharmacology

Notes: It is shown that the Löwdin orthogonalization gives the unique minimum for the functional φ measuring the least squares distance between the given orbitals and the orthogonalized orbitals. Furthermore a much stronger result is obtained, namely that φ has only one local minimum, which is attained at the Löwdin orthogonalization and which is global. This justifies certain computer programs that compute Löwdin orthogonalization via minimization procedures. Finally there is a discussion of replacing the least squares metric by other metrics. The Löwdin orthogonalization turns out to be optimal with respect to all the commonly encountered norms.