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Rank One Perturbations at Infinite Coupling

Gesztesy, F. and Simon, B. (1995) Rank One Perturbations at Infinite Coupling. Journal of Functional Analysis, 128 (1). pp. 245-252. ISSN 0022-1236. doi:10.1006/jfan.1995.1030.

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We discuss rank one perturbations A_α = A + α(φ,·)φ, α ∈R , A ≥ 0 self-adjoint. Let dμα(x) be the spectral measure defined by (φ, (A_α - z)^(−1) φ) = ∫ dμ_α(x)/(x - z). We prove there is a measure dρ_∞ which is the weak limit of (1 + α^2) dμ_α(x) as α → ∞. If φ is cyclic for A, then A_∞, the strong resolvent limit of A_α, is unitarily equivalent to multiplication by x on L^2(R, dρ_∞). This generalizes results known for boundary condition dependence of Sturm-Liouville operators on half-lines to the abstract rank one case.

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Simon, B.0000-0003-2561-8539
Additional Information:© 1995 Academic Press. Received February 7, 1994. Communicated by L. Gross. This material is based on work supported by the National Science Foundation under Grant DMS-9101715. The government has certain rights in this material. F. G. is indebted to the Department of Mathematics at Caltech for its hospitality and support during the summer of 1993 where some of this work was done.
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Official Citation:F. Gesztesy, B. Simon, Rank One Perturbations at Infinite Coupling, Journal of Functional Analysis, Volume 128, Issue 1, 1995, Pages 245-252, ISSN 0022-1236, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81203
Deposited By: Ruth Sustaita
Deposited On:06 Sep 2017 21:49
Last Modified:15 Nov 2021 19:41

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