Rank One Perturbations at Infinite Coupling

We discuss rank one perturbations Aα = A + α(φ,·)φ, α ∈R , A ≥ 0 self-adjoint. Let dμα(x) be the spectral measure defined by (φ, (Aα - z)−1 φ) = ∫ dμα(x)/(x - z). We prove there is a measure dρ∞ which is the weak limit of (1 + α2) dμα(x) as α → ∞. If φ is cyclic for A, then A∞, the strong resolvent...

Full description

Saved in:
Bibliographic Details
Published in:Journal of functional analysis 1995-02, Vol.128 (1), p.245-252
Main Authors: Gesztesy, F., Simon, B.
Format: Article
Language:English
Citations: Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We discuss rank one perturbations Aα = A + α(φ,·)φ, α ∈R , A ≥ 0 self-adjoint. Let dμα(x) be the spectral measure defined by (φ, (Aα - z)−1 φ) = ∫ dμα(x)/(x - z). We prove there is a measure dρ∞ which is the weak limit of (1 + α2) dμα(x) as α → ∞. If φ is cyclic for A, then A∞, the strong resolvent limit of Aα, is unitarily equivalent to multiplication by x on L2(R, dρ∞). This generalizes results known for boundary condition dependence of Sturm-Liouville operators on half-lines to the abstract rank one case.
ISSN:0022-1236
1096-0783
DOI:10.1006/jfan.1995.1030