• Corpus ID: 119039501

# The Krein spectral shift and rank one perturbations of spectra

@article{Poltoratski1996TheKS,
title={The Krein spectral shift and rank one perturbations of spectra},
author={Alexei Poltoratski},
journal={arXiv: Spectral Theory},
year={1996}
}
• A. Poltoratski
• Published 3 January 1996
• Mathematics
• arXiv: Spectral Theory
We use recent results on the boundary behavior of Cauchy integrals to study the Krein spectral shift of a rank one perturbation problem for self-adjoint operators. As an application, we prove that all self-adjoint rank one perturbations of a self-adjoint operator are pure point if and only if the spectrum of the operator is countable. We also study pairs of pure point operators unitarily equivalent up to a rank one perturbation and give various examples of rank one perturbations of singular…
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## References

SHOWING 1-9 OF 9 REFERENCES
Singular continuous spectrum under rank one perturbations and localization for random hamiltonians
• Mathematics
• 1986
We consider a selfadjoint operator, A, and a selfadjoint rank-one projection, P, onto a vector, φ, which is cyclic for A. In terms of the spectral measure dμAφ, we give necessary and sufficient
Pure point spectrum under 1-parameter perturbations and instability of Anderson localization
We consider a selfadjoint operator,A, and a selfadjoint rank-one projection,P, onto a vector, φ, which is cyclic forA. We study the set of all eigenvalues of the operatorAt=A+tP (t∈∝) that belong to
On the distributions of boundary values of Cauchy integrals
We use new methods to give short proofs to some known results on the distributions of boundary values of Cauchy integrals. We also indicate some further generalizations. INTRODUCTION Let W be an
Spectral analysis of rank one perturbations and applications
A review or Lhe general lheory of ~df-lVljoillt operatoro or lhe form A + on where n is nlll].;: one is presellled, Applicl1liu,,~ ine!llde proofs of loealizatiun fur S'chr(idingcr operators.