# 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} }

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…

## 15 Citations

Survival Probability¶in Rank-One Perturbation Problems

- Mathematics
- 2001

Abstract: A finite complex Borel measure μ on the unit circle or on the real line is called Rajchman if its Fourier coefficients tend to 0 as n→∞. In quantum dynamics the self-adjoint operators…

The Spectral Shift Operator

- Mathematics
- 1999

We introduce the concept of a spectral shift operator and use it to derive Krein’s spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz…

Equivalence up to a rank one perturbation

- Mathematics
- 2000

This note is devoted to the spectral analysis of rank one perturbations of unitary and self-adjoint operators. We study the following question: given two cyclic (i.e., having simple spectrum)…

Cyclicity in rank‐1 perturbation problems

- MathematicsJ. Lond. Math. Soc.
- 2013

It is shown that for a fixed non-zero vector the property of being a cyclic vector is not rare, in the sense that for any family of rank-one perturbations of self-adjoint or unitary operators acting on the space, that vector will be cyclic for every operator from the family, with a possible exception of a small set with respect to the parameter.

On Matrix–Valued Herglotz Functions

- Mathematics
- 1997

We provide a comprehensive analysis of matrix–valued Herglotz functions and illustrate their applications in the spectral theory of self–adjoint Hamiltonian systems including matrix–valued…

Canonical systems and finite rank perturbations of spectra

- Mathematics
- 1996

We use Rokhlin's Theorem on the uniqueness of canonical systems to find a new way to establish connections between Function Theory in the unit disk and rank one perturbations of self-adjoint or…

Families of Spectral Measures with Mixed Types

- Mathematics
- 2002

Consider a family of Sturm-Liouville operators H θ on the half-axis defined as
$${{H}_{\theta }}u = - u'' + q(x)u 0 \leqslant x < \infty $$
with the boundary condition
$$ u(0)\cos \theta +…

Institute for Mathematical Physics Equivalence up to a Rank One Perturbation Equivalence up to a Rank One Perturbation

- Mathematics, Physics
- 2009

We prove that any two unitary operators with simple singular spectrum which contains the whole circle are unitarily equivalent up to a rank one operator.

Traces of operators

- Mathematics
- 2006

This survey is devoted to the history and the current state of the theory of regularized traces of linear operators. The main focus is on operators with discrete spectrum. Several appendices are…

## 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

- Mathematics
- 1994

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

- Mathematics
- 1996

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

- Physics
- 1995

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.…