Rank one perturbations and singular integral operators

TL;DR

This paper studies rank one perturbations of self-adjoint operators, revealing their spectral representation as singular integral operators, and establishes new two-weight estimates for the Hilbert transform with applications to spectral theory.

Contribution

It introduces a novel spectral representation for rank one perturbed operators as singular integral operators and proves uniform boundedness of regularized Cauchy transforms, linking spectral measures to spectral types.

Findings

Spectral representation of perturbed operators as singular integral operators.

Uniform boundedness of regularized Cauchy transforms between spectral measures.

A criterion for pure absolutely continuous spectrum based on spectral measure density.

Abstract

We consider rank one perturbations $A_\alpha=A+\alpha(\cdot,\varphi)\varphi$ of a self-adjoint operator $A$ with cyclic vector $\varphi\in\mathcal H_{-1}(A)$ on a Hilbert space $\mathcal H$. The spectral representation of the perturbed operator $A_\alpha$ is given by a singular integral operator of special form. Such operators exhibit what we call 'rigidity' and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms $T_\varepsilon$ are uniformly (in $\varepsilon$) bounded operators from $L^2(\mu)$ to $L^2(\mu_\alpha)$, where $\mu$ and $\mu_\alpha$ are the spectral measures of $A$ and $A_\alpha$, respectively. As an application, a sufficient condition for $A_\alpha$ to have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of $A$ with respect to $\varphi$. Some examples, like Jacobi matrices and Schr\"odinger operators with $L^2$ potentials are considered.