  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.
