\author{R. del Rio, S. Fuentes, A. Poltoratski}
\title{Coexistence of spectra in rank-one perturbation problems.}
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ABSTRACT.  \begin{abstract} 
We study the behavior of spectral functions corresponding to selfadjoint operators of the form 
$A + \lambda \langle \varphi, \cdot \rangle \varphi $. The focus is on the coexistence of absolutely continuous and singular spectra for values of the real parameter $\lambda $ in a given set $B$. For almost all points of $B$ it is possible to construct a family of rank one perturbations with mixed spectra. 
\end{abstract}