Point spectrum and mixed spectral types for rank one perturbations

We consider examples $A_{\lambda }= A+\lambda (\varphi , \,\cdot \,)\varphi$ of rank one perturbations with $\varphi$ a cyclic vector for $A$. We prove that for any bounded measurable set $B\subset I$, an interval, there exist $A, \varphi$ so that $\{E\in I \mid \text{some $A_{\lambda }$ has $E$ as an}$
eigenvalue$\}$ agrees with $B$ up to sets of Lebesgue measure zero. We also show that there exist examples where $A_{\lambda }$ has a.c. spectrum $[0,1]$ for all $\lambda$, and for sets of $\lambda$'s of positive Lebesgue measure, $A_{\lambda }$ also has point spectrum in $[0,1]$, and for a set of $\lambda$'s of positive Lebesgue measure, $A_{\lambda }$ also has singular continuous spectrum in $[0,1]$.