Del Rio R., Simon B.
Point Spectrum and Mixed Spectral Types for Rank One Perturbations
(22K, AMSTeX)

ABSTRACT.  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 exists $A, \varphi$ 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 $[0,1]$.