96-330 Del Rio R., Simon B.

Point Spectrum and Mixed Spectral Types for Rank One Perturbations (22K, AMSTeX) Jul 8, 96

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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 $[0,1]$.

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