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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Point spectrum and mixed spectral types
for rank one perturbations


Authors: Rafael del Rio and Barry Simon
Journal: Proc. Amer. Math. Soc. 125 (1997), 3593-3599
MSC (1991): Primary 47A55
MathSciNet review: 1416082
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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 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 $[0,1]$.


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Additional Information

Rafael del Rio
Affiliation: IIMAS-UNAM, Apdo. Postal 20-726, Admon. No. 20, Deleg Alvaro Obregon, 01000 Mexico, Mexico

Barry Simon
Affiliation: IIMAS-UNAM, Apdo. Postal 20-726, Admon. No. 20, Deleg Alvaro Obregon, 01000 Mexico, Mexico; Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03997-X
PII: S 0002-9939(97)03997-X
Received by editor(s): July 3, 1996
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The Government has certain rights in this material.
The first author was partially supported by CONACYT, Project 400316-5-0567PE
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 R. Del Rio and B. Simon