Point spectrum and mixed spectral types for rank one perturbations
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- by Rafael del Rio and Barry Simon PDF
- Proc. Amer. Math. Soc. 125 (1997), 3593-3599
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 $\{ E\in I \mid {}$ 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 $[0,1]$.References
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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
- MR Author ID: 189013
- 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
- © Copyright 1997 R. Del Rio and B. Simon
- Journal: Proc. Amer. Math. Soc. 125 (1997), 3593-3599
- MSC (1991): Primary 47A55
- DOI: https://doi.org/10.1090/S0002-9939-97-03997-X
- MathSciNet review: 1416082