del Rio, Rafael and Simon, Barry (1997) Point spectrum and mixed spectral types for rank one perturbations. Proceedings of the American Mathematical Society, 125 (12). pp. 3593-3599. ISSN 0002-9939. doi:10.1090/S0002-9939-97-03997-X. https://resolver.caltech.edu/CaltechAUTHORS:20180719-132519253
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Abstract
We consider examples A_ λ = A + λ(ϕ, •)ϕ of rank one perturbations with ϕ a cyclic vector for A. We prove that for any bounded measurable set B ⊂ I, an interval, there exist A, ϕ so that {E ∈ I | some A_ λ has E as an eigenvalue} agrees with B up to sets of Lebesgue measure zero. We also show that there exist examples where A_ λ has a.c. spectrum [0,1] for all λ, and for sets of λ's of positive Lebesgue measure, A_ λ also has singular continuous spectrum in [0,1].
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| Additional Information: | © 1997 R. Del Rio and B. Simon. Received by the editors July 3, 1996. 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. R.d.R. would like to thank Professor Alejandro Bravo for very useful discussions. | |||||||||
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| Issue or Number: | 12 | |||||||||
| Classification Code: | 1991 Mathematics Subject Classification. Primary 47A55. | |||||||||
| DOI: | 10.1090/S0002-9939-97-03997-X | |||||||||
| Record Number: | CaltechAUTHORS:20180719-132519253 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20180719-132519253 | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 88005 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | George Porter | |||||||||
| Deposited On: | 19 Jul 2018 20:48 | |||||||||
| Last Modified: | 16 Nov 2021 00:22 |
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