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Point spectrum and mixed spectral types for rank one perturbations

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. 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].


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1090/S0002-9939-97-03997-XDOIArticle
https://www.jstor.org/stable/2162259JSTORArticle
ORCID:
AuthorORCID
Simon, Barry0000-0003-2561-8539
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.
Funders:
Funding AgencyGrant Number
NSFDMS-9401491
Consejo Nacional de Ciencia y Tecnología (CONACYT)400316-5-0567PE
Issue or Number:12
Classification Code:1991 Mathematics Subject Classification. Primary 47A55.
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:03 Oct 2019 20:02

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