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Operators with Singular Continuous Spectrum, VII. Examples with Borderline Time Decay

Simon, Barry (1996) Operators with Singular Continuous Spectrum, VII. Examples with Borderline Time Decay. Communications in Mathematical Physics, 176 (3). pp. 713-722. ISSN 0010-3616. https://resolver.caltech.edu/CaltechAUTHORS:20180315-112637401

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Abstract

We construct one-dimensional potentials V(x) so that if H = - d^2/dx^2 + V(x) on L^2(ℝ), then H has purely singular spectrum; but for a dense set D, φ є D implies that │φ,ℯ^(-itH)φ)│ ≦ C_φ│t│^(-1/2)In(│t│) for │t│ > 2. This implies the spectral measures have Hausdorff dimension one and also, following an idea of Malozemov-Molchanov, provides counterexamples to the direct extension of the theorem of Simon-Spencer on one-dimensional infinity high barriers.


Item Type:Article
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URLURL TypeDescription
https://doi.org/10.1007/BF02099257DOIArticle
https://link.springer.com/article/10.1007/BF02099257PublisherArticle
https://projecteuclid.org/euclid.cmp/1104286122PublisherArticle
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ORCID:
AuthorORCID
Simon, Barry0000-0003-2561-8539
Additional Information:© 1996 Springer-Verlag. Received: 10 January 1995, in revised form: 30 May 1995. Communicated by A. Jaffe. 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.
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Funding AgencyGrant Number
NSFDMS-9401491
Subject Keywords:Neural Network; Statistical Physic; Complex System; Nonlinear Dynamics; Time Decay
Issue or Number:3
Record Number:CaltechAUTHORS:20180315-112637401
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20180315-112637401
Official Citation:Simon, B. Commun.Math. Phys. (1996) 176: 713. https://doi.org/10.1007/BF02099257
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85332
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:16 Mar 2018 14:53
Last Modified:03 Oct 2019 19:29

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