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. doi:10.1007/BF02099257. 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.
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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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Subject Keywords: | Neural Network; Statistical Physic; Complex System; Nonlinear Dynamics; Time Decay | |||||||||||||||
Issue or Number: | 3 | |||||||||||||||
DOI: | 10.1007/BF02099257 | |||||||||||||||
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: | 15 Nov 2021 20:27 |
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