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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. doi:10.1007/BF02099257.

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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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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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Subject Keywords:Neural Network; Statistical Physic; Complex System; Nonlinear Dynamics; Time Decay
Issue or Number:3
Record Number:CaltechAUTHORS:20180315-112637401
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Official Citation:Simon, B. Commun.Math. Phys. (1996) 176: 713.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85332
Deposited By: Ruth Sustaita
Deposited On:16 Mar 2018 14:53
Last Modified:15 Nov 2021 20:27

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