Abstract
We construct one-dimensional potentialsV(x) so that if\(H = - \frac{{d^2 }}{{dx^2 }} + V(x)\) onL 2(ℝ), thenH has purely singular spectrum; but for a dense setD, φ εD implies that |ϕ,e -itHϕ|≦C ϕ|t|-1/2 ln(|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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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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Simon, B. Operators with singular continuous spectrum, VII. Examples with borderline time decay. Commun.Math. Phys. 176, 713–722 (1996). https://doi.org/10.1007/BF02099257
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DOI: https://doi.org/10.1007/BF02099257