95-277 Simon B.
Operators with Singular Continuous Spectrum, VII. Examples with Borderline Time Decay (29K, AMSTeX) Jun 15, 95
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Abstract. We construct one-dimensional potentials $V(x)$ so that if $H= -\frac{d^2}{dx^2}+V(x)$ on $L^{2}(\Bbb R)$, then $H$ has purely singular spectrum; but for a dense set $D$, $\varphi\in D$ implies that $|(\varphi, e^{-itH}\varphi)|\leq C_{\varphi} |t|^{-1/2}\ln (|t|)$ 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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