 95277 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 onedimensional 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 MalozemovMolchanov, provides
counterexamples to the direct extension of the theorem of
SimonSpencer on onedimensional infinity high barriers.
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