Simon B.
Operators with Singular Continuous Spectrum, 
VII. Examples with Borderline Time Decay
(29K, AMSTeX)

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.