- 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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