Operators with singular continuous spectrum,VII. Examples with borderline time decay

We construct one-dimensional potentialsV(x) so that if onL ²(), thenH has purely singular spectrum; but for a dense setD, D implies that |,e ^-itH |C |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.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.