Operators with Singular Continuous Spectrum. Pt. 7. Exammples with Borderline Time Decay

We construct one-dimensional potentials V(x) so that if H=-d2/dx2+V(x) on L2(R), then H has purely singular spectrum; but for a dense set D, φ element of D implies that vertical stroke (φ, e-itHφ) vertical stroke ≤ C φ vertical stroke t vertical stroke -1/2 ln(vertical stroke t vertical stroke) for vertical stroke t vertical stroke >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. (orig.)