Operators with Singular Continuous Spectrum, VII. Examples with Borderline Time Decay

Abstract

We construct one-dimensional potentials V(x) so that if H = - d^2/dx^2 + V(x) on L^2(ℝ), then H has purely singular spectrum; but for a dense set D, φ є D implies that │φ,ℯ^(-itH)φ)│ ≦ C_φ│t│^(-1/2)In(│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.