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Operators with singular continuous spectrum, VII. Examples with Borderline Time Decay

By Barry Simon


We construct one-dimensional potentials V (x) sothatifH = − d2 + V (x) on dx2 L2 (R), then H has purely singular spectrum; but for a dense set D, ϕ ∈ D implies that |(ϕ, e−itH ϕ) | ≤Cϕ|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

Year: 2012
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