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Published March 1996 | public
Journal Article

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.

Additional Information

© 1996 Springer-Verlag. Received: 10 January 1995, in revised form: 30 May 1995. Communicated by A. Jaffe. 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.

Additional details

Created:
August 22, 2023
Modified:
October 18, 2023