, Volume 253, Issue 1, pp 221-252
Date: 27 Aug 2004

Dynamical Analysis of Schrödinger Operators with Growing Sparse Potentials

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Abstract

We consider discrete half-line Schrödinger operators H with potentials of the form V(n)=S(n)+Q(n). Here Q is any compactly supported real function, if n=L N and S(n)=0 otherwise, where η ∈ (0,1) and L N is a very fast growing sequence. We study in a rather detailed manner the time-averaged dynamics exp(−itH)ψ for various initial states ψ. In particular, for some ψ we calculate explicitly the “intermittency function” β ψ (p) which turns out to be nonconstant. The dynamical results obtained imply that the spectral measure of H has exact Hausdorff dimension η for all boundary conditions, improving the result of Jitomirskaya and Last.

Communicated by B. Simon
Acknowledgement I would like to thank F. Germinet for useful discussions.