Dynamical analysis of Schrödinger operators with growing sparse potentials
Résumé
We consider Scrödinger operators in l^2(Z^+) with potentials of the form V(n)=S(n)+Q(n). Here S is a sparse potential: S(n)=n^{1-\eta \over 2 \eta}, 0<\eta <1, for n=L_N and S(n)=0 else, where L_N is a very fast growing sequence. The real function Q(n) is compactly supported. We give a rather complete description of the (time-averaged) dynamics exp(-itH) \psi for different initial states \psi. In particular, for some \psi we calculate explicitely the "intermittency function" \beta_\psi^- (p) which turns out to be nonconstant. As a particular corollary of obtained results, we show that the spectral measure restricted to (-2,2) has exact Hausdorff dimension \eta for all boundary conditions, improving the result of Jitomirskaya and Last.
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