Serguei Tcheremchantsev
Dynamical analysis of Schr\"odinger operators with growing sparse 
potentials
(80K, LaTeX)

ABSTRACT.  We consider Scr\"odinger 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.