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.