Kiselev A., Last Y., Simon B.
Modified Pr\"ufer and EFGP Transforms and the Spectral Analysis of
One-Dimensional Schr\"odinger Operators
(106K, AMSTeX)
ABSTRACT. Using control of the growth of the transfer matrices, we discuss
the spectral analysis of continuum and discrete half-line
Schr\"odinger operators with slowly decaying potentials. Among
our results we show if $V(x) = \sum^\infty_{n=1} a_n W(x-x_n)$
where $W$ has compact support and $x_n / x_{n+1} \to 0$, then
$H$ has purely a.c.~(resp.~purely s.c.) spectrum on $(0,\infty)$
if $\sum a^2_n < \infty$ (resp.~$\sum a^2_n = \infty$). For
$\lambda n^{-1/2} a_n$ potentials where $a_n$ are independent, identically distributed random variables with $E(a_n) =0$,
$E(a^2_n)=1$, and $\lambda < 2$, we find singular continuous
spectrum with explicitly computable fractional Hausdorff dimension.