C. Remling
Schr\"odinger operators with decaying potentials:
some counterexamples
(84K, LaTeX 2e)

ABSTRACT.  Consider the one-dimensional Schr\"odinger operator
$H=-d^2/dx^2 +V(x)$ on $L_2(0,\infty)$. It's known that
if $|V(x)| \le C(1+x)^{-\alpha}$ with $1/2 < \alpha \le 1$,
then $H$ has absolutely continuous spectrum on $[0,\infty)$,
with possibly also some embedded singular spectrum there.
Here, we construct examples which show that recently
obtained results on the singular spectrum are in fact
optimal.