Christ M., Kiselev A.
Absolutely continuous spectrum for one-dimensional Schr\"odinger
operators with slowly decaying potentials: some optimal results.
(80K, LATeX)

ABSTRACT.  We prove new results on stability of the absolutely continuous spectrum
of one-dimensional Schr\"odinger operators. In particular, our results 
imply that the absolutely continuous spectrum of free and periodic 
Schr\"odinger operators is preserved under all perturbations 
V(x) satisfying |V(x)| \leq C(1+x)^{-\alpha}, \alpha>1/2. This 
result is known to be optimal on the power scale. We derive a
new general criterion for stability of the absolutely continuous
spectrum. We also consider more general potentials than power
decaying. In all cases we show that the main term of the asymptotic
behavior of generalized eigenfunctions has WKB form for almost
all energies.