97-365 Christ M., Kiselev A.
Absolutely continuous spectrum for one-dimensional Schr\"odinger operators with slowly decaying potentials: some optimal results. (80K, LATeX) Jun 21, 97
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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.

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