97-361 Christ, M., Kiselev, A., Remling, C.

The absolutely continuous spectrum of one-dimensional Schr\"odinger operators with decaying potentials. (14K, LATeX) Jun 19, 97

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Abstract. This is an announcement of the proof of some optimal results on the presevation of the absolutely continuous spectrum under perturbations by decaying potentials. We show that if |V(x)| \leq C(1+x)^{-\alpha} with \alpha > 1/2, the whole positive semi-axis is an essential support of the absolutely continuous spectrum. This result is optimal on the power scale. We also derive a new general criterion for the stability of the a.c. spectrum. Another result is that if limsup_{x \goto \infty}x|V(x)| < C, the spectrum is purely a.c. on ((2C/\pi)^{2},\infty). This is also optimal.

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