First KdV Integrals¶and Absolutely Continuous Spectrum¶for 1-D Schrödinger Operator

Abstract:

We consider 1-D Schrödinger operators on L ²(R ₊) with slowly decaying potentials. Under some conditions on the potential, related to the first integrals of the KdV equation, we prove that the a.c. spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable. Examples show that for special classes of sparse potentials these results can not be improved.