Inverse spectral analysis with partial information on the potential. I. The case of an A.C. component in the spectrum

We consider operators -d2 /dx2+V in L2(R) with the sole hypothesis that V is limit point at ±∞ and that -d2 /dx2+V in L2((0,∞)) has some absolutely continuous component S+ in its spectrum. We prove that V on (-∞, 0) is completely determined by knowledge of V on (0,∞) and by the reflection coefficient R+(λ) for scattering from right incidence and energies λ element of S, where S subset or equal S+ has positive Lebesgue measure. (orig.)