INVERSE SPECTRAL-ANALYSIS WITH PARTIAL INFORMATION ON THE POTENTIAL .1. THE CASE OF AN AC COMPONENT IN THE SPECTRUM

We consider operators -d(2)/dx(2) + V in L(2)(R) with the sole hypothe sis that V is limit point at +/-infinity and that -d(2)/dx(2) + V in L (2)((0, infinity)) has some absolutely continuous component S-+ in its spectrum. We prove that V on (-infinity, 0) is completely determined by knowledge of V on (0, infinity) and by the reflection coefficient R (+)(lambda) for scattering from right incidence and energies lambda is an element of S, where S subset of or equal to S-+ has positive Lebes gue measure.