Local Inverse Scattering at a Fixed Energy for Radial Schrodinger Operators and Localization of the Regge Poles

We study inverse scattering problems at a fixed energy for radial Schrodinger operators on , . First, we consider the class of potentials q(r) which can be extended analytically in such that , . If q and are two such potentials and if the corresponding phase shifts and are super-exponentially close, then . Second, we study the class of potentials q(r) which can be split into q(r) = q (1)(r) + q (2)(r) such that q (1)(r) has compact support and . If q and are two such potentials, we show that for any fixed when if and only if for almost all . The proofs are close in spirit with the celebrated Borg-Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in with , , we show that the Regge poles are confined in a vertical strip in the complex plane.