Inverse scattering and distribution of resonances on the real line

Abstract: We study aspects of scattering theory for the Schrödinger operator on the real line. In the first part of the thesis we consider potentials supported by a half-line, and we are interested in the inverse problem of reconstruction of the potential from the knowledge of values of the reflection coefficient at equidistributed points on the positive imaginary axis. Under the assumption of exponential decay of the potential, Hölder type stability estimates for this problem are obtained. In the second part of the thesis we study the distribution of scattering poles for the class of super-exponentially decaying potentials. Sharp upper bounds on the counting function of the poles in discs are derived and the density of resonances in strips is estimated. We also obtain estimates on the width of a pole-free strip and derive bounds on the location of the poles.