Using mixed data in the inverse scattering problem

Abstract

Consider the fixed-$\ell$ inverse scattering problem. We show that the zeros of the regular solution of the Schrödinger equation, $r_n(E)$, which are monotonic functions of the energy, determine a unique potential when the domain of the energy is such that the $r_n(E)$ range from zero to infinity. This suggests that the use of the mixed data of phase-shifts $\{\delta ({\ell }_0,k),k\ge k_0\}\cup \{\delta (\ell ,k_0),\ell \ge {\ell }_0\}$, for which the zeros of the regular solution are monotonic in both domains, and range from zero to infinity, offers the possibility of determining the potential in a unique way.

9 pages, 2 figures. Talk given at the Conference of Inverse Quantum Scattering Theory, Hungary, August 2007