On the determination of potentials without bound state data (Q1891046)

One considers the quantum mechanical inverse scattering problem in one space dimension, clearly the problem of recovering a potential \(V(x)\) from scattering data. The known basic result states that the so-called \(S\)-matrix, together with bound state information, determines the potential uniquely. The paper examines under which conditions the \(S\)-matrix alone is sufficient for unique recovery of \(V\). Some uniqueness results are stated in this way, and their proofs are based on a transformation to an equivalent ``time problem''. A potential recovery algorithm is given, but as pointed out by the author himself, it does not converge for a wide class of potentials.