Determining potentials from the scattering map of the time-dependent Schrödinger equation

For a time dependent Schrödinger equation, the scattering map is the map sending the asymptotic profile of a solution as $t\to -\mathrm{\infty }$ to its asymptotic profile as $t\to +\mathrm{\infty }$. In this paper we show that, for a certain class of metrics, the scattering maps and Poisson operators associated to two Schrödinger operators on the same curved space only differ by a compact operator on a critical level if and only if the two potentials are equal.