Mathematical Sciences: Inverse Scattering Theory

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Work to be done during the period of this award will focus on higher dimensional inverse scattering problems and on related one dimensional problems. The underlying idea for work of this nature is that of constructing obstacles from data measured as particles pass the obstacle. The transition from the future of a solution of, say, the wave equation, to the past is called the scattering transform. Knowledge of either the transform or its spectrum is considered to be central to understanding any wave or generalized wave which experiences scattering. This work will consider extending results on time dependent Schrodinger equations with slowly decaying potentials in which the inverse equation admits simple poles. In higher dimensional settings the advent of multiple poles cannot be ruled out. However, their structure is not understood. The question of whether or not they may accumulate will also be studied. One method to be used will consider solutions for time dependent Schrodinger equations which are not necessarily bounded in time. This method has been used to achieve the complete solution (for all potentials) for the one dimensional nxn inverse scattering problem as well as for the one dimensional n-th order scalar equation. Several classical equations and systems are expected to be analyzed using results developed through this investigation. They include the Zakharov-Shabat system and the KPI equation.