Mathematical Sciences: Well-Posed Inverse Problems

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The object of this work, the study of inverse problems in mathematical analysis, is one of recovering physical properties of an object or phenomenon from data obtained by remote observations. The problems are well-posed in the sense that the data sets are not so burdensome as to rule out any solutions, but sufficient for the reconstruction. The problem of determining a quantum mechanical potential from its scattering amplitude goes back to the beginning of quantum mechanics. But little attention is given to finding data sets for which the problem is well-posed in the sense that the mapping from the potential to the scattering data is continuously invertible. One likely candidate for such a data set is backscattering data, the subject of this research. Much successful work has been accomplished on the question in three dimensions, though mainly confined to small potentials. The present work seeks to remove this assumption and look for more general results, possibly giving up some degree of well-posedness. Work will continue on two-dimensional backscattering as well, where a type of singularity in a parameter makes backscattering a much more untractable problem, even assuming compact support of the potential.Inverse problems in the context of this project are formulated in terms of classical equations, such as the Schrodinger equation. These equations may represent wave phenomena involving magnetic and electric potentials, acoustics and other type of waves in the presence of obstacles. Although much of the research focuses on the mathematical analysis of the inverse mapping, its connections with the physical world are easily traced.