A great deal of work has been done on the 2-dimensional conductivity problem by studying a first order system related to the conductivity equation and applying complex analysis methods. In this dissertation, we study the 2-dimensional homogeneous Schrodinger equation, which is best known in quantum physics as a probability distribution for the position of a particle, and the 2-dimensional Lame system of linear elasticity, which describes the deformation of a body due to external forces. We derive first order systems related to each of these problems. For the Schrodinger equation, we obtain a system of two equations, (D - Q)psi = 0, where Q is an integral operator, i.e. a non-local potential. For the Lame system, we obtain a system of four equations, (D - Q0 - Q-1)psi = 0, where Q0 is defined locally and Q-1 is an integral operator. With appropriate integrability assumptions and small norm conditions for the parameters, we find series solutions to both systems in appropriate function spaces, and relate those solutions to solutions of the original problems. We determine additional assumptions that allow for an asymptotic expansion of these solutions. For the Schrodinger equation, we show that the nonphysical scattering data is in L2. Finally, our main result is showing that the map which takes the potential to the scattering data is continuous, for potentials that are compactly supported.
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