Mathematical Sciences: Some Problems in Partial DifferentialEquations

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Work will be done on problems arising in several areas of partial differential equations. Efforts will be made to characterize the spectrum of an elliptic differential operator, a fundamental question in both mathematics and physics. Special attention will be given to multiparticle Schrodinger operators. In this context, focus will be placed on predicting the finiteness or infiniteness of bound states for N-body atomic Schrodinger operators using geometric methods. These operators model an atom with an infinitely heavy nucleus and N - 1 electrons moving about the nucleus. Work will also be done investigating the spectral and scattering theory for certain reduced wave equations which are obtained from acoustic equations by separation the time variable. Emphasis will be placed on wave equations with singular coefficients which are strongly discontinuous or do not behave well at infinity. Applications of this research may be made to the analysis of wave propagation such as occurs in acoustic imaging. A third direction this work will take deals with the establishment of the classical Muntz approximation theorem for sets of positive measure. The approach to resolving this longstanding conjecture will use recent work relating properties of an associated discontinuos coefficient wave equation boundary value problem. A solution to this basic question will demonstrate that the so-called bang-bang principle in the theory of time optimal control for the heat equation is also valid.