Mathematical Sciences: Harmonic Analysis Special Functions and Separation of Variables

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Miller 9400533 This project continues a detailed mathematical study into the relationship between symmetries of the various fundamental partial differential and q-difference equations of mathematical physics; the possible coordinate systems in which these equations permit separation of variables and properties of the special function solutions which so arise. The work divides into several parts. The first concerns the problem of determining a practical definition of variable separation that applies to all (systems of) partial differential equations and the structure theory of possible separation types. Second is the problem of intrinsic characterization of separable coordinates, i.e., a coordinate- free characterization of coordinates. For Hamilton-Jacobi and Schrodinger equations this problem is essentially solved, but for systems of equations such as the Dirac and linear elasticity equations, it remains open, although progress has been made. Third, there is the problem of determining the special functions that arise when variables are separated in a particular equation and establishing the properties of these functions. Finally, there are general applications of these and related group theoretic methods to problems that arise in other science and engineering areas, such as quantum mechanics and radar/sonar. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. One particularly important form of approximation is through the use of special functions. The study of special functions related to solutions of partial differential equations lies at t he heart of this research project. ***