Mathematical Sciences: Harmonic Analysis, Special Functions and Separation of Variables

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Separation of variables is the best known method of reducing partial differential equations to simpler form. It is taught in undergraduate mathematics courses and is often viewed as nothing more than a serendipitous accident. There are, in fact, deep mathematical and physical reasons why separation of variables works and why it actually applies across a large class of equations. The purpose of this work is to continue a long-term investigation into the circumstances and applicable methods for treating equations which can be decomposed. The work divides into several parts. First is the problem of determining a general definition of variable separation that applies to all (systems of) partial differential equations and working out the structure theory of possible separation types. Second is the problem of intrinsic characterization of separable coordinates. For Hamilton-Jacobi and Schrodinger equations this problem is essentially solved. For systems of equations such as the Dirac and linear elasticity equations, it remains open. Work will continue in this area. Finally, there is the problem of determining the special functions that arise when variables are separated in a particular equation and establishing properties of these functions. Particular emphasis is placed on those properties which are a consequence of the symmetry of the associated equation.