Symmetry and Separation of Variables

This book is concerned with the relationship between the symmetries of a linear partial differential equation of mathematical physics, the coordinate systems in which the equation admits solutions via separation of variables, and the properties of the special functions that arise in this manner. It is shown how the method of separation of variables can be used to provide a group-theoretic foundation for much of special function theory. Special functions included in this treatment are the functions of Mathieu, Ince, Lame and others. A group-theoretic machine can be constructed which describes the various separable coordinate systems and expansion theorems relating distinct separable solutions. Indeed, for the most important linear equations the separated solutions can be characterized as eigenfunctions of a set of commuting second-order symmetric operators in the enveloping algebra. The problem of expanding one set of separable solutions in terms of another reduces to a problem in the representation theory of the Lie symmetry algebra. This book constitutes an important step in the group-theoretic approach to special functions. It is clearly written and should be accessible to a broad spectrum of readers.