Separation of Variables and Superintegrability

Separation of variables methods for solving partial differential equations are of immense theoretical and practical importance in mathematical physics, being the most powerful tool known for obtaining explicit solutions of the partial differential equations.

This book provides an up-to-date presentation of the theory and its relation to superintegrability. Collating and presenting the results scattered in the literature in a unified and accessible manner, that the authors have prepared, is an invaluable resource for mathematicians and mathematical physicists in particular, as well as science, engineering, geological and biological researchers interested in explicit solutions. Some of the key features of the book include: a general definition of separation of variables that is intuitive and also permits constructive computation; distinction between regular and non-regular separation, a source of confusion in the literature; presentation of an updated version of the geometric and algebraic Bôcher theory of variable separation of Laplace equations and its relation to the analytic Eisenhart theory; application of generalized Inönü-Wigner contractions of Lie algebras to separation of variables and superintegrability theory; computation of symmetry operators, computation of separation equations, etc; and unification of much of the theory of separation of variables, superintegrability and the special functions of mathematical physics through the study of R-separability of Laplace equations.

Historically, the most important applications of the theory have been to classical and quantum integrable systems on Riemannian spaces where the possible separable systems are related to higher-order symmetry of the underlying spaces. However, the authors demonstrate that the theory also applies to equations where there is no underlying symmetry present, and to situations where the separation is neither additive or multiplicative, but functional.