Multiparameter spectral theory and separation of variables

Multiparameter spectral theory generalizes the classical spectral theory of linear operators to n linear operators linked by n spectral parameters. The spectral parameters arise as separation constants when, for example, separation of variables techniques are used to solve boundary-value problems for partial differential equations. In this paper, we discuss some of the fundamental aspects of the theory such as solvability, commutativity and abstract representation theory. Many of these concepts are of importance to modern developments in separation of variables techniques. The ideas are illustrated in application to differential equations and special functions of mathematical physics