Sturm—Liouville Operators

Abstract

Sturm—Liouville equations arise in many applications of electromagnetics, including in the formulation of waveguiding problems using scalar potentials, and using scalar components of vector fields and potentials. Sturm— Liouville equations are also encountered in separation-of-variables solutions to Laplace and Helmholtz equations, making a connection with certain special functions and classical polynomials. Spectral theory of the Sturm— Liouville operator is well developed and useful and is intertwined with the spectral theory of compact, self-adjoint operators through the inverse and resolvent operators. Accordingly, eigenfunctions of the regular Sturm— Liouville operator are found to be complete in certain weighted spaces of Lebesgue square-integrable functions, and generalizations to accommodate a continuous spectrum in the singular case are possible.