7. Sturm-Liouville Problems

The solution of ordinary second-order differential equations has played a fundamental role in the evolution of mathematical physics, starting with the eigenvibrations of a string, and culminating in the atomic vibrations of Schrödinger's wave equation. The separation of the fundamental differential operators of physics into functions of a single variable leads to a large class of important second order equations. While the solution of these equations cannot be given in closed form, except in special cases, it is possible to obtain an eminently useful approximation in terms of a mere quadrature. In the study of this method we encounter a certain refinement which permits us to deduce expressions in terms of elementary functions, which approximate some of the fundamentally important function classes of mathematical physics (such as the Bessel functions, and the Legendre, Hermite, and Laguerre type of polynomials), with a remarkably high degree of accuracy.