4.1 Introduction

A regular Sturm–Liouville system, or simply S–L system, refers to the following boundary value problem (BVP):

with prescribed boundary conditions at the end points a and b of the given interval [a, b]. Here p is a positive C1 function, q and ρ are continuous functions and ρ > 0, all defined on a bounded interval [a, b], and λ is a real parameter. When p vanishes somewhere in the interval or the interval under consideration is unbounded, the problem is termed as singular. Singular BVPs are more difficult to deal with. The interested reader may refer to Ref. [15].

It turns out that the non-trivial solutions to the BVP exist only for a discrete set of values of the parameter λ, tending to infinity. The situation may thus be compared with the eigenvalues of a matrix, considered as a linear operator on a finite-dimensional space. The main difference is that we are now working in an infinite dimensional space. In analogy with matrices, the discrete set of the values of the parameter for which the non-trivial solutions exist is called the eigenvalues of the BVP, and the corresponding non-trivial solutions are called the eigenfunctions.

Again, continuing with the similarity with matrices, we know that any vector in ℝn can be written as a unique linear combination of eigenvectors of a real, symmetric matrix. In this case, the eigenvectors are also orthogonal; we discuss more regarding orthogonality in the next section. Surprisingly, these concepts can be extended to the operator. In the case of a BVP too, one may express an arbitrary function as a linear combination of the corresponding eigenfunctions. Since these are (infinite) power series, we need to discuss the convergence and so on, and the tools required come from functional analysis (Hilbert space).

The S–L systems arise in many physical problems, for example, in the following situation. Consider the longitudinal vibrations of an elastic bar of local stiffness ρ(ξ) and density ρ(ξ).