Spectral Theory of Self-Adjoint Ordinary Differential Operators

Introduction: Many of the properties of the ordinary Fourier series expansion of a given function are shared by the orthogonal expansion in terms of eigenfunctions of a second order ordinary differential operator. Let p = p(x) and q = q(x) be real-valued functions such that p, p', and q are continuous, and p(x) > 0, on a finite interval a ≤ x ≤ b. Let λ be a complex parameter. The classical Strum-Liouville theory [9, section 27; 4, Chapter 7; 21, Chapter 1]¹ is concerned with solutions of the differential equation -(py') + qy = λ, which satisfy certain real boundary conditions whose form need not be given here. These solutions, the so-called eigenfunctions, exist only for certain values of λ, the corresponding eigenvalues constitute a countable set of real numbers which cluster only at ∞. The corresponding eigenfunctions constitute an orthogonal system on [a,b] which is complete in L²(a, b). Thus the Parseval relation is also valid.