19 - Sturm—Liouville Theory

Abstract

This chapter elaborates Sturm–Liouville theory applicable to second-order linear ordinary differential operators. Many of the differential equations of mathematical physics are Sturm–Liouville equations. Sturm–Louville equations arise naturally, for instance, when separation of variables is applied to the wave equation, the potential equation or the diffusion equation. The chapter also defines the Sturm–Liouville operator. The differential equation with boundary conditions − (x2y′) ′ = 0; u(l) = 0; u(e) = 0 for x ∈ [1,e] is a regular Sturm–Liouville problem with unmixed boundary conditions, so the eigenfunctions are complete. The chapter highlights that the regular Sturm–Liouville equation, written in the form d2z/dt2 − r(t)z + λz = 0 with the boundary conditions z(0) = z(L) = 0 has the asymptotic eigenvalues and eigenfunctions.