The Periodic Sturm–Liouville Equations

Abstract

We begin with a brief introduction the periodic Sturm–Liouville equations \([p(x)y'(x)]' + [\lambda w(x) - q(x)] y(x) = 0\). After reviewing some elementary knowledge of the theory of a general class of second-order linear homogeneous ordinary differential equations, we introduce two basic Sturm theorems on the zeros of solutions of the equations. Then, we study the theory of the equations with periodic coefficients—the Floquet theory—and learn basic concepts such as the normalized solutions and the discriminant. In the major part of the chapter, we study how the discriminant \(D(\lambda )\) of a periodic Sturm–Liouville equation changes as \(\lambda \) changes, and reach an understanding that solutions of a periodic Sturm–Liouville equation in various \(\lambda \) ranges are determined by \(D(\lambda )\) and the eigenvalues have a band structure. Finally, several theorems on the zeros of solutions of the periodic Sturm–Liouville equations are given. The formalisms of solutions in various \(\lambda \) ranges and the theorems on the zeros of solutions play an essential role in the theory in this book.