Chapter 2 - Homogeneous Equation

Abstract

This chapter is devoted to the study of the general theory of the homogeneous Hill's equation. After an introductory section, in Section 2.2 we present the classical Sturm comparison theorem which will lead us to the description of the spectrum of Dirichlet problem in Section 2.3 and the spectra of mixed and Neumann problems in Section 2.4. In Section 2.5 we present the main result of this chapter: the classical Oscillation Theorem, which describes the structure of the intervals of stability and instability of Hill's equation in terms of the eigenvalues of the related periodic and antiperiodic problems. Finally, in Section 2.6 we present the order relation between the eigenvalues of the previous boundary value problems.