Oscillations

Abstract

The Floquet theory of Chapter 1 gives an overview of the global growth properties of the solutions of periodic systems. For the purposes of spectral analysis of formally symmetric systems, the oscillations or rotations of the real-valued solutions are also of similar importance. The tool for studying oscillations in Sturm-Liouville and Dirac systems is the Prüfer transform, which is introduced in section 2.2 and then used to analyse the boundary-value problems with separated boundary conditions on the period interval in section 2.3. When conjoined with the results on the periodic and semi-periodic boundary-value problems, this leads to the observation that the oscillations of the solutions of periodic systems have a linear growth asymptotic. The growth rate is a continuous, monotone increasing function of the real spectral parameter. It is known as the rotation number and is connected, by the physical interpretation of the equations, to the quasimomentum and the integrated density of states. In the special case of Hill’s equation, the oscillation properties can be equivalently studied by counting zeros of solutions.