Asymptotics

Abstract

The main purpose of this chapter is to examine the nature of the instability intervals–first their asymptotic lengths as they recede to infinity, and second the more specialised situation when all but a finite number of them are absent. To deal with the lengths we require asymptotic estimates for the eigenvalues λ _n and μ _n as n→∞, and the theory of Chapters 1 and 2 provides two methods to produce these estimates. Our method of choice in this chapter is based on the Prüfer transformation and oscillation theory of Chapter 2, and the other method (which we shall also touch on) uses a direct examination of the discriminant D(λ) as λ→∞. A feature of the asymptotic estimates is that they become increasingly accurate the more times that the coefficients p(x), q(x) and w(x) are differentiable, and we develop this theme in sections 3.3-3.7.