Second-Order Equations on the Real Line

Abstract

In this chapter we consider equations of the form

$$y''+p(x, \lambda )y'+q(x,\lambda )y=0$$

and systems of two first-order equations for real x. We establish asymptotic formulae for the solutions under the assumption that the equations have no turning points. The coefficients of the equations are assumed to be sufficiently smooth. The asymptotic formulae presented in this chapter are variously called the WKB-approximation (or WKB-asymptotic behaviour), quasi-classical approximation, short-wave approximation, high-frequency approximation.