WKB approach to zero distribution of solutions of linear second order differential equations

Abstract

Given a second-order linear differential equation y″( z)+ S( z) y( z)=0, the distribution of zeros of its solutions is defined by ν=∑ y(z)=0 δ z , where δ z stands for the Dirac delta at the point z. Some techniques of approximation of the restriction of ν to R directly from S( z) are considered. In particular, for the WKB method error bounds are provided and some related results established. In the second part, formulas for the appropriate scaling in the holonomic case are given. As an illustration, we obtain the asymptotic distribution of the real zeros of some families of polynomials.