Asymptotic solutions and eigenvalues of second-order differential equations with poles and turning points

Abstract

We consider the following system of differential equations dy dt = iρ 1 R1(t) x, dx dt = (iρR2(t) + q(t) iρR1(t)) y (∗) on a finite interval I = [a, b]. In this paper, we transform (∗) to the equation with poles and turning points of first order. Using of the asymptotic estimates provided in [2] for a special fundamental system of solutions of Sturm-Liouville equation, we study the asymptotic estimates for a special fundamental sys-tem of solutions of the corresponding differential equation and determine the asymptotic distribution of the eigenvalues