On the asymptotic behavior of solutions of the Sturm-Liouville equation with an oscillating potential

Abstract

In the article we consider the Sturm-Liouville equation −y ′′+(q(x)+h(x))y = 0, 0 ≤ x < +∞, where q(x) satisfies the conditions of regularity of growth at infinity and h(x) is a fast-oscillating function. Sufficient conditions are found under which the asymptotics of solutions of the equation is determined only by the function q(x). A new method for obtaining asymptotic formulas for solutions is proposed, consisting in the standard replacement of the equation by a system of first-order equations followed by the application of the Hausdorff identity [1]