Asymptotic integration of a certain second-order linear delay differential equation

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• 作者：Pavel Nesterov
• 关键词：Asymptotics ; Delay differential equation ; Oscillating coefficients ; Oscillation of solutions ; Levinson’s theorem ; Method of averaging ; Schrödinger equation
• 刊名：Monatshefte für Mathematik
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：182
• 期：1
• 页码：77-98
• 全文大小：
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081
• 卷排序：182

文摘

We construct the asymptotic formulas for solutions of a certain linear second-order delay differential equation as independent variable tends to infinity. When the delay equals zero this equation turns into the so-called one-dimensional Schrödinger equation at energy zero with Wigner–von Neumann type potential. The question of interest is how the behaviour of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients.