Asymptotics for Solutions of Harmonic Oscillator with Integral Perturbation

Abstract

We construct the asymptotics for solutions of a harmonic oscillator with integral perturbation when the independent variable tends to infinity. The distinctive feature of the considered integral perturbation is the oscillatory decreasing character of its kernel. We assume that the integral kernel is degenerate. This makes it possible to reduce the initial integro-differential equation to an ordinary differential system. To get the asymptotic formulas for the fundamental solutions of the obtained ordinary differential system, we use a special method proposed for the asymptotic integration of linear dynamical systems with oscillatory decreasing coefficients. By the use of special transformations, we reduce the ordinary differential system to the so-called L-diagonal form. We then apply the classical Levinson’s theorem to construct the asymptotics for the fundamental matrix of the L-diagonal system. The obtained asymptotic formulas allow us to reveal the resonant frequencies, i.e., the frequencies of the oscillatory component of the kernel that give rise to unbounded oscillations in the initial integro-differential equation. It appears that these frequencies differ slightly from the resonant frequencies that occur in the adiabatic oscillator with the sinusoidal component of the time-decreasing perturbation.