On the Cauchy Problem for a Linear Harmonic Oscillator with Pure Delay

TL;DR

This paper provides an explicit solution representation for a delayed harmonic oscillator's Cauchy problem using a special delay exponential function, highlighting spectral differences caused by delay without imposing positivity conditions.

Contribution

It introduces a novel explicit solution formula for a linear oscillator with pure delay, utilizing a delay exponential function and addressing spectral complexity.

Findings

Spectrum becomes infinite with delay, unlike the finite spectrum without delay.

Explicit solution representation is achieved without positivity constraints.

The approach advances understanding of delayed differential equations in oscillatory systems.

Abstract

In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace so that the solution of the homogeneous problem can be written as a linear combination of these two eigenfunctions. As opposed to that, in the presence even of a small delay, the spectrum is infinite and a finite sum representation is not possible. Using a special function referred to as the delay exponential function, we give an explicit solution representation for the Cauchy problem associated with the linear oscillator with pure delay. In contrast to earlier works, no positivity conditions are imposed.