Behavior for large time of an infinite chain of harmonic oscillators with defects

TL;DR

This paper investigates the long-time behavior of an infinite harmonic chain with defects, deriving dispersive bounds for solutions in energy-weighted norms to understand how irregularities affect wave propagation over time.

Contribution

It provides new insights into the dispersive properties and long-time dynamics of infinite harmonic chains with localized defects, extending previous models without irregularities.

Findings

Derived dispersive bounds for solutions in energy-weighted norms

Analyzed the impact of defects on wave propagation over long times

Established conditions for stability and decay in irregular chains

Abstract

An infinite irregular harmonic chain of particles is considered. We assume that some particles (``defects'') in the chain have masses and force constants of interaction different from the masses and the interaction constants of the other particles. We study the Cauchy problem for this model. The main goal is to study the long-time behavior and derive the dispersive bounds for the solutions in the energy weighted norms.