Spectral asymptotics and Lam\'e spectrum for coupled particles in periodic potentials

TL;DR

This paper investigates the spectral properties of coupled particles in periodic potentials, revealing asymptotic periodicity in their spectrum and connecting the linearized problem to Lamé equations, especially for sinusoidal potentials.

Contribution

It uncovers a hidden asymptotic periodicity in the spectrum of coupled particles and links the linearization around sinusoidal potentials to Lamé equations, showing most instability zones collapse.

Findings

Spectral asymptotic periodicity in coupled particle motion.

Linearization around sinusoidal potentials relates to Lamé equations.

Most instability zones collapse for one-harmonic potentials.

Abstract

We make two observations on the motion of coupled particles in a periodic potential. Coupled pendula, or the space-discretized sine-Gordon equation is an example of this problem. Linearized spectrum of the synchronous motion turns out to have a hidden asymptotic periodicity in its dependence on the energy; this is the gist of the first observation. Our second observation is the discovery of a special property of the purely sinusoidal potentials: the linearization around the synchronous solution is equivalent to the classical Lam\`e equation. As a consequence, {\it all but one instability zones of the linearized equation collapse to a point for the one-harmonic potentials}. This provides a new example where Lam\'e's finite zone potential arises in the simplest possible setting.