Spectral Asymptotics and Lame Spectrum for Coupled Particles in Periodic Potentials

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 Lame equation. As a consequence, all but one instability zones of the linearized equation collapse to a point for the one-harmonic potentials. This provides a new example where Lame's finite zone potential arises in the simplest possible setting.