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arXiv · 2105.07577

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

Ki Yeun Kim·Mark Levi·Jing Zhou
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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.

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Ki Yeun Kim, Mark Levi, Jing Zhou. 2021-05-17. Spectral asymptotics and Lam\'e spectrum for coupled particles in periodic potentials. https://doi.org/10.1007/s10884-021-10108-z

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