Spectral asymptotics and Lamé spectrum for coupled particles in periodic potentials
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913860110254080 |
|---|---|
| author | Kim, Ki Yeun Levi, Mark Zhou, Jing |
| author_facet | Kim, Ki Yeun Levi, Mark Zhou, Jing |
| contents | 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 Lamè 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 Lamé's finite zone potential arises in the simplest possible setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_07577 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Spectral asymptotics and Lamé spectrum for coupled particles in periodic potentials Kim, Ki Yeun Levi, Mark Zhou, Jing Dynamical Systems Mathematical Physics Classical Analysis and ODEs 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 Lamè 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 Lamé's finite zone potential arises in the simplest possible setting. |
| title | Spectral asymptotics and Lamé spectrum for coupled particles in periodic potentials |
| topic | Dynamical Systems Mathematical Physics Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2105.07577 |