Rational integrability of trigonometric polynomial potentials on the flat torus |
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Authors: | Thierry Combot |
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Institution: | 1.Scuola Normale Superiore,Centro di Ricerca Matematica Ennio De Giorgi, Laboratorio Fibonacci,Piazza Cavalieri,Pisa,Italy |
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Abstract: | We consider a lattice ? ? ? n and a trigonometric potential V with frequencies k ∈ ?. We then prove a strong rational integrability condition on V, using the support of its Fourier transform. We then use this condition to prove that a real trigonometric polynomial potential is rationally integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of rationally integrable potentials in dimensions 2 and 3 and recover several integrable cases. After a complex change of variables, these potentials become real and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high-degree first integrals are explicitly integrated. |
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