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This is joint work with Mauricio Garcia Arroyo (Paris-Dauphine).We consider a simplified model of Polyacetylene introduced by Su,Schrieffer and Cheeger in 1979.This model has been extensively studied in the physics literature by numerical methods,and generalized to other polymers.It involves a quantum gas containing finitely or infinitely many electrons interacting with a finite or infinite chain of nuclei.Since these nuclei are much heavier than the electrons,they are described by Classical Mechanics.