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Introduce several KAM theorems for infinite dimensional Hamiltonian withshort range and discuss the relationship between spectra of linearized operatorand invariant tori. Especially, introduce a KAM theorem in the paper(Cummun.Math. Phys. 226, 61-100 (2002)) which shows that there are rich KAM tori for aclass of Hamiltonian with short range and with linearized operator of purepoint spectra. Here are also presented several open problems.

KAM theorems and open problems for infinite dimensional Hamiltonian with short range