  Introduce several KAM theorems for infinite dimensional Hamiltonian with
short range and discuss the relationship between spectra of linearized operator
and 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 a
class of Hamiltonian with short range and with linearized operator of pure
point spectra. Here are also presented several open problems.
