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

TL;DR

This paper develops KAM theorems for infinite dimensional Hamiltonian systems with short-range interactions, exploring spectral properties and invariant tori, and presents open problems in the field.

Contribution

It introduces new KAM theorems for infinite dimensional Hamiltonians with short-range interactions and analyzes the spectral relationship with invariant tori.

Findings

Existence of rich KAM tori for Hamiltonians with pure point spectra

Established links between spectral properties and invariant structures

Presented open problems for future research

Abstract

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.