On the Kolmogorov theorem for some infinite-dimensional Hamiltonian systems of short range

In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighborhood of a given $I^{0}$ for nearly integrable and analytic Hamiltonian system $H(I,\theta) = H_{0}(I)+ \epsilon H_{1}(I,\theta)$ of infinite degree of freedom and of short range. That is to say, we will give an extension of the original Kolmogorov theorem to the infinite-dimensional case of short range. The proof is based on the approximation of finite-dimensional Kolmogorov theorem and an improved KAM machinery which works for the normal form depending on initial $I^{0}$.