KAM for Hamiltonian partial differential equations with weaker Spectral Asymptotics

In this paper, we establish an abstract infinite dimensional KAM theorem dealing with normal frequencies in weaker spectral asymptotics \Omega_{i}(\xi)=i^d+o(i^{d})+o(i^{\delta}), where $d>0, \delta<0$, which can be applied to a large class of Hamiltonian partial differential equations in high dimensions. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for Schr\"odinger equations, the Klein-Gordon equations with exponential nonlinearity and other equations of any spatial dimension.