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In this paper, we establish an abstract infinite dimensional KAM theoremdealing 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 beapplied to a large class of Hamiltonian partial differential equations in highdimensions. As a consequence, it is proved that there exist many invariant toriand thus quasi-periodic solutions for Schr\"odinger equations, the Klein-Gordonequations with exponential nonlinearity and other equations of any spatialdimension.

KAM for Hamiltonian partial differential equations with weaker Spectral Asymptotics