KAM for Hamiltonian partial differential equations with weaker Spectral Asymptotics

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In this paper, we establish an abstract infinite dimensional KAM theorem dealing with normal frequencies in weaker spectral asymptotics Ωᵢ(ξ)=iᵈ+o(iᵈ)+o(iδ), where d>0, δ<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 Schrödinger equations, the Klein-Gordon equations with exponential nonlinearity and other equations of any spatial dimension.