An efficient method with no numerical diagonalization of a huge Hamiltonianmatrix and calculation of a tedious Green's function is proposed to acquire theexact energy spectrum and dynamical conductivity in a gated AA-stacking$N$-layer Graphene (AANLG) with the intrinsic spin-orbital coupling (SOC). $2N\times 2N$ tight-binding Hamiltonian matrix, velocity operator and Green'sfunction representation of an AANLG are simultaneously reduced to $N$ $2\times2$ diagonal block matrices through a proper transformation matrix. A gatedAANLG with intrinsic SOC is reduced to $N$ graphene-like layers. The energyspectrum of a graphene-like layer is $E= \varepsilon _{\bot}\pm\varepsilon_{||}$. $ \varepsilon _{\bot}$ depends on the interlayerinteraction, gated voltage and layer number. $ \varepsilon_{||}=\sqrt{E_{MG}^2+\Delta^2}$, where $E_{MG}$ is the energy spectrum of a monolayer graphene and $\Delta$ is the magnitude of intrinsic SOC. More importantly, by inserting thediagonal block velocity operator and Green's function representation in theKubo formula, the exact dynamical conductivity of an AANLG is shown to be$\sigma = \Sigma_{j=1} ^N \sigma_j$, the sum of the dynamical conductivity of$N$ graphene-like layers. The analytical form of $\sigma_j$ is presented andthe dependence of $\sigma_j$ on $\varepsilon_{\bot}$, $\Delta$, and chemicalpotential is clearly demonstrated. Moreover, the effect of Rashba SOC on theelectronic properties of an AANLG is explored with the exact energy spectrumpresented.
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