The Hamiltonian in any theory of gravity is a boundary term, consequently in AdS the Hamiltonian is a boundary term. From the AdS/CFT duality, we have the same spectrum of states on both the AdS and the CFT side. If we write down the spectral decomposition of a general Hamiltonian as $$H := \sum_i E_i |E_i\rangle \langle E_i|,$$ firstly is it correct to conclude that both $H_{AdS}$ and $H_{CFT}$ admit the same spectral decomposition, i.e. the spectrum of energies and eigenstates is the same for both?

Given this, is it possible to address questions about bulk using the spectrum of the CFT Hamiltonian? More precisely, is it possible to work out the exact expression for $H_{AdS}$ in the bulk using $H_{CFT}$?



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