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The Lehmann representation of the Green function to a system with $N$ identical particles can be write as

$$G(\textbf{x}, \textbf{x}', E) = \sum_{n} \frac{\langle \Psi_{0}^{N} | \psi(\textbf{x})| \Psi_{n}^{N+1} \rangle \langle \Psi_{n}^{N+1} | \psi^{\dagger}(\textbf{x}')| \Psi_{0}^{N} \rangle}{E + E_{0}^{N} - E_{n}^{N+1} + i\eta} + ...$$

There is another term but let's ignore it. My question is simple: the sum is over all the eigenstates $\Psi_{n}^{N+1}$ of the hamiltonian of a system with $N+1$ particles. Can't the spectrum of this hamiltonian have a continuous part? In this case, shouldn't we integrate instead of adding?

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In QM, sums over states are understood to imply integration when the states are continuous.

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