Lehmann representation of Green function

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

$$G(textbf{x}, textbf{x}’, E) = sum_{n} frac{langle Psi_{0}^{N} | psi(textbf{x})| Psi_{n}^{N+1} rangle langle Psi_{n}^{N+1} | psi^{dagger}(textbf{x}’)| Psi_{0}^{N} rangle}{E + E_{0}^{N} – E_{n}^{N+1} + ieta} + …$$

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?