where the sum runs over all states | i ⟩ ⟩ other than the vacuum | Ω Ω ⟩ ⟩ itself. Whether this integral converges or not depends on the large- E behavior of the spectral density ρ ρ Ω Ω ( E ) . In turn, this depends on the short-distance behavior of the two-point correlation function of the operator V . Indeed, we can write