In the non-interacting case, ψ ψ k † † ∣ ∣ α α ′ ⟩ ⟩ is an eigenstate with (grand-canonical) energy E α α ′ + ξ ξ k , where ξ ξ k = ϵ ϵ k − − μ μ is the single-particle dispersion relation measured with respect to the chemical potential. The spectral density therefore becomes ρ ρ 0 ( k , ω ω ) = 1 Z 2 π π δ δ ( ξ ξ k − − ω ω ) ∑ ∑ α α ′ ⟨ ⟨ α α ′ ∣ ∣ ψ ψ k ψ ψ k † † ∣ ∣ α α ′ ⟩ ⟩ ( 1 − − ζ ζ e − − β β ξ ξ k ) e − − β β E α α ′ .