As β → ∞, the spectral density becomes ρ ρ ( k , ω ω ) = 2 π π ∑ ∑ α α [ δ δ ( E α α − − E 0 − − ω ω ) | ⟨ α α ∣ ∣ ψ ψ k † † ∣ ∣ 0 ⟩ | 2 − − ζ ζ δ δ ( E 0 − − E α α − − ω ω ) | ⟨ 0 ∣ ∣ ψ ψ k † † ∣ ∣ α α ⟩ | 2 ] where α = 0 corresponds to the ground state. Note that only the first (second) term contributes when ω is positive (negative).