Spectral Representation and Moments of a Green’s function Deriving the spectral representation

⟨{c_{k}(t), c_{k}^{†}(0)}⟩ = ∑_{nm} e^{−βϵ_{n}} {⟨n |c_{k}(t)| m⟩ ⟨m |c_{k}^{†}(0)| n⟩ + ⟨n |c_{k}^{†}(0)| m⟩ ⟨m |c_{k}(t)| n⟩}

(1)

∫_{−∞}^{∞} dte^{iωt} ⟨{c_{k}(t), c_{k}^{†}(0)}⟩ = ∑_{nm} e^{−βϵ_{n}}2πδ(ω − (ϵ_{m} − ϵ_{n})) |⟨m |c_{k}^{†}| n⟩|^{2}

+ ∑_{nm} e^{−βϵ_{n}}2πδ(ω − (ϵ_{n} − ϵ_{m})) |⟨m |c_{k}| n⟩|^{2}

= 2πA(k, ω)

Deriving the moments of the spectra using the equation of motion

i\frac{∂}{∂t}c_{k} = [c_{k}, H]  (2)
2πA(k, ω)ω^{p} = ∫_{−∞}^{∞} \frac{1}{i^{p}} [\frac{∂^{p}}{∂t^{p}} e^{iωt}] ⟨\{c_{k}(t), c_{k}^{†}(0)\}⟩ dt
= ∫_{−∞}^{∞} e^{iωt} ⟨{[[c_{k}, H] , · · ··^{ptimes}, H] , c_{k}^{†}(0)}⟩ dt

Therefore

∫_{−∞}^{∞} dωA(k, ω)ω^{p} = ⟨{[[c_{k}, H] , · · ··^{ptimes}, H] , c_{k}^{†}(0)}⟩  (3)

and in particular

∫_{−∞}^{∞} dωA(k, ω)ω^{p+q} = ⟨{[[c_{k}, H] , · · ··^{ptimes}, H] , [H, · · ··^{qtimes} [H, c_{k}^{†}]]}⟩ (4)