Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 337 / 1 / ii / Solution

Let $H|a⟩=E_a|a⟩$, $p_a=e^{-E_a/T}$, and take $A$ Hermitian. Inserting energy eigenstates givesOn the support of the delta function, $E_b-E_a=\omega$ andThis has the same sign as $\omega$. Every remaining factor is nonnegative, henceThis thermal spectral-positivity statement also shows that $\mathrm{Im}{G}_R$ is odd after pairing $a,b$.