 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-337/1/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

H ∣ a ⟩ = E_{a} ∣ a ⟩

p_{a} = e^{- E_{a} / T}

, and take

A

Hermitian. Inserting energy eigenstates gives

Im G_{R} (ω) = - π \sum_{a, b} (p_{a} - p_{b}) ∣ A_{ab} ∣^{2} δ (ω - E_{b} + E_{a}) .

On the support of the delta function,

E_{b} - E_{a} = ω

and

p_{a} - p_{b} = p_{a} (1 - e^{- ω / T}) .

This has the same sign as

ω

. Every remaining factor is nonnegative, hence

ω Im G_{R} (ω) \leq 0 .

This thermal spectral-positivity statement also shows that

Im G_{R}

is odd after pairing

a, b

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