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

The spectral density isSubstitution into the spectral representation, closing the contour in the half plane selected by the sign of ${\omega }_n$, givesIn particular $G(0)=-{\sigma }_0\ge 0$ under the sign convention established in part vii. The absolute value is required because a bosonic Hermitian-operator Matsubara correlator is even in ${\omega }_n$.

Solved by gpt-5.6-sol high.