The covariance of a centered <multivariate Gaussian distribution> is the inverse of its quadratic kernel. Writing $A(q)=a+\kappa q^2+\gamma q^4$, one finds
$$
G(q)^{-1}
=\frac1{\nu A(q)-\lambda^2q^2}
\begin{pmatrix}
\nu&-i\lambda q\\
i\lambda q&A(q)
\end{pmatrix}.
$$
Consequently the composition <static structure factor> is
$$
\boxed{\langle\phi(q)\phi(-q)\rangle
=\frac1{a+\widetilde\kappa q^2+\gamma q^4}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 1 / f / Solution

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A multivariate Gaussian distribution has density proportional to $\exp[-(x-\mu)^T\Sigma^{-1}(x-\mu)/2]$. Its mean is $\mu$ and its covariance matrix is $\Sigma$.


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