+ \hat{K} (\hat{T}^{\dagger})^m\right),$$ where the $n$th iteration of the operator $\hat{T}^n$ is related to waves undergoing $n$ reflections at $\partial \Omega$. The operator $\hat{K} $ in (\[G0\]) contains the diagonal contribution of the double sum and is defined as $$\begin{aligned}
 \label{K}
\hat{K} = \sum_{n=0}^\infty \hat{K}_n &=& \sum_{n=0}^{\infty}  \hat{T}^{n} \hat{\Gamma}_0 (\hat{T}^{\dagger})^{n} .\end{aligned}$$ It contains the smooth part of the correlation function as will be shown below. The terms $\hat{K}_n$ can be interpreted as the $n$th iteration or reflection contribution to the smooth part of the correlation function. Note that some damping is implicitly assumed here in order for these sums to converge.

We finally remark that, once the correlation function has been characterised in a boundary representation, as set out in this section, one can use Green function identities to propagate this information to the interior. See [@GCT15] for a more detailed discussion of such propagation of correlation functions from a straight boundary, including the treatment of evanescent components, for example.

Relation to classical phase space densities
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The quantities $\hat\Gamma_0$, $\hat\Gamma$ and $\hat{K}$ can be related to phase space densities using Wigner transformation after additional averaging as demonstrated below and in \[app:A\]. The WDF of an operator $\hat\Gamma$ is defined 