derive explicit expressions for the variance around the ensemble mean in terms of classical phase space quantities. The results are again validated numerically, here for a fully chaotic quantum map, the perturbed cat map [@SCC94].

Transfer operator formulation and correlation functions {#sec:trans}
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The transfer operator $\hat{T}$
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The wave problem under consideration is described in an operator formulation using transfer operators $\hat{T}$ defined on a $(d-1)$-dimensional manifold, or surface of section (SOS), where $d$ is the dimension of the underlying space. As a concrete example, we consider the Helmholtz equation in a $d$-dimensional region $\Omega \subset \mathbb{R}^d$ $$\label{helm}
-\nabla^2\psi - k^2\psi = \psi_0.$$ Boundary conditions given on $\partial \Omega$ are assumed to take the form of a linear relationship between the solution $\psi$ itself and its normal derivative $\partial \psi/ \partial n$. We will in general use $\partial \Omega$ as the SOS although other choices are possible. We include an inhomogeneous source term $\psi_0$ here, which drives the wave dynamics. An operator formulation on the boundary is naturally given in terms of boundary integral equations, which yield relations between the wave function $\psi$ and its normal derivative on the boundary. For our purposes, it is more convenient to follow 