[@CHT13] and decompose the boundary field into incoming and outgoing components $$\label{psim}
\ket{\psi} = \ket{\psi_-} + \ket{\psi_+}$$ as depicted in Fig. \[sec:inout\]. Note that we use the ket notation for the solutions restricted to the SOS given here by $\partial \Omega$.

![\[sec:inout\] Boundary field decomposed into incoming and outgoing component. ](fig/inoutfig.pdf)

The solution to the inhomogeneous wave problem can now be cast in terms of incoming waves at boundaries, $$\label{prop}
(1 - \hat{T}) \ket{\psi_-} = \ket{\psi_-^0},$$ where $\ket{\psi_-^0}$ is the source wave field incoming on the boundary. Transfer operators can be defined for generic wave systems and for generalised SOSs other than $\partial \Omega$, see [@Bog92; @TS07a]. An explicit construction of the exact $\hat{T}$ is, however, often not straightforward and implementations have been presented for model systems only [@Pro94; @RS95; @Pro96; @CHT13]. General expressions for $\hat{T}$ can be given explicitly in the semiclassical limit $k$ (or $\omega$) $\to \infty$ in terms of the so-called Bogomolny transfer operator [@Bog92; @Dor92], $$\label{BoT}
T({x},{x}';\omega) \approx \left(\frac{k}{2\pi \rmi}\right)^{(d-1)/2}
\sum_{\stackrel{{\rm rays}}{{x}'\to {x}}} 
D(x,x') \exp\left(\rmi k S({x},{x}';\omega) - \rmi \mu
\frac{\pi}{2} \right)$$ with $$\label{BoT1}
D(x,x') = \sqrt{\left|\det \left(\frac{\partial^2 S}{\partial x \partial x'}\right)
\right|} .$$ The sum in (\[BoT\]) is over all rays passing through the interior 