= Solution
In the two constant regions define
$$
\omega_\pm=\sqrt{k_z^2+a_\pm^2|\mathbf k|^2}.
$$
The normalized <positive-frequency solution>[positive-frequency modes] are
$$
\boxed{u_\pm(\eta)=\frac{e^{-i\omega_\pm\eta}}{\sqrt{2\omega_\pm}}},
$$
up to the common spatial Fourier normalization. They satisfy the unit Wronskian condition
$$
i(u_\pm^*u_\pm'-u_\pm u_\pm^{*\prime})=1,
$$
which is the mode form of the <Klein-Gordon inner product>. Because the mode equation contains no delta function at $\eta=0$, both $\Phi$ and $\Phi'$ are continuous there.
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