be changed to $\delta\to \delta(k) \equiv \tau_B-\tau_H(k)$.

### Between region III and region IV

The matching conditions are $$\begin{aligned}
v_3(\Delta) = v_4 (\Delta), \quad v'_3(\Delta) = v'_4 (\Delta) \, .\end{aligned}$$ In matrix form this yields $$\begin{aligned}
\begin{pmatrix} 
e^{ik\Delta} & e^{-ik\Delta} \\ i k e^{ik\Delta} & -i k e^{-ik\Delta}  
\end{pmatrix} \begin{pmatrix}
c_{31} \\ c_{32}
\end{pmatrix} = \begin{pmatrix}
\delta^{1-q_1} & \delta^{q_1} \\ -(1-q_1) \delta^{-q_1} &
-q_1\delta^{q_1-1}
\end{pmatrix} \begin{pmatrix}
c_{41} \\ c_{42}
\end{pmatrix}.\end{aligned}$$

### Between region IV and region V

The matching conditions in this case are $$\begin{aligned}
v_4(\tau_B-\epsilon) = v_5 (\tau_B+\epsilon),\quad
v'_4(\tau_B-\epsilon) = v'_5 (\tau_B+\epsilon) \, .\end{aligned}$$ In matrix form this yields $$\begin{aligned}
\begin{pmatrix}
\epsilon^{1-q_1} & \epsilon^{q_1} \\ -(1-q_1) \epsilon^{-q_1} & -q_1
\epsilon^{q_1-1}
\end{pmatrix} \begin{pmatrix}
c_{41} \\ c_{42}
\end{pmatrix} = \begin{pmatrix}
\epsilon^{1-q_2} & \epsilon^{q_2} \\ (1-q_2) \epsilon^{-q_2} & q_2
\epsilon^{q_2-1}
\end{pmatrix} \begin{pmatrix}
c_{51} \\ c_{52}
\end{pmatrix}.\end{aligned}$$

Analysis and Results for the Power Spectrum {#sec4}
===========================================

Combining the results of the previous section we find that the final mode coefficients can be written in terms of the initial ones via $$\begin{aligned}
\nonumber
& \mathcal C_5 = 
\begin{pmatrix}
c_{51} \\ c_{52}
\end{pmatrix},\end{aligned}$$ where[^5]

$$\begin{aligned}
c_{51}&=&\frac{c_{11}}{(1-2q_2)(1-2q_1)2k(1-2q_1)} \left[ a_{11} (1 -
  2 (q_1-1) q_1 - 2 (q_2 -1 ) q_2) \right.  \nonumber \\ &-&
  \left. a_{12} 