\dg F + [(\de_{y}-iktU')^{2}, \dg] \Psi, \label{eq:eqforDPsi}\end{aligned}$$ Using the commutator relation $[A^2,B]=2A[A,B]+[[A,B],A]$, and that $$\begin{aligned}
\label{eq:comm0}
[\de_y-iktU',\dg]=\dc'\de_y+iktU''\dc=\dc^{(1)}\de_y- \frac{U''}{U'}\dc\left(\de_y-iktU'\right)\end{aligned}$$ we compute $$\begin{aligned}
\label{eq:comm1}
[(\de_{y}-iktU')^{2}, \dg] \Psi &= 2 (\de_{y}-iktU')[\de_{y}-iktU', \dg] \Psi + [\de_{y}-iktU',[\de_{y}-iktU', \dg]] \Psi \notag\\
    &=- 2 (\de_{y}-iktU') \frac{U''}{U'}\dc\left(\de_y-iktU'\right)\Psi+ 2 (\de_{y}-iktU') \dc^{(1)} \de_{y} \Psi \notag\\
    &\quad+ \left[(\de_{y}-iktU'), \frac{U''}{U'}\dc\left(\de_y-iktU'\right)\right]\Psi + [(\de_{y}-iktU'),\dc^{(1)} \de_{y}]\Psi. \end{aligned}$$ The first term is already in the desired form. Also, we simply rewrite the second term as $$\begin{aligned}
\label{eq:comm2}
2 (\de_{y}-iktU') \dc^{(1)} \de_{y} \Psi =2 (\de_{y}-iktU') \frac{\dc^{(1)}}{\dc} \dg \Psi\end{aligned}$$ For the third term, we note that the operator $\de_{y}-iktU'$ commutes with itself and we hence obtain $$\begin{aligned}
\label{eq:comm3}
\left[(\de_{y}-iktU'), \frac{U''}{U'}\dc\left(\de_y-iktU'\right)\right]\Psi 
&=\left[(\de_{y}-iktU'), \frac{U''}{U'}\dc\right]\left(\de_y-iktU'\right)\Psi\notag\\
&= \left(\frac{U''}{U'}\dc\right)'\left(\de_y-iktU'\right)\Psi,\end{aligned}$$ while for the fourth term we directly compute that $$\begin{aligned}
\label{eq:comm4}
[(\de_{y}-iktU'),\dc^{(1)} \de_{y} \Psi]
&=\dc^{(2)}\de_y\Psi- \frac{U''}{U'}\dc^{(1)}\left(\de_y-iktU'\right)\Psi\notag\\
&=\frac{\dc^{(2)}}{d}\dg\Psi- \frac{U''}{U'}\dc^{(1)}\left(\de_y-iktU'\right)\Psi.\end{aligned}$$ Collecting all of the above, we obtain $$\begin{aligned}
&\left(-k^{2}+(\de_{y}-iktU')^{2}-2 (\de_{y}-iktU') \frac{\dc^{(1)}}{\dc}-\frac{\dc^{(2)}}{d}\right)\dg \Psi \notag\\
&\qquad\qquad= \dg F +\left[ \left(\frac{U''}{U'}\dc\right)'- \frac{U''}{U'}\dc^{(1)}- 2 (\de_{y}-iktU') \frac{U''}{U'}\dc\right]\left(\de_y-iktU'\right)\Psi\end{aligned}$$ We then split $\dg \Psi$ into a non-homogenous solution $\tPhi$ with homogeneous Dirichlet boundary conditions and the linear combination of homogeneous solutions to , namely $$\begin{aligned}
\label{eq:homoline}
\dg \Psi=\tPhi+ (\dg\Psi|_{y=\a}) \e^{{ikt(U(y)-U(\a))}}\tilde{h}_{\a} +(\dg\Psi|_{y=b})    \e^{{ikt(U(y)-U(\b))}}\tilde{h}_{\b},\end{aligned}$$ where $\tilde{h}_{\a}$ and $\tilde{h}_{\b}$ are defined in . Furthermore, using integration by parts and recalling from that $\Psi$ vanishes at the 