solutions to , with the solutions to the WDW analog of . To start with, we note, that the Eq.  is a $2$nd order difference equation, however, due to the specific properties of $\Theta$ characteristic for each of the prescriptions considered here, the whole solution is determined just by the single value $e_k(v=4)$ (see [@aps-imp; @acs; @mmo] for the details on respective prescriptions).

To analyze the solution it is more convenient to rewrite that equation in the $1$st order form [@simon]. For that we introduce the vector notation, defining $$\label{eq:eig-vec}
  \vec{e}_k(v):= \left( \begin{array}{l} e_k(v) \\ e_k(v-4) \end{array} \right) .$$ Using it we can rewrite in the following form $$\label{eq:eig-1st}
  \vec{e}_k(v+4) = {\boldsymbol{A}}(v)\,\vec{e}_k(v) ,$$ where the matrix ${\boldsymbol{A}}$ can be expressed (with use of the notation introduced in Eq. ) as $$\label{eq:A-def}
  {\boldsymbol{A}}(v) = \left( \begin{array}{cc} 
                     \frac{f_o(v)-\omega^2(k)}{f_+(v)} & -\frac{f_-(v)}{f_+(v)} \\
                      1 & 0
                   \end{array} \right) .$$ To relate $e_{k}$ with $\ub{e}_{\pm k}$ we note that the value of $e_{k}$ at each pair of consecutive points $v,v+4$ can be encoded as values of the (specific for the chosen pair of points) combination of $\ub{e}_{\pm k}$, that is $$\label{eq:eig-coeff}
  \vec{e}_k(v) = {\boldsymbol{B}}(v) \vec{\chi}_k(v) ,$$ where the transformation matrix ${\boldsymbol{B}}$ is defined as follows 