\$\frac{d^2}{dr^2} k(r) + \frac{2}{r} \frac{d}{dr} k(r) = \frac{2}{r^2}K^2(r)k(r) -4 v^2λ(1-k^2(r))k(r)\$\$\frac{d^2}{dr^2} h(r) + \frac{2}{r} \frac{d}{dr} h(r) = \frac{2}{r^2}K^2(r)h(r) -4 v^2λ(1-h^2(r))k(r)\$
\$k(0) = 0; K(0) = 1; k(∞) = 1; K(∞) = 0\$\$h(0) = 0; K(0) = 1; h(∞) = 1; K(∞) = 0\$
\$\frac{d}{dr}F_L(r) + \frac{1+K(r)}{r} F_L(r) +f_g\frac{v}{\sqrt{2}} k(r)F_R(r) = ε G_L(r)\$\$\frac{d}{dr}F_L(r) + \frac{1+K(r)}{r} F_L(r) +f_g\frac{v}{\sqrt{2}} h(r)F_R(r) = ε G_L(r)\$
\$\frac{d}{dr}G_L(r) + \frac{1-k(r)}{r} G_L(r) +f_g\frac{v}{\sqrt{2}} k(r)G_R(r) = -ε F_L(r)\$\$\frac{d}{dr}G_L(r) + \frac{1-h(r)}{r} G_L(r) +f_g\frac{v}{\sqrt{2}} h(r)G_R(r) = -ε F_L(r)\$
\$\frac{d}{dr}F_(r) + \frac{2}{r} F_R(r) +f_g\frac{v}{\sqrt{2}} k(r)F_L(r) = -ε G_R(r)\$\$\frac{d}{dr}F_(r) + \frac{2}{r} F_R(r) +f_g\frac{v}{\sqrt{2}} h(r)F_L(r) = -ε G_R(r)\$
\$\frac{d}{dr}G_R(r) + f_g\frac{v}{\sqrt{2}} k(r)G_L(r) = ε F_R(r)\$\$\frac{d}{dr}G_R(r) + f_g\frac{v}{\sqrt{2}} h(r)G_L(r) = ε F_R(r)\$
In the first set of equations, \$r\$ is a variable, \$K(r)\$ and \$k(r)\$\$h(r)\$, are functions of \$r\$ and \$v\$, \$g\$ and \$λ\$ are parameters. After solving the first set, I want to put \$K(r)\$ and \$k(r)\$\$h(r)\$ in the second set which is an eigensystem with the eigenvalue \$ε\$, where \$f_g\$ is a parameter.
$$\begin{bmatrix}
\begin{array}{cc|cc|cc|cc}
0 && \frac{d}{dr} + \frac{1 + k}{r} && 0 && f_g\frac{v}{\sqrt{2}}k \\ \hline
-\frac{d}{dr} + \frac{k-1}{r} && 0 && -f_g\frac{v}{\sqrt{2}}k && 0 \\ \hline
0 && -f_g\frac{v}{\sqrt{2}}k && 0 && -\frac{d}{dr} - \frac{2}{r}
\\ \hline
f_g \frac{v}{\sqrt{2}}k && 0 && \frac{d}{dr} && 0
\end{array}
\end{bmatrix}
\begin{bmatrix}G_L \\\ F_L \\\ G_R \\\ F_R\end{bmatrix}
= ε \begin{bmatrix}G_L \\\ F_L \\\ G_R \\\ F_R\end{bmatrix}
$$$$\begin{bmatrix}
\begin{array}{cc|cc|cc|cc}
0 && \frac{d}{dr} + \frac{1 + h}{r} && 0 && f_g\frac{v}{\sqrt{2}}h \\ \hline
-\frac{d}{dr} + \frac{h-1}{r} && 0 && -f_g\frac{v}{\sqrt{2}}h && 0 \\ \hline
0 && -f_g\frac{v}{\sqrt{2}}h && 0 && -\frac{d}{dr} - \frac{2}{r}
\\ \hline
f_g \frac{v}{\sqrt{2}}h && 0 && \frac{d}{dr} && 0
\end{array}
\end{bmatrix}
\begin{bmatrix}G_L \\\ F_L \\\ G_R \\\ F_R\end{bmatrix}
= ε \begin{bmatrix}G_L \\\ F_L \\\ G_R \\\ F_R\end{bmatrix}
$$
