Endpoint derivative map for an evanescent wave layer (${\psi }^{\prime \prime }(z)={\alpha }^2\psi (z)$)

For ${\psi }^{\prime \prime }={\alpha }^2\psi$ on $-1<z<1$ with endpoint values $P=\psi (1)$ and $Q=\psi (-1)$, the interior derivatives are ${\psi }^{\prime }(1)=\alpha [\coth (2\alpha )P-\mathrm{csch}(2\alpha )Q]$ and ${\psi }^{\prime }(-1)=\alpha [\mathrm{csch}(2\alpha )P-\coth (2\alpha )Q]$. Fitting the hyperbolic-function solution proves the map. It turns matching conditions into a finite-dimensional interface system, with exponentially small cross-interface coupling at large $\alpha$.