= Endpoint derivative map for an evanescent wave layer
{title2=$\psi''(z)=\alpha^2\psi(z)$}
For $\psi''=\alpha^2\psi$ on $-1<z<1$ with endpoint values $P=\psi(1)$ and $Q=\psi(-1)$, the interior derivatives are $\psi'(1)=\alpha[\coth(2\alpha)P-\operatorname{csch}(2\alpha)Q]$ and $\psi'(-1)=\alpha[\operatorname{csch}(2\alpha)P-\coth(2\alpha)Q]$. 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 $\alpha$.
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