= Half-line delta-well amplitude enhancement
{title2=$|A|^2=\lambda^2x^2/[(1-\cos2x)^2+(\lambda x-\sin2x)^2]$}
A hard wall at the origin and an attractive <delta potential> at distance $a$ form a leaky quantum cavity. For unit incident amplitude from outside, the interior <standing wave> has coefficient $A=-i\lambda x/[1-e^{2ix}+i\lambda x]$, where $x=ka$ and $\lambda=\hbar^2/(mqa)>0$. Its <modulus> squared is the title expression. A <resonance pole> $p_n=x_n-i\gamma_n$ produces an enhancement with a locally <Lorentzian line shape>, proportional to $[(x-x_n)^2+\gamma_n^2]^{-1}$. At fixed low $n$ and small $\lambda$, $x_n\simeq n\pi(1+\lambda/2+\lambda^2/4)$ and $\gamma_n\simeq(\lambda n\pi/2)^2$.
The coefficient can exceed one because it measures a standing-wave amplitude, while the net interior <probability current> is zero. Conservation of <probability current> gives unit reflected flux: no current passes through the hard wall. Near zero, $|A|^2\to\lambda^2/(2-\lambda)^2$ when $\lambda\ne2$; the threshold case $\lambda=2$ requires a separate limit.
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