= Odd bound state and resonance of two delta barriers
{title2=$2\kappa=-U_0(1-e^{-2\kappa a})$}
For equal delta potentials at $\pm a$, odd parity makes the interior wave proportional to $\sin(kx)$. The derivative jump gives the odd-channel denominator $k\cot(ka)+U_0-ik$. At $k=i\kappa$, its zero is the displayed equation. The function $2\kappa/(1-e^{-2\kappa a})$ increases strictly from $1/a$, proving exactly one odd bound state when $U_0a<-1$. For $U_0a=u\gg1$, the outgoing pole near $ka=\pi$ is $\pi-\pi/u+(\pi-i\pi^2)/u^2+O(u^{-3})$, with positive resonance width.
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