Odd bound state and resonance of two delta barriers ($2\kappa =-U_0(1-e^{-2\kappa a})$)

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 $U_{0}a<-1$. For $U_{0}a=u\gg 1$, the outgoing pole near $ka=\pi$ is $\pi -\pi /u+(\pi -i{\pi }^2)/u^2+O(u^{-3})$, with positive resonance width.