Odd-parity S-matrix of a symmetric double-delta potential ($S_{--}$)

For positive wavenumber $k$, the odd channel of the symmetric double-delta potential hasFor real $k$ its numerator and denominator are complex conjugates, making the channel unitary. Its poles are the zeros of $k{e}^{-ika}+U_0\sin (ka)$.

For $U_0<0$, an odd bound state has $k=i\kappa$ with $\kappa >0$ and obeysA nonzero positive solution exists exactly when $-U_{0}a>1$; equality is the zero-energy threshold.

For $U_{0}a\gg 1$, the lowest odd scattering resonance pole lies near the first odd mode trapped between the barriers:in its real part and to leading nonzero order in its imaginary part. Writing $E={\hslash }^2{k}^2/(2m)=E_R-i\Gamma /2$, the survival probability decays as $e^{-\Gamma t/\hslash }$.