Consider the quantum mechanical scattering of a particle of mass $m$ in one dimension off a parity-symmetric potential, $V(x)=V(-x)$. State the constraints imposed by parity, unitarity and their combination on the components of the $S$-matrix in the parity basis,

$$S=\left(\begin{array}{cc} S_{++} & S_{+-} \\ S_{-+} & S_{--} \end{array}\right)$$

For the specific potential

$$V=\hbar^{2} \frac{U_{0}}{2 m}\left[\delta_{D}(x+a)+\delta_{D}(x-a)\right]$$

show that

$$S_{--}=e^{-i 2 k a}\left[\frac{\left(2 k-i U_{0}\right) e^{i k a}+i U_{0} e^{-i k a}}{\left(2 k+i U_{0}\right) e^{-i k a}-i U_{0} e^{i k a}}\right]$$

For $U_{0}<0$, derive the condition for the existence of an odd-parity bound state. For $U_{0}>0$ and to leading order in $U_{0} a \gg 1$, show that an odd-parity resonance exists and discuss how it evolves in time.