(a) In one dimension, a particle of mass $m$ is scattered by a potential $V(x)$ where $V(x) \rightarrow 0$ as $|x| \rightarrow \infty$. For wavenumber $k>0$, the incoming $(\mathcal{I})$ and outgoing $(\mathcal{O})$ asymptotic plane wave states with positive $(+)$ and negative $(-)$ parity are given by

$$\begin{array}{rr} \mathcal{I}_{+}(x)=e^{-i k|x|}, & \mathcal{I}_{-}(x)=\operatorname{sign}(x) e^{-i k|x|} \\ \mathcal{O}_{+}(x)=e^{+i k|x|}, & \mathcal{O}_{-}(x)=-\operatorname{sign}(x) e^{+i k|x|} \end{array}$$

(i) Explain how this basis may be used to define the $S$-matrix,

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

(ii) For what choice of potential would you expect $S_{+-}=S_{-+}=0$ ? Why?

(b) The potential $V(x)$ is given by

$$V(x)=V_{0}[\delta(x-a)+\delta(x+a)]$$

with $V_{0}$ a constant.

(i) Show that

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

where $U_{0}=2 m V_{0} / \hbar^{2}$. Verify that $\left|S_{--}\right|^{2}=1$. Explain the physical meaning of this result.

(ii) For $V_{0}<0$, by considering the poles or zeros of $S_{--}(k)$, show that there exists one bound state of negative parity if $a U_{0}<-1$.

(iii) For $V_{0}>0$ and $a U_{0} \gg 1$, show that $S_{--}(k)$ has a pole at

$$k a=\pi+\alpha-i \gamma$$

where $\alpha$ and $\gamma$ are real and

$$\alpha=-\frac{\pi}{a U_{0}}+O\left(\frac{1}{\left(a U_{0}\right)^{2}}\right) \quad \text { and } \quad \gamma=\left(\frac{\pi}{a U_{0}}\right)^{2}+O\left(\frac{1}{\left(a U_{0}\right)^{3}}\right)$$

Explain the physical significance of this result.