7.2 For this problem, you will use properties of the reflection \(A_{R}\) and transmission \(A_{T}\) amplitudes that we derived in this chapter:

\[
\begin{align*}
&A_{R}=\frac{m V_{0}}{k^{2}-m V_{0}+\mathrm{i} k \sqrt{k^{2}-2 m V_{0}} \cot \left(\frac{a}{\hbar}\sqrt{k^{2}-2 m V_{0}}\right)}  \tag{7.128} \\
&A_{T}=\frac{k e^{-\mathrm{i} \frac{ka}{\hbar} \sqrt{k^{2}-2 m V_{0}}}}{k \sqrt{k^{2}-2 m V_{0}} \cos\left(\frac{a}{\hbar}\sqrt{k^{2}-2 m V_{0}}\right)-\mathrm{i}\left(k^{2}-m V_{0}\right) \sin \left(\frac{a}{\hbar} \sqrt{k^{2}-2 m V_{0}}\right)}
\end{align*}
\]


We'll explore the properties of these amplitudes for complex-valued momentum \(k \in \mathbb{C}\).

(a) Show that the poles of the reflection and transmission amplitudes are located at the same value of \(k\).

(b) Under what conditions on \(a\) and \(V_{0}\) will there be no poles in the reflection and transmission amplitudes? What does this mean for the well, physically, from our discussion of bound states?

(c) In this chapter, we had studied the limit of these amplitudes when the potential becomes narrow, but let's now consider the limit of fixed width \(a\) but very

deep bottom of the potential: \(V_{0}<0\) and \(\left|V_{0}\right| \rightarrow \infty\). Where are the poles in these amplitudes now, in this limit? Have we seen this before?