A particle in one dimension of mass $m$ and energy $E=\hbar^{2} k^{2} / 2 m(k>0)$ is incident from $x=-\infty$ on a potential $V(x)$ with $V(x) \rightarrow 0$ as $x \rightarrow-\infty$ and $V(x)=\infty$ for $x>0$. The relevant solution of the time-independent Schrödinger equation has the asymptotic form

$$\psi(x) \sim \exp (i k x)+r(k) \exp (-i k x), \quad x \rightarrow-\infty$$

Explain briefly why a pole in the reflection amplitude $r(k)$ at $k=i \kappa$ with $\kappa>0$ corresponds to the existence of a stable bound state in this potential. Indicate why a pole in $r(k)$ just below the real $k$-axis, at $k=k_{0}-i \rho$ with $k_{0} \gg \rho>0$, corresponds to a quasi-stable bound state. Find an approximate expression for the lifetime $\tau$ of such a quasi-stable state.

Now suppose that

$$V(x)= \begin{cases}\left(\hbar^{2} U / 2 m\right) \delta(x+a) & \text { for } x<0 \\ \infty & \text { for } x>0\end{cases}$$

where $U>0$ and $a>0$ are constants. Compute the reflection amplitude $r(k)$ in this case and deduce that there are quasi-stable bound states if $U$ is large. Give expressions for the wavefunctions and energies of these states and compute their lifetimes, working to leading non-vanishing order in $1 / U$ for each expression.

[ You may assume $\psi=0$ for $x \geqslant 0$ and $\lim _{\epsilon \rightarrow 0+}\left\{\psi^{\prime}(-a+\epsilon)-\psi^{\prime}(-a-\epsilon)\right\}=U \psi(-a)$.]