A particle of mass $m$ scatters on a localised potential well $V(x)$ in one dimension. With reference to the asymptotic behaviour of the wavefunction as $x \rightarrow \pm \infty$, define the reflection and transmission amplitudes, $r$ and $t$, for a right-moving incident particle of wave number $k$. Define also the corresponding amplitudes, $r^{\prime}$ and $t^{\prime}$, for a left-moving incident particle of wave number $k$. Derive expressions for $r^{\prime}$ and $t^{\prime}$ in terms of $r$ and $t$.

(a) Define the $S$-matrix, giving its elements in terms of $r$ and $t$. Using the relation

$$|r|^{2}+|t|^{2}=1$$

(which you need not derive), show that the S-matrix is unitary. How does the S-matrix simplify if the potential well satisfies $V(-x)=V(x)$ ?

(b) Consider the potential well

$$V(x)=-\frac{3 \hbar^{2}}{m} \frac{1}{\cosh ^{2}(x)}$$

The corresponding Schrödinger equation has an exact solution

$$\psi_{k}(x)=\exp (i k x)\left[3 \tanh ^{2}(x)-3 i k \tanh (x)-\left(1+k^{2}\right)\right]$$

with energy $E=\hbar^{2} k^{2} / 2 m$, for every real value of $k$. [You do not need to verify this.] Find the S-matrix for scattering on this potential. What special feature does the scattering have in this case?

(c) Explain the connection between singularities of the S-matrix and bound states of the potential well. By analytic continuation of the solution $\psi_{k}(x)$ to appropriate complex values of $k$, find the wavefunctions and energies of the bound states of the well. [You do not need to normalise the wavefunctions.]