Suppose $p=p(x)$ is a smooth, real-valued, function of $x \in \mathbb{R}$ which satisfies $p(x)>0$ for all $x$ and $p(x) \rightarrow 1, p_{x}(x), p_{x x}(x) \rightarrow 0$ as $|x| \rightarrow \infty$. Consider the Sturm-Liouville operator:

$$L \psi:=-\frac{d}{d x}\left(p^{2} \frac{d \psi}{d x}\right)$$

which acts on smooth, complex-valued, functions $\psi=\psi(x)$. You may assume that for any $k>0$ there exists a unique function $\varphi_{k}(x)$ which satisfies:

$$L \varphi_{k}=k^{2} \varphi_{k}$$

and has the asymptotic behaviour:

$$\varphi_{k}(x) \sim \begin{cases}e^{-i k x} & \text { as } x \rightarrow-\infty \\ a(k) e^{-i k x}+b(k) e^{i k x} & \text { as } x \rightarrow+\infty\end{cases}$$

(a) By analogy with the standard Schrödinger scattering problem, define the reflection and transmission coefficients: $R(k), T(k)$. Show that $|R(k)|^{2}+|T(k)|^{2}=1$. [Hint: You may wish to consider $W(x)=p(x)^{2}\left[\psi_{1}(x) \psi_{2}^{\prime}(x)-\psi_{2}(x) \psi_{1}^{\prime}(x)\right]$ for suitable functions $\psi_{1}$ and $\left.\psi_{2} \cdot\right]$

(b) Show that, if $\kappa>0$, there exists no non-trivial normalizable solution $\psi$ to the equation

$$L \psi=-\kappa^{2} \psi$$

Assume now that $p=p(x, t)$, such that $p(x, t)>0$ and $p(x, t) \rightarrow 1, p_{x}(x, t), p_{x x}(x, t) \rightarrow$ 0 as $|x| \rightarrow \infty$. You are given that the operator $A$ defined by:

$$A \psi:=-4 p^{3} \frac{d^{3} \psi}{d x^{3}}-18 p^{2} p_{x} \frac{d^{2} \psi}{d x^{2}}-\left(12 p p_{x}^{2}+6 p^{2} p_{x x}\right) \frac{d \psi}{d x}$$

satisfies:

$$(L A-A L) \psi=-\frac{d}{d x}\left(2 p^{4} p_{x x x} \frac{d \psi}{d x}\right)$$

(c) Show that $L, A$ form a Lax pair if the Harry Dym equation,

$$p_{t}=p^{3} p_{x x x}$$

is satisfied. [You may assume $L=L^{\dagger}, A=-A^{\dagger}$.]

(d) Assuming that $p$ solves the Harry Dym equation, find how the transmission and reflection amplitudes evolve as functions of $t$.