The first <Born approximation for a one-dimensional reflection coefficient> follows by replacing the Jost solution inside its Volterra scattering equation by the free wave $e^{ikx}$:
$$
\boxed{
R(k)
=\frac{\epsilon}{2ik}
\int_{-\infty}^{\infty}e^{-2ikz}q(z)\,dz
+O(\epsilon^2)
}.
$$

To check the inverse problem, any shallow discrete state contributes only at $O(\epsilon^2)$ pointwise, so to first order the Marchenko input is
$$
F(s)
=\frac1{2\pi}\int_{-\infty}^{\infty}R(k)e^{iks}\,dk.
$$
Differentiating and inserting the Born approximation gives
$$
\begin{aligned}
F'(s)
&=\frac{\epsilon}{4\pi}
\int_{\mathbb R}\int_{\mathbb R}
e^{ik(s-2z)}q(z)\,dz\,dk+O(\epsilon^2)\\
&=\frac{\epsilon}{4}q(s/2)+O(\epsilon^2),
\end{aligned}
$$
where the second line is the <Fourier inversion theorem>. At first order the quadratic integral term in the Marchenko equation may be dropped, so
$$
K(x,y)=-F(x+y)+O(\epsilon^2).
$$
The reconstruction formula therefore returns
$$
u(x)
=-2\frac d{dx}K(x,x)
=4F'(2x)+O(\epsilon^2)
=\boxed{\epsilon q(x)+O(\epsilon^2)}.
$$
\b[Thus direct and inverse scattering agree to first order.]

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 33C / iii / Solution

{c}

For a weak potential $u=\epsilon q$, the first <Born approximation> in the KdV Schrödinger convention is
$$
R(k)=\frac{\epsilon}{2ik}\int_{\mathbb R}e^{-2ikx}q(x)\,dx+O(\epsilon^2).
$$

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