Example: an edge-case
Let us consider a more advanced example where the inverse Born approximation does not work.
We study the following (quite complicated) non-linear differential operator
$$H_4u := u^{(4)} + q_\alpha(x)\frac{\sin(|u|)}{|u|}u' + V|u|^2u,$$
where $V$ is the characteristic function of the interval $[0,0.5]$ multiplied by $0.5\mathrm{i}$, where $\mathrm{i}$ is the imaginary unit and
$q_\alpha(x)=\frac{1}{2}((x+2.5)^\alpha - x-2.5)$ if $-2.5 < x <-1.5$ and $q_\alpha=0$ otherwise.
Here the function $q_\alpha$ belongs to the Sobolev space $W_1^1(\mathbb{R})$ for all $0<\alpha \leq 1$.
However, the limiting function $q$ as $\alpha\to 0$ has a jump discontinuity at origin and hence does not belong to $W_1^1(\mathbb{R})$.
According to our theory [1] we expect the Born approximation to recover the combination $\beta = -\frac{\sin(1)}{2}q_\alpha' + \mathrm{H}(V)$, where $\mathrm{H}$ is the Hilbert transform of $V$.
(This Hilbert transform appears in the reconstruction, because the function $V$ is complex valued.)
Numerically the inverse Born approximation for this operator works reasonably well when $\alpha > 0.3$, but it starts to fail when $\alpha < 0.3$.
This shows that the assumption that $q\in W_1^1(\mathbb{R})$ is quite natural.
Below is an animation of what happens to the reconstructions as $\alpha\to 0$.
[1] Tyni T and Serov V, Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line, Inverse Problems & Imaging 13:1 (2019), 159-175.
Let us consider a more advanced example where the inverse Born approximation does not work.
We study the following (quite complicated) non-linear differential operator
$$H_4u := u^{(4)} + q_\alpha(x)\frac{\sin(|u|)}{|u|}u' + V|u|^2u,$$
where $V$ is the characteristic function of the interval $[0,0.5]$ multiplied by $0.5\mathrm{i}$, where $\mathrm{i}$ is the imaginary unit and
$q_\alpha(x)=\frac{1}{2}((x+2.5)^\alpha - x-2.5)$ if $-2.5 < x <-1.5$ and $q_\alpha=0$ otherwise.
Here the function $q_\alpha$ belongs to the Sobolev space $W_1^1(\mathbb{R})$ for all $0<\alpha \leq 1$.
However, the limiting function $q$ as $\alpha\to 0$ has a jump discontinuity at origin and hence does not belong to $W_1^1(\mathbb{R})$.
According to our theory [1] we expect the Born approximation to recover the combination $\beta = -\frac{\sin(1)}{2}q_\alpha' + \mathrm{H}(V)$, where $\mathrm{H}$ is the Hilbert transform of $V$.
(This Hilbert transform appears in the reconstruction, because the function $V$ is complex valued.)
Numerically the inverse Born approximation for this operator works reasonably well when $\alpha > 0.3$, but it starts to fail when $\alpha < 0.3$.
This shows that the assumption that $q\in W_1^1(\mathbb{R})$ is quite natural.
Below is an animation of what happens to the reconstructions as $\alpha\to 0$.