If we consider the Schrödinger equation
$$(\Delta+k^2-V(x))\psi(k,x)=0,\qquad(k\in\mathbb{R}, x\in\mathbb{R}^{n})$$
with smooth (and compactly supported) scattering potential $V:\mathbb{R}^n\to\mathbb{R}$, then in the case of single scattering, we have
$$\psi_{\text{sc}}=G\ast V\psi_{\text{inc}},$$
where $\psi_{\text{sc}}$ and $\psi_{\text{inc}}$ represent the scattered and incident wave functions respectively and $G\in\mathcal{D}'(\mathbb{R}^n)$ is the fundamental solution of the Helmholtz operator subject to the Sommefeld radiation condition.
Moreover, since $\psi=\psi_{\text{inc}}+\psi_{\text{sc}}$ in $\mathbb{R}^n\setminus\operatorname{supp}V$ (i.e. away from the scatterer), we may write the Lippmann-Schwinger equation for single scattering as
$$\psi=\psi_{\text{inc}}+G\ast V\psi_{\text{inc}},$$ and solving for $V$ would be the inverse scattering problem (for single scattering).
Now, in the case of multiple scattering, we get an iteration of the Lippmann-Schwinger equation; in particular we have the Born series
$$\psi=\psi_{\text{inc}}+G\ast V\psi_{\text{inc}}-G\ast( V\psi_{\text{inc}}(G\ast V\psi_{\text{inc}}))+\cdots$$
which converges when we assume that it satisfies the weak scattering condition
$$\| G\ast V\|<1.$$
I would like to solve the inverse scattering problem now in one dimension and in three dimensions. I have found some literature (although not much) which details how you may solve it in the three dimensional setting; in particular, when we assume that $V=C_1 V_1+C_2 V_2+\cdots$, where $C_n$ are real valued coefficients. However, I have not found anything as of yet for the one dimensional case.
Hence I was wondering if someone could provide some good references which deal with this problem? Mathematician friendly references are especially welcome! Plus, an insightful exposition here on Stack Exchange would also be appreciated.