# Inverse Scattering Transform (IST) for the Linear Schrödinger Equation

I know that the Inverse Scattering Transform (IST) has been employed to solve, for instance, the KdV equation and I believe also other nonlinear PDEs, such as the NLS.

However, if we consider the linear Schrödinger equation with scattering potential $V$, whereby we have multiple scattering. That is:

$$\left[-\frac{\hbar^2}{2m}\nabla^2+V(\vec{x})\right]\psi(\vec{x})=E\psi(\vec{x}),$$

where $\psi$ is the wave field. And by "multiple" scattering, I mean that we can make the following "ansatz":

$\psi=\psi_0+\psi_1+\psi_2+...$

where $\psi_0\propto e^{ikz}$ is the incoming plane wave and $\psi_1+\psi_2+...=\psi_s$ represents the scattered part perturbing the original wave function. Then we can rewrite this in terms of the Born series (essentially an iteration of the Lippmann-Schwinger equation which we'd have if we only had "single scattering", i.e. "$\psi=\psi_0+\psi_1$", since each $_i\in\mathbb{N}$ represents a "scattering") to get:

$$|\psi\rangle=|\psi_0\rangle+G_0(E)V|\psi_1\rangle+[G_0(E)V]^2\psi_2\rangle+...$$

where $G_0$ are Green's functions. This is a nonlinear problem. In the case of single scattering, we can put the inverse Fourier transform to good use. However, in the case of multiple scattering, Roger G. Newton, for example, formulated a method for an exact solution to the inverse scattering problem in $\mathbb{R}^3$. There are also similar procedures for solving in the inverse scattering problem in $\mathbb{R}^1$.

My question is: are the procedures employed by Roger G. Newton et al essentially Inverse Scattering Transforms? I felt quite inclined to ask this question since I only ever hear of IST being employed on NLS; never its linear analogue. Moreover, if not, is it possible to formulate the inverse scattering problem of the Schrödinger equation in such a way that the method by which we solve it would employ the IST?

I apologise if the notation is a bit off. I'm really a mathematician but I tried my best to formulate this question "a la physics".