Inverse scattering up to smooth functions for the Schrödinger equation in one dimension (Q1354743)

For the Schrödinger equation in one dimension an explicit formula for the reconstruction of the potential from the class \[ W^{n,1} (\mathbb{R})= \left\{f(x):\;{d^m \over dx^m} f(x) \in L^1(\mathbb{R}), \quad m=0,1, \dots, n\right\} \] is given through its reflection coefficient upto a function from the class \(W^{n+1,1} (\mathbb{R})\). The necessary and sufficient conditions for the reflection coefficient of the potential from the class \(L^1(\mathbb{R})\) in order that this potential belongs to the class \(W^{n,1}\) are given. It is shown that the scattering matrix of the exponentially decreasing potential from the class \(W{n,1} (\mathbb{R})\) does not determine in general the potential upto a function from the class \(C^{n+\varepsilon} (\mathbb{R})\), \(\varepsilon >0\).