The "Action" Variable is not an Invariant for the Uniqueness in the Inverse Scattering Problem
A. Kheifets, P. Yuditskii
Abstract
We give a simple example of non-uniqueness in the inverse scattering for Jacobi matrices: roughly speaking S-matrix is analytic. Then, multiplying a reflection coefficient by an inner function, we repair this matrix in such a way that it does uniquely determine a Jacobi matrix of Szegö class; on the other hand the transmission coefficient remains the same. This implies the statement given in the title.
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