The "Action" Variable is not an Invariant for the Uniqueness in the Inverse Scattering Problem

Details The "Action" Variable is not an Invariant for the Uniqueness in the Inverse Scattering Problem The "Action" Variable is not an Invariant for the Uniqueness in the Inverse Scattering Problem

2002-02-13

arxiv.org/abs/math/0202128v1

Mathematics

Spectral Theory

Scientific paper

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 Szeg\"o class; on the other hand the transmission coefficient remains the same. This implies the statement given in the title.

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