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Examples of non-uniqueness in one-dimensional inverse scattering for which \(T(k)=O(k^ 3)\) and \(O(k^ 4)\) as k\(\to 0\). (English) Zbl 0625.35074

An example of a one-parameter family of potentials in one-dimensional non-relativistic quantum mechanics is given such that all the potentials in the family have the same scattering matrix at all energies and the transmission coefficient behaves like \(T(k)=O(k^ 3)\) as \(k\to 0\). An example where \(T(k)=O(k^ 4)\) as \(k\to 0\) is also given.

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
81U20 \(S\)-matrix theory, etc. in quantum theory
35P25 Scattering theory for PDEs
35R30 Inverse problems for PDEs
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