Abstract
We study one dimensional scattering problems governed by the Schrödinger equation or by the impedance equation. In the absence of bound states, there exists a class of potentials (resp. impedances) that is bijectively related with a class of spectral data, ie reflection coefficients as a function of energy. On the other hand, it is known in larger classes several examples of different potentials that are consistent with a given reflection coefficient and no true bound state. In the lecture below it is shown that
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(1)
these ambiguities are related with a Darboux-type transformation which is defined on very wide classes of potentials (resp. impedances), leaves invariant the Schrödinger (resp. impedance) equation whereas the reflection coefficient is flipped, and depends on one arbitrary parameter, so that we obtain a one parameter family of “equivalent” potentials (resp. impedances).
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(2)
If potentials classes (resp. classes of impedance factors) are defined by their leading asymptotic behavior [ℓ±(ℓ± + 1)x-2], resp. (xp, xq), the transformation take potential (resp. am impedance factor) from one class to another one.
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(3)
The transformation leaves the transmission coefficient invariant but introduces or suppresses zero energy bound states or half bound states, so that it is not.
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References
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© 1986 Birkhäuser Verlag Basel
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Sabatier, P.C. (1986). Reconstruction Ambiguities of Inverse Scattering on the Line. In: Cannon, J.R., Hornung, U. (eds) Inverse Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 77. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7014-6_14
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DOI: https://doi.org/10.1007/978-3-0348-7014-6_14
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