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Inverse scattering with partial information on the potential. (English) Zbl 1012.34080

The inverse problem of recovering the real potential with finite first moment appearing in the Schrödinger equation on the line from scattering data consisting of (i) a reflection coefficient, (ii) all of the bound state energies, (iii) the potential on a finite interval, and (iv) all but one of the bound state norming constants. It is shown that such an inverse problem has at most two solutions. In the particular case where the missing norming constant in the scattering data corresponds to the lowest energy bound state, necessary and sufficient conditions are given for such nonuniqueness to occur. Some illustrative examples are worked out in detail.

MSC:

34L25 Scattering theory, inverse scattering involving ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81U40 Inverse scattering problems in quantum theory
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