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Bound states and inverse scattering for the Schrödinger equation in one dimension. (English) Zbl 0822.34070

Summary: The one-dimensional Schrödinger equation is considered when the potential and its first moment are absolutely integrable. When the potential has support contained on the left (right) half-line, it is uniquely constructed by using only the reflection coefficient from the right (left). The bound state norming constants determine whether the potential has support contained on a half-line or on the full-line. The bound state energies and the unique set of norming constants yielding the potential with support contained on the left (right) half-line are completely determined by the reflection coefficient from the right (left). An explicit example is provided.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81U40 Inverse scattering problems in quantum theory
34A55 Inverse problems involving ordinary differential equations
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