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Darboux transformation for the Schrödinger equation with steplike potentials. (English) Zbl 0977.81012

Summary: The one-dimensional Schrödinger equation is considered when the potential is asymptotic to a positive constant on the right half line. The corresponding Darboux transformation is established by showing how the scattering solutions, the scattering coefficients, and the potential change when bound states are added or removed. The scattering coefficients are represented as certain integrals, from which their properties can be directly extracted.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
34L25 Scattering theory, inverse scattering involving ordinary differential operators
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