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Scattering for 1D Schrödinger equation with energy-dependent potentials and the recovery of the potential from the reflection coefficient. (English) Zbl 1050.81706

Summary: We consider the 1D Schrödinger equation with a potential proportional to energy. When the spatial part of the potential is twice continuously differentiable, is less than 1 everywhere, and satisfies a certain integrability condition, we compute the scattering matrix. For the same class of potentials, under the further assumption that the potential is non-negative, we obtain the potential from one of the reflection coefficients.

MSC:

81U05 \(2\)-body potential quantum scattering theory
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
35J10 Schrödinger operator, Schrödinger equation
35P25 Scattering theory for PDEs
35R30 Inverse problems for PDEs
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