Scattering and inverse scattering for the 1-D Schrödinger equation with energy-dependent potentials.(English)Zbl 0758.35055

Summary: The one-dimensional Schrödinger equation with a potential $$k^ 2V(x)$$ proportional to energy is studied. This equation is equivalent to the wave equation with variable speed. When $$V(x)<1$$, is bounded below, and satisfies two integrability conditions, the scattering matrix is obtained and its asymptotics for small and large energies are established. The inverse scattering problem of recovering $$V(x)$$ when the scattering matrix is known is also solved. By proving that all the solutions of a key Riemann-Hilbert problem have the same asymptotics for large energy, it is shown that the potential obtained is unique.

MSC:

 35P25 Scattering theory for PDEs 35R30 Inverse problems for PDEs 81U40 Inverse scattering problems in quantum theory 35Q40 PDEs in connection with quantum mechanics
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References:

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