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.


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


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