Abstract
We discuss the direct and inverse scattering theory for the nonlinear Schrödinger equation \(-\triangle u(x)+h(x,|u(x)|)u(x)=k^{2}u(x),\quad x \epsilon\mathbb{R}^{3}\) where h is a very general and possibly singular combination of potentials. We prove first that the direct scattering problem has a unique bounded solution. We establish also the asymptotic behaviour of scattering solutions. A uniqueness result and a representation formula is proved for the inverse scattering problem with general scattering data. The method of Born approximation is applied for the recovery of jumps in the unknown function from general scattering data and fixed angle data.
Keywords
Mathematics Subject Classification (2010). Primary 35P25; Secondary 35R30.
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Harju, M., Serov, V. (2014). Three-dimensional Direct and Inverse Scattering for the Schrödinger Equation with a General Nonlinearity. In: Cepedello Boiso, M., Hedenmalm, H., Kaashoek, M., Montes Rodríguez, A., Treil, S. (eds) Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, vol 236. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0648-0_16
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DOI: https://doi.org/10.1007/978-3-0348-0648-0_16
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