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Inverse scattering for the nonlinear Schrödinger equation II. Reconstruction of the potential and the nonlinearity in the multidimensional case. (English) Zbl 0986.35125
Summary: We solve the inverse scattering problem for the nonlinear Schrödinger equation on \({\mathbb{R}}^n, n \geq 3\): \[ i \frac{\partial}{\partial t}u(t,x)= -\Delta u(t,x)+V_0(x)u(t,x) + \sum_{j=1}^{\infty} V_j(x)|u|^{2(j_0+j)} u(t,x). \]
We prove that the small-amplitude limit of the scattering operator uniquely determines \(V_{j}, j=0,1, \dots\). Our proof gives a method for the reconstruction of the potentials \(V_{j}, j=0,1, \dots\). The results of this paper extend our previous results for the problem on the line.

MSC:
35R30 Inverse problems for PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
81U40 Inverse scattering problems in quantum theory
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