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The Schrödinger operator in the plane. (English) Zbl 0797.35140

Summary: The \(\overline \partial\) method of scattering and inverse scattering is adapted to the Schrödinger operator \(\partial^ 2/ \partial \zeta^ 2 + \partial^ 2/ \partial \eta^ 2-q (\zeta,\eta)\) in \(\mathbb{R}^ 2\) with small potential \(q(\zeta,\eta)\). Eigenfunctions of eigenvalue zero are studied. One may determine the \(\overline \partial\) scattering data from the leading coefficients of the asymptotic expansions of these eigenfunctions at large values of \((\zeta,\eta)\) or by taking the \(\partial_{\overline z}\) derivatives of these eigenfunctions. There are four scattering data to be used to solve the inverse problem but they can be reduced to one. The relations between small potentials and small \(\overline \partial\) scattering data are discussed.

MSC:

35Q40 PDEs in connection with quantum mechanics
35P25 Scattering theory for PDEs
35R30 Inverse problems for PDEs
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
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