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arXiv · 0710.3524

Construction of potentials using mixed scattering data

M. Lassaut·S. Y. Larsen·S. A. Sofianos·J. C. Wallet
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Abstract

The long-standing problem of constructing a potential from mixed scattering data is discussed. We first consider the fixed-$\ell$ inverse scattering problem. We show that the zeros of the regular solution of the Schrödinger equation, $r_{n}(E)$ which are monotonic functions of the energy, determine a unique potential when the domain of energy is such that the $r_{n}(E)$'s range from zero to infinity. The latter method is applied to the domain $\{E \geq E_0, \ell=\ell_0 \} \cup \{E=E_0, \ell \geq \ell_0 \}$ for which the zeros of the regular solution are monotonic in both parts of the domain and still range from zero to infinity. Our analysis suggests that a unique potential can be obtained from the mixed scattering data $\{δ(\ell_0,k), k \geq k_0 \} \cup \{δ(\ell,k_0), \ell \geq \ell_0 \}$ provided that certain integrability conditions required for the fixed $\ell$-problem, are fulfilled. The uniqueness is demonstrated using the JWKB approximation.

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M. Lassaut, S. Y. Larsen, S. A. Sofianos, J. C. Wallet. 2008-09-09. Construction of potentials using mixed scattering data. https://doi.org/10.1088/0266-5611%2F24%2F5%2F055014

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