Inverse scattering with rational scattering coefficients and wave propagation in nonhomogeneous media.(English)Zbl 1096.34006

Kaashoek, Marinus A. (ed) et al., Recent advances in operator theory and its applications. The Israel Gohberg anniversary volume. Selected papers of the 14th international workshop on operator theory and its applications, IWOTA 2003, Cagliari, Italy, June 24–27, 2003. Basel: Birkhäuser (ISBN 3-7643-7290-7/hbk). Operator Theory: Advances and Applications 160, 1-20 (2005).
The authors are concerned with the explicit recovery of the potential $$V(x)$$ associated with the one-dimensional Schrödinger equation
$\psi ^{\prime \prime }(k,x)+k^{2}\psi (k,x)=V(x)\psi (k,x)$
with $$x\in {\mathbb R}.$$ The main assumption is that scattering data, the reflection coefficients $$R(k)$$ and $$L(k),$$ are rational functions in $$\mathbb{C}^{+}$$. Here, instead of using the classical Marchenko’s integral equation, the authors follow the analysis due to T. Aktosun, M. Klaus and C. van der Mee [Integral Equations Oper. Theory 15, No. 6, 879–900 (1992; Zbl 0790.47012)].
When the potential belongs to the Fadeev class, i.e., $$\int_{\mathbb{R}}\left( 1+\left| x\right| \right) \left| V(x)\right| dx<\infty ,\;\;V(x)\in \mathbb{R},$$ and with no bound states, then the right and left Fadeev functions $$m_r(k,x)$$ and $$m_{\ell}(k,x)$$ can be expressed by means of the poles and residues of the reflection coefficient $$R$$. A simplification leads to a neat formula involving the ratio of two determinants. Obviously, the scheme allows numerical experiments on which the authors briefly comment and compare with other methods at the end of their paper.
For the entire collection see [Zbl 1077.47003].

MSC:

 34A55 Inverse problems involving ordinary differential equations 81U40 Inverse scattering problems in quantum theory 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

Zbl 0790.47012