Inverse scattering for the 1-D Helmholtz equation (Q285482)

The aim of the paper is to give a more elementary proof in the inverse scattering problem for the equation \[ u''+\frac{k^{2}}{c^{2}}u=0 \text{ on } \mathbb{R}. \] The function \(c\) is a measurable, bounded and bounded from below by a positive constant function and \(c-1 \in L^{1}(\mathbb{R})\). The main result states that the function \(c\) is uniquely determined by the reflection coefficient \(R_{2}(k)\) of the scattering matrix. The proof uses a uniqueness result for Nevanlinna functions. Let us mention that uniqueness in the inverse scattering problem for the more general equation \[ u''+qu=k^{2}wu, \] \(q\geq0\), \(w\) real and may change the sign, was proved in [\textit{C. Bennewitz} et al., J. Differ. Equations 253, No. 8, 2380--2419 (2012; Zbl 1323.34096)].