Some remarks concerning the scattering theory for the Sturm-Liouville operator
The author discusses scattering theory for the Sturm-Liouville operator \[ L = \frac {1}{y} \Big( - \frac{d^2}{dx^2} + q \Big), \] which depends on the parameter \(q \in \mathbb C_0(\mathbb R)\) and a strictly positive weight function \(y \in \mathbb C_0(\mathbb R)\), which is assumed to be fixed. A scattering hypothesis on the function \(q\) is given and the set of scattering data is described. The relation between the scattering coefficients and the potential \(q\) is studied, and then used to solve the inverse scattering problem, that is, the reconstruction of \(q\) when \(y\) is fixed and a set of scattering data is given. In addition, several relations concerning the spectral properties of \(L\) and the solutions of the related eigenvalue equation are established.
scientific article; zbMATH DE number 7491611
scientific article