Energy dependent scattering theory (Q1891550)

The authors consider the one-dimensional Schrödinger equation \(d^2 \psi/dx^2 + k^2 \psi = [ip (x) + q(x)] \psi\), where \(p(x)\) is purely imaginary and \(q(x)\) is real. The inverse problem is to recover \(p(x)\) and \(q(x)\) from an appropriate set of scattering data. When there are no bound states, in the radial and one dimensional cases, a solution of this inverse problem is obtained by solving a pair of coupled linear integral equations and a pair of coupled differential equations [\textit{M. Jaulent} and \textit{C. Jean}, Ann. Inst. Henri Poincaré, N. Ser., Sect. A 25, 105-118 (1976; Zbl 0357.34018) and 119-137 (1976; Zbl 0357.34019)]. In this paper it is shown that the method of Jaulent and Jean can be simplified by replacing the step of solving the coupled pair of differential equations by an easier step. A special solution of the inverse problem is given when there is one bound state. By proving the solvability of the inverse problem the authors also obtain a global existence result for a pair of coupled nonlinear evolution equations.