Inverse scattering for the 1-D Helmholtz equation

We prove a uniqueness result for Nevanlinna functions. and this result is then used to give an elementary proof of the uniqueness in the inverse scattering problem for the equation $u" + \frac{k^2}{c^2}u=0$ on $\mathbb R$. Here $c$ is a real positive measurable function that is bounded from below by a positive constant, and is close to $1$ at $\pm \infty$.