Regularity of reflection coefficients (or more generally the scattering transform)

Question

Consider the Schrodinger operator $L(q) = -\partial_x^2 + q(x)$ where the potential $q$ is a real-valued function of a real variable which decays sufficiently rapidly at $\pm \infty$.

We define the scattering data in the usual way, as follows:

The essential spectrum of $L(q)$ is the positive real axis $[0,\infty)$ and it has multiplicity two. The Jost functions $f_\pm(\cdot,k)$ corresponding to $L(q)$ solve $L(q)f_\pm = k^2f_\pm$ with $f_\pm(x,k) \sim e^{ikx}$ as $x \to \pm \infty$.

The reflection coefficients $R_\pm(k)$ are defined so that $f_\pm = \bar{f_\mp} + R_\pm(k)f_\mp$ where here overbar denotes complex conjugate. The intuition is that $R$ measures the amount of energy which is reflected back to spatial $\infty$ when a wave with spatial frequency $k$ that originates at spatial $\infty$ interacts with the potential $q$.

The scattering transform (the map from $q$ to the scattering data, of which $R_+$ is a part) and its inverse are important in the theory of integrable PDE.

My question is the following: What is the regularity of the map $q \mapsto R_+$? Is it continuously differentiable?

To answer this question, we first must specify the spaces that $q$ and $R_+$ live in. I don't really care so long as they are reasonable spaces, for example take $q$ in weighted $H^1$ where the weight enforces a rapid decay at $\pm \infty$.

This post imported from StackExchange MathOverflow at 2014-09-13 08:15 (UCT), posted by SE-user Aaron Hoffman