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 $pm infty$.
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