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 $\pm \infty$.

  • $\begingroup$ An indirect comment: the forward and inverse directions of the IST have both been shown to be one-to-one. I also know that weighted Sobolev spaces play a role, because the NLS equation, at least, is well-posed in that setting. It may, however, be a category error to impose that space specifically on the potentials and the reflection coefficients. $\endgroup$ – Adrian Keister Feb 20 '18 at 20:28
  • $\begingroup$ @AdrianKeister Would you please provide a reference for the bijectivity of the Scattering/Inverse Scattering Transforms? $\endgroup$ – Matt Rosenzweig Apr 1 '18 at 20:13
  • $\begingroup$ @MHR Unfortunately, I don't have access to research materials at all, now. However, I would guess that Ablowitz and Segur, "Solitons and the Inverse Scattering Transform", SIAM, Philadephia, 1982, would at least have a reference, if not a proof. $\endgroup$ – Adrian Keister Apr 2 '18 at 13:41

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