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

+ 5 like - 0 dislike
105 views

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
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.