[Paper Review] Inverse scattering at high energies for the multidimensional Newton equation in a long range potential

Alexandre Jollivet|arXiv (Cornell University)|Jun 16, 2013

Spectral Theory in Mathematical Physics14 references3 citations

TL;DR

This paper establishes inverse scattering theory for the multidimensional Newton equation with a long-range potential decaying as $ r^{-\alpha} $ for $ \alpha \in (0,1] $. It proves that high-energy scattering data uniquely determine the short-range part of the potential up to knowledge of the long-range tail, using estimates on scattering solutions and a Born approximation, extending Novikov's 1999 framework to long-range interactions in higher dimensions.

ABSTRACT

We define scattering data for the Newton equation in a potential $V\\in C^2(\\R^n,\\R)$, $n\\ge2$, that decays at infinity like $r^{-\\alpha}$ for some $\\alpha\\in (0,1]$. We provide estimates on the scattering solutions and scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of the potential up to the knowledge of the long range tail of the potential. The Born approximation at fixed energy of the scattering data is also considered. We then change the definition of the scattering data to study inverse scattering in other asymptotic regimes. These results were obtained by developing the inverse scattering approach of [Novikov, 1999].

Motivation & Objective

• To develop an inverse scattering framework for the multidimensional Newton equation with long-range potentials decaying as $ r^{-\alpha} $, $ \alpha \in (0,1] $.
• To establish uniqueness of the short-range potential component from high-energy scattering data, given knowledge of the long-range tail.
• To extend Novikov's inverse scattering approach to long-range interactions in $ n \geq 2 $ dimensions.
• To analyze the Born approximation at fixed energy and define scattering data suitable for asymptotic regimes beyond high energy.

Proposed method

• Decompose the potential into long-range $ V^l $ and short-range $ V^s $ components satisfying specific decay estimates for $ |j| \leq 2 $.
• Define scattering solutions asymptotic to free trajectories in the past and future, using solutions $ z_\pm(v, \cdot) $ of the long-range equation $ \ddot{z} = F^l(z) $.
• Construct the scattering map $ S $ from initial data $ (v_-, x_-) \in \mathbb{R}^n \setminus \mathcal{B}(0,\mu) \times \mathbb{R}^n $, with $ \mu $ depending on potential decay constants.
• Use integral equations and contraction mapping arguments to estimate the scattering data $ a_{sc}, b_{sc} $, and derive bounds on their deviation from free motion.
• Apply the Born approximation at fixed energy to approximate scattering data, analyzing error terms via $ L^1 $-type estimates in time and space.
• Introduce modified scattering data definitions to study inverse scattering in alternative asymptotic regimes, such as high-energy limits.

Experimental results

Research questions

• RQ1 Can high-energy scattering data uniquely determine the short-range part of a potential that decays as $ r^{-\alpha} $, $ \alpha \in (0,1] $, in $ n \geq 2 $ dimensions?
• RQ2 How does the presence of a long-range potential affect the uniqueness and reconstruction of the short-range component in inverse scattering?
• RQ3 What is the behavior of the scattering data and its Born approximation at high energies for such potentials?
• RQ4 How can the scattering data be redefined to enable inverse scattering analysis in different asymptotic regimes?
• RQ5 What are the precise $ L^1 $-type estimates on the deviation of scattering solutions from free trajectories, and how do they depend on energy and spatial decay?

Key findings

• The scattering data at high energies uniquely determine the short-range part $ V^s $ of the potential, provided the long-range part $ V^l $ is known.
• Estimates on the scattering solutions and data are derived, showing $ |\dot{y}_\pm(t)| + |y_\pm(t)| \to 0 $ as $ t \to \pm\infty $, with explicit bounds on the deviation from free motion.
• The Born approximation at fixed energy is shown to approximate the scattering data with error terms controlled by $ \mathcal{O}\left( \frac{1}{|v_-|^{2\alpha+1}} \right) $, depending on decay parameters.
• The scattering map $ S $ is continuous and defined almost everywhere on $ \mathbb{R}^n \setminus \mathcal{B}(0,\mu) \times \mathbb{R}^n $, with the image determined by the restriction to the set $ \mathcal{M} $ of orthogonal initial data.
• Error bounds for the scattering data are derived via integral estimates, with terms decaying as $ \mathcal{O}\left( \frac{1}{|v_-|^{2\alpha+1}} \right) $, confirming high-energy stability.
• Modified definitions of scattering data allow for inverse scattering analysis in asymptotic regimes beyond high energy, extending the applicability of the framework.