Arkiv för Matematik

  • Ark. Mat.
  • Volume 37, Number 1 (1999), 141-169.

Small angle scattering and X-ray transform in classical mechanics

Roman G. Novikov

Full-text: Open access

Abstract

We consider the Newton equation $$\ddot{x} = F(x), \quad F(x) = -∇ \upsilon (x), \quad x \in \mathbf{R}^d,\\ \text{where }\upsilon \in C^2(\mathbf{R}^d, \mathbf{R}), |∂^j_x v(x) |\le c_{|j|}(1+|x|)^{-(\alpha+|j|)}$$ for |j|≤2 and some α>1.

We give estimates and asymptotics for scattering solutions and scattering data for the equation (*) for the case of small angle scattering. We show that scattering data at high energies uniquely determine the X-ray transforms PF and Pv. Applying results on inversion of the X-ray transform P we obtain that for d≥2 scattering data at high energies uniquely determine F and v. For the case of potentials with compact support we give a connection between boundary value data and scattering data and for d≥2 we obtain, using known results, a uniqueness theorem in the inverse scattering problem at fixed energy.

Article information

Source
Ark. Mat., Volume 37, Number 1 (1999), 141-169.

Dates
Received: 28 October 1997
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898620

Digital Object Identifier
doi:10.1007/BF02384831

Mathematical Reviews number (MathSciNet)
MR1673429

Zentralblatt MATH identifier
1088.70009

Rights
1999 © Institut Mittag-Leffler

Citation

Novikov, Roman G. Small angle scattering and X -ray transform in classical mechanics. Ark. Mat. 37 (1999), no. 1, 141--169. doi:10.1007/BF02384831. https://projecteuclid.org/euclid.afm/1485898620


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