Journal of Shanghai Jiaotong University (Science)

, Volume 23, Issue 1, pp 146–157 | Cite as

Focal Points at Infinity for Short-Range Scattering Trajectories



Classical scattering trajectories are known to form a Lagrangian manifold in euclidean phase space, which allows the classification of local focal points for sufficiently small dimensions. For the case of a short-range potential, we show that the natural description of focal points at infinity is a Lagrangian manifold in the cotangent bundle of the sphere and establish the relationship between focal points at infinity and the projection singularities of that manifold.

Key words

Lagrangian manifold classical scattering theory short-range potential 

CLC number

O 17 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    ROBERT D, TAMURA H. Asymptotic behaviour of scattering amplitudes in semi-classical and low energy limits [J]. Annales-Institut Fourier, 1989, 39(1): 155–192.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    DEREZIŃSKI J, GÉRARD C. Scattering theory of classical and quantum N-particle systems [M]. Berlin: Springer-Verlag, 1997.CrossRefMATHGoogle Scholar
  3. [3]
    PROTAS Y N. Quasiclassical asymptotics of the scattering amplitude for the scattering of a plane wave by inhomogeneities of the medium [J]. Mathematics of the USSR-Sbornik, 1983, 45(4): 487–506.CrossRefMATHGoogle Scholar
  4. [4]
    VAINBERG B R. Asymptotic methods in equations of mathematical physics [M]. New York: Gordon and Breach Science Publishrs, 1989.MATHGoogle Scholar
  5. [5]
    ARNOL’D V I. Integrals of rapidly oscillating functions and singularities of projections of Lagrangian manifolds [J]. Functional Analysis and Its Applications, 1972, 6(3): 61–62.MathSciNetGoogle Scholar
  6. [6]
    ARNOL’D V I. Normal forms for functions near degenerate critical points, the Weyl groups of A k, D k, E k and Lagrangian singularities [J]. Functional Anallysis and Its Applications, 1972, 6(4): 254–272.CrossRefGoogle Scholar
  7. [7]
    ARNOL’D V I. Critical points of smooth functions and their normal forms [J]. Russian mathematical Surveys, 1975, 30(5): 3–65.MathSciNetMATHGoogle Scholar
  8. [8]
    REED M, SIMON B. Methods of modern mathematical physics III: Scattering theory [M]. New York: Academic Press, 1979.MATHGoogle Scholar

Copyright information

© Shanghai Jiaotong University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Michigan - Shanghai Jiao Tong University Joint InstituteShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Institute for MathematicsUniversity of PotsdamPotsdamGermany

Personalised recommendations