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Focal Points at Infinity for Short-Range Scattering Trajectories

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Abstract

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.

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References

  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.

    Article  MathSciNet  MATH  Google Scholar 

  2. DEREZIŃSKI J, GÉRARD C. Scattering theory of classical and quantum N-particle systems [M]. Berlin: Springer-Verlag, 1997.

    Book  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  4. VAINBERG B R. Asymptotic methods in equations of mathematical physics [M]. New York: Gordon and Breach Science Publishrs, 1989.

    MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  7. ARNOL’D V I. Critical points of smooth functions and their normal forms [J]. Russian mathematical Surveys, 1975, 30(5): 3–65.

    MathSciNet  MATH  Google Scholar 

  8. REED M, SIMON B. Methods of modern mathematical physics III: Scattering theory [M]. New York: Academic Press, 1979.

    MATH  Google Scholar 

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Correspondence to Horst Hohberger.

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Hohberger, H., Klein, M. Focal Points at Infinity for Short-Range Scattering Trajectories. J. Shanghai Jiaotong Univ. (Sci.) 23, 146–157 (2018). https://doi.org/10.1007/s12204-018-1920-2

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  • DOI: https://doi.org/10.1007/s12204-018-1920-2

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