Journal of Soviet Mathematics

, Volume 49, Issue 5, pp 1225–1236 | Cite as

Eikonal approximation for fast-decreasing potentials. I

  • D. R. Yafaev


The Schrödinger equation with a potential gq(x), decreasing faster at infinity than any power of ¦x¦−1, is considered at energy k2. The complete asymptotic expansion of its wave function is constructed for k → ∞ and g≤C2−γ, γ > 0. This expansion is used to derive the asymptotics of the forward scattering amplitude and of the total scattering cross section.


Wave Function Asymptotic Expansion Scattering Cross Section Forward Scatter Scattering Cross 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature cited

  1. 1.
    W. Hunziker, “Potential scattering at high energies,” Helv. Phys. Acta,36, 838–856 (1963).Google Scholar
  2. 2.
    V. S. Buslaev, “Trace formulas and certain asymptotic estimates of the kernel of the resolvent for the Schrödinger operator in three-dimensional space,” in: Problems of Mathematical Physics, No. 1 [in Russian], Leningrad State Univ. (1966), pp. 82–101.Google Scholar
  3. 3.
    V. S. Buslaev, “On the asymptotic behavior of the spectral characteristics of exterior problems for the Schrödinger operator,” Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 1, 149–235 (1975).Google Scholar
  4. 4.
    A. Jensen, “The scattering cross section and its Born approximation at high energies,” Helv. Phys. Acta,53, 398–403 (1980).Google Scholar
  5. 5.
    B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics [in Russian], Moscow State Univ. (1982).Google Scholar
  6. 6.
    K. Yajima, “The quasiclassical limit of scattering amplitude. Finite range potentials,” in: Schrodinger Operators, Lecture Notes in Math., No. 1159, Springer, New York (1985).Google Scholar
  7. 7.
    D. R. Yafaev, “The eikonal approximation and the asymptotics of the total scattering cross section for the Schrödinger equation,” Ann. Inst. H. Poincaré, Sect. A,44, No. 4, 397–425 (1986).Google Scholar
  8. 8.
    A. V. Sobolev, “The quasiclassical asymptotic behavior of the scattering cross section for asymptotically homogeneous potentials,” J. Sov. Math.,40, No. 5 (1988).Google Scholar
  9. 9.
    D. R. Yafaev, “On the quasiclassical asymptotics of the scattering amplitude and cross section,” Dokl. Akad. Nauk SSSR,292, No. 2, 334–337 (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • D. R. Yafaev

There are no affiliations available

Personalised recommendations