Theoretical and Mathematical Physics

, Volume 193, Issue 1, pp 1420–1428 | Cite as

Constructive scattering theory

Article

Abstract

We consider a problem of factoring the scattering matrix for Schrödinger equation on the real axis. We find the elementary factorization blocks in both the finite and infinite cases and establish a relation to the matrix conjugation problem. We indicate a general scheme for constructing a large class of scattering matrices admitting a quasirational factorization.

Keywords

scattering theory model equation factorization quasirational solution 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. B. Shabat, “Difference Schrödinger equation and quasisymmetric polynomials,” Theor. Math. Phys., 184, 1067–1077 (2015).CrossRefMATHGoogle Scholar
  2. 2.
    A. B. Shabat, “Scattering theory for delta-type potentials,” Theor. Math. Phys., 183, 540–552 (2015).CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    A. B. Shabat, “Inverse spectral problem for delta potentials,” JETP Lett., 102, 620–623 (2015).ADSCrossRefGoogle Scholar
  4. 4.
    P. Koosis, The Logarithmic Integral: I (Cambridge Stud. Adv. Math., Vol. 12), Cambridge Univ. Press, Cambridge (1997).MATHGoogle Scholar
  5. 5.
    V. Bargman, “Remarks on the determination of a central field of force from the elastic scattering phase shifts,” Phys. Rev., 75, 301–303 (1949).ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. B. Shabat, “Rational interpolation and solitons,” Theor. Math. Phys., 179, 637–648 (2014).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    M. Sh. Badakhov and A. B. Shabat, “Darboux transformations in the inverse scattering problem,” Ufa Math. Journal, 8, No. 4, 42–51 (2016).CrossRefMathSciNetGoogle Scholar
  8. 8.
    R. Ch. Kulaev and A. B. Shabat, “Inverse Scattering problem for finite potentials in the space of Borel measures [in Russian],” Preprint, SMI VSC RAS, Vladikavkaz (2016).Google Scholar
  9. 9.
    A. B. Shabat, “Functional Cantor equation,” Theor. Math. Phys., 189, 1712–1717 (2016).CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    A. B. Shabat, “Inverse-scattering problem for a system of differential equations,” Funct. Anal. Appl., 9, 244–247 (1975).CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsMoscowRussia

Personalised recommendations