Theoretical and Mathematical Physics

, Volume 190, Issue 1, pp 77–90 | Cite as

A test for the existence of exceptional points in the Faddeev scattering problem

Article

Abstract

Exceptional points are values of the spectral parameter for which the homogeneous Faddeev scattering problem has a nontrivial solution. We formulate a criterion for existence of exceptional points that belong to a given path. For this, we use measurements at the endpoints of the path. We also study the existence or absence of exceptional points for small perturbations of conductive potentials of arbitrary shape and show that problems with absorbing potentials do not have exceptional points in a neighborhood of the origin.

Keywords

exceptional point Faddeev Green’s function conductive potential 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. D. Faddeev, “Increasing solutions of the Schrödinger equation,” Sov. Phys. Dokl., 10, 1033–1035 (1966).ADSGoogle Scholar
  2. 2.
    A. I. Nachman, “Global uniqueness for a two-dimensional inverse boundary value problem,” Ann. Math., 143, 71–96 (1996).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    R. G. Novikov, “Multidimensional inverse spectral problem for the equation −Δψ + (v(x) − Eu(x))ψ = 0,” Funct. Anal. Appl., 22, 263–272 (1988).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    R. G. Novikov, “The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator,” J. Funct. Anal., 103, 409–463 (1992).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    B. R. Vainberg, “Principles of radiation, limit absorption, and limit amplitude in the general theory of partial differential equations,” Russian Math. Surv., 21, 115–193 (1966).ADSCrossRefGoogle Scholar
  6. 6.
    M. Music, “The nonlinear Fourier transform for two-dimensional subcritical potentials,” Inverse Problems and Imaging, 8, 1151–1167 (2001).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    R. Beals and R. R. Coifman, “Multidimensional scattering and nonlinear partial differential equations,” in: Pseudodifferential Operators and Applications (Proc. Symp. Pure Math., Vol. 43, F. Trèves, ed.), Amer. Math. Soc., Providence, R. I. (1985), pp. 45–70.CrossRefGoogle Scholar
  8. 8.
    R. G. Novikov and G. M. Henkin, “The \(\bar \partial \) -equation in the multidimensional inverse scattering problem,” Russian Math. Surv., 42, 109–180 (1987).ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    M. Music, P. Perry, and S. Siltanen, “Exceptional circles of radial potentials,” Inverse Problems, 29, 045004 (2013).ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    S. Siltanen and J. Tamminen, “Exceptional points of radial potentials at positive energies,” arXiv:1307.2037v1 [math.NA] (2013).Google Scholar
  11. 11.
    P. G. Grinevich and S. P. Novikov, “Two-dimensional ‘inverse scattering problem’ for negative energies and generalized-analytic functions: I. Energies below the ground state,” Funct. Anal. Appl., 22, 19–27 (1988).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    E. Lakshtanov, R. Novikov, and B. Vainberg, “A global Riemann–Hilbert problem for two-dimensional inverse scattering at fixed energy,” arXiv:1509.06495v1 [math-ph] (2015).Google Scholar
  13. 13.
    P. G. Grinevich and S. V. Manakov, “Inverse scattering problem for the two-dimensional Schrödinger operator, the \(\bar \partial \) -method, and nonlinear equations,” Funct. Anal. Appl., 20, 94–103 (1986).CrossRefMATHGoogle Scholar
  14. 14.
    S. Siltanen, Annales Academiae Scientiarum Fennicae Mathematica. Mathematica. Dissertationes, Vol. 121, Helsinki Univ. of Technology, Helsinki (1999).Google Scholar
  15. 15.
    M. G. Zaidenberg, S. G. Krein, P. A. Kuchment, and A. A. Pankov, “Banach bundles and linear operators,” Russian Math. Surv., 30, 115–175 (1975).ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Y. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions (Math. Its Appl., Vol. 384), Kluwer, Dordrecht (1996).CrossRefMATHGoogle Scholar
  17. 17.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators, Acad. Press, New York (1982).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Department of MathematicsAveiro UniversityAveiroPortugal
  2. 2.Department of Mathematics and StatisticsUniversity of North CarolinaCharlotteUSA

Personalised recommendations