Inverse Scattering Problems For Schrödinger Operators with Magnetic and Electric Potentials

  • G. Eskin
  • J. Ralston
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 90)

Abstract

Consider the Schrödinger equation in R n , n ≥ 3, with magnetic potential A(x) = (A 1(x),…,A n (x)) and electric potential V(x):
$$ {\left( {\frac{1}{i}\frac{\partial }{{\partial x}} + A\left( x \right)} \right)^{2}}u + V\left( x \right)u = {k^{2}}u$$
(1.1)
or equivalently
$$ - \Delta u - 2i\sum\limits_{{j = 1}}^{n} {Aj} \left( x \right)\frac{{\partial u}}{{\partial {x_{j}}}} + q\left( x \right)u = {k^{2}}u$$
(1.2)
where We will assume that the potentials A and V are real-valued and exponentially decreasing, i.e.

Keywords

Acoustics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A]
    Agmon, S., Spectral properties of Schrödinger operators and scattering theory, Annali di Pisa, Serie IV, 2 (1975), 151–218.MathSciNetMATHGoogle Scholar
  2. [ER]
    Eskin, G. and Ralston, J., Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy, Commun. Math. Phys. 173 (1995), 173–199.MathSciNetCrossRefGoogle Scholar
  3. [F]
    Faddeev, L. D., The inverse problem of quantum scattering II, J.Sov. Math. 5 (1976), 334–396.MATHCrossRefGoogle Scholar
  4. [H]
    Hörmander L., Uniqueness theorems for second order elliptic differential equations. Comm. in PDE 8 (1983), 21–64.MATHCrossRefGoogle Scholar
  5. [I-N]
    Isakov V. and Nachman A., Global uniqueness in a two-dimensional semilinear elliptic inverse problem, (1994), Preprint.Google Scholar
  6. [LP]
    Lax, P. and Phillips, R. S., Scattering Theory, Academic Press, San Diego, 1967 (revised 1990).MATHGoogle Scholar
  7. [NK]
    Novikov, R. G., Khenkin, G. M. The „¯-equation in the multidimensional inverse scattering problem, Russ. Math. Surv., 42 (1987), 109–180.MathSciNetMATHCrossRefGoogle Scholar
  8. [No]
    Novikov, R. G., The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential, Comm. Math. Phys., 161 (1994), 569–595.MathSciNetMATHCrossRefGoogle Scholar
  9. [NSU]
    Nakamura, G., Sun, Z., Uhlmann, G., Global Identifiability for an Inverse Problem for the Schrödinger Equation in a Magnetic Field, to appear in Math. Annalen.Google Scholar
  10. [RS]
    Ramm, A. and Stefanov, P. Fixed energy inverse scattering problem for non-compactly supported potentials, Math. Comput.Modeling, vol. 18, No. 1 (1993), 57–64.MathSciNetMATHCrossRefGoogle Scholar
  11. [Sh]
    Shiota, T. An inverse problem for the wave equation with first order perturbation, Amer. J. Math. 107 (1985), 241–251.MathSciNetMATHCrossRefGoogle Scholar
  12. [SU]
    Sylvester, J. and Uhlmann, G., A global Uniqueness Theorem for an Inverse Boundary Value Problem, Ann. of Math., 125 (1987), 153–169.MathSciNetMATHCrossRefGoogle Scholar
  13. [Su]
    Sun, Z., Personal Communication.Google Scholar
  14. [U]
    Uhlmann, G., Inverse Boundary Value Problems and Applications, Méthodes Semi-Classique I, Astérisque, 207 (1992), 153–211.MathSciNetGoogle Scholar
  15. [Sy]
    Sylvester, J., The Cauchy data and the scattering amplitude, Commun. in P.D.E. 19 (1994), 1735–1741.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • G. Eskin
    • 1
  • J. Ralston
    • 1
  1. 1.Department of MathematicsUCLALos AngelesUSA

Personalised recommendations