Resolvent Estimates for Schrödinger-Type and Maxwell Equations with Applications

  • Matania Ben-Artzi
  • Jonathan Nemirovsky

Abstract

It is the purpose of this paper to present some recent results concerning properties of solutions to time dependent Schrödinger-type equations of the form,
$$ \begin{array}{*{20}c} {\frac{1}{i}\frac{{du}}{{dt}} = (H_0 + V)u,} \\ {u(0) = u_0 \in L^2 (\mathbb{R}^n ).} \\ \end{array} $$
(0.1)

Keywords

Anisotropy 

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References

  1. 1.
    S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa 2 (1975), 151–218.MathSciNetMATHGoogle Scholar
  2. 2.
    E. Balslev and B. Helffer, Limiting absorption principle and resonances for the Dirac operator, Advances in applied mathematics, (1992), 13, (2), 186–215.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    M. Ben-Artzi, Unitary equivalence and scattering theory for Stark-like Hamiltonians, J. Math. Phys. 25 (1984), 951–964.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    M. Ben-Artzi, Regularity and smoothing for some equations of evolution, in “Nonlinear Partial Differential Equations and Their Applications,” H. Brezis, J. L. Lions (Eds.), Vol. II, pp. 1-12, Pittman Publ. 1994.Google Scholar
  5. 5.
    M. Ben-Artzi and Y. Dermenjian and J. C. Guillot, Acoustic waves in perturbed stratified fluids: A spectral theory, Comm. in PDE, (1989), 14, (4), 479–517.MathSciNetMATHGoogle Scholar
  6. 6.
    M. Ben-Artzi and A. Devinatz, Resolvent estimates for a sum of tensor products with applications to the spectral theory of differential operators, J. d’Analyse Math. 43 (1984), 215–250.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    M. Ben-Artzi and A. Devinatz, “The limiting absorption principle for partial differential operators,” Memoirs Amer. Math. Soc. #364, 1987.Google Scholar
  8. 8.
    M. Ben-Artzi and A. Devinatz, Local smoothing and convergence properties for Schrödinger type equations, J. Fuc. Anal. 101 (1991), 231–254.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    M. Ben-Artzi and A. Devinatz, Regularity and decay of solutions to the Stark evolution equation, J. Func. Anal, (to appear).Google Scholar
  10. 10.
    M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schrödinger equation, J. d’Analyse Math. 58 (1992), 25–37.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    M. Ben-Artzi and J. Nemirovsky, Remarks on relativistic Schrödinger operators and their extensions, Ann. Inst. H. Poincaré-Phys. Théorique, 67 (1), (1997), 29–39.MathSciNetMATHGoogle Scholar
  12. 12.
    M. Sh. Birman and M. Z. Solomyak, L 2 theory of the Maxwell operator in arbitrary domains, Usp. Math. Nauk., 6, 42 (1987), 61–76.MathSciNetGoogle Scholar
  13. 13.
    P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), 413–439.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana U. Math. J. 40 (1991), 33–69.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    O. A. Ladyzhenskaya, “The mathematical theory of viscous incompressible flow”, (1969), 23-31, Gordon and Breach Science publishers.Google Scholar
  16. 16.
    J. Nemirovsky, Limiting Absorption Principle and Spectral Analysis for Electro-magnetic Waves in Stratified Anisotropic Media, Ph. D. thesis (Hebrew University).Google Scholar
  17. 17.
    M. Reed and B. Simon, “Methods of Modern Mathematical Physics III —Scattering theory,” Academic Press, 1972.Google Scholar
  18. 18.
    T. Umeda, Radiation conditions and resolvent estimates for relativistic Schrödinger operators, Ann. Inst. H. Poincaré-Phys. Théorique 63 (1995), 277–296.MathSciNetMATHGoogle Scholar
  19. 19.
    I. Šole and Českoslov, Časopis pro Fysiku, (1953), 3, p. 366.Google Scholar
  20. 20.
    K. Yajima, Spectral and scattering theory for Schrödinger operators with Stark-effect, J. Fac. Sci. Univ. Tokyo, Sec. 1A, 26 (1979), 377–389.MathSciNetMATHGoogle Scholar
  21. 21.
    A. Yariv and P. Yeh, “Optical Waves in Crystals”, (1984), Wiley-Interscience Publication.Google Scholar
  22. 22.
    P. Yeh, “Optical Waves in Layered Media”, 1988, Wiley-Interscience Publication.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Matania Ben-Artzi
    • 1
  • Jonathan Nemirovsky
    • 1
  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael

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