Exponential Estimates of Eigenfunctions of Matrix Schrödinger and Dirac Operators

  • V. Rabinovich
  • S. Roch
Part of the Operator Theory: Advances and Applications book series (OT, volume 210)


The paper is devoted to exponential estimates of eigenfunctions of the discrete spectrum of matrix Schrödinger operators with variable potentials, and Dirac operators for nonhomogeneous media with variable light speed and variable electric and magnetic potentials, For the study of exponential estimates we apply methods developed in our recent paper [25].

Mathematics Subject Classification (2000)

Primary 58J10 Secondary 81Q10 


Uniformly elliptic systems exponential estimates limit operators Schrödinger and Dirac operators 


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  1. [1]
    S. Agmon, Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 4 (1975), 151–218.MathSciNetGoogle Scholar
  2. [2]
    S. Agmon, Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations. Princeton University Press, Princeton, 1982.zbMATHGoogle Scholar
  3. [3]
    M.S. Agranovich, Elliptic Operators on Smooth Manifolds. In: Itogi Nauki i Tekhniki, Sovremennie Problemi Matematiki, Fundamentalnie Napravlenia, V. 63, Partial Differential Equations-6 (Russian), 5 130.Google Scholar
  4. [4]
    E. Buzano, Super-exponential decay of solutions to differential equations inr. in: Modern Trends in Pseudo-Differential Operators, Operator Theory: Adv. Appl. 172, Birkhäuser, Basel 2007, 117–133.CrossRefGoogle Scholar
  5. [5]
    M. Cappiello, T. Gramchev, L. Rodino, Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients. J. Functional Anal. 237 (2006), 2, 634–654.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    M. Cappiello, T. Gramchev, L. Rodino, Gelfand-Shilov spaces, pseudo-differential operators and localization operators. in: Modern Trends in Pseudo-Differential Operators, Operator Theory: Adv. Appl. 172, Birkhäuser, Basel 2007, 297–312.CrossRefGoogle Scholar
  7. [7]
    H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin, Heidelberg, New York 1987.Google Scholar
  8. [8]
    P.A.M. Dirac, The Principles of Quantum Mechanics. Oxford: Clarendon Press, 1958.zbMATHGoogle Scholar
  9. [9]
    G. Esposito, Dirac Operators and Spectral Geometry, Cambridge Lecture Notes in Physics, Cambridge, University Press, 1998.Google Scholar
  10. [10]
    V. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics, Springer, Berlin Heidelberg New York, 1998.zbMATHGoogle Scholar
  11. [11]
    V.B. Berestecskiy, E.M. Lifshitz, L.P. Pitaevsky, Quantum Electrodynamics, 3 Edition, Nauka, Moskow, 1989 (in Russian).Google Scholar
  12. [12]
    R. Froese, I. Herbst, Exponential bound and absence of positive eigenvalue for N-body Schrödinger operators. Comm. Math. Phys. 87 (1982), 429–447.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    R. Froese, I. Herbst, M. Hoffman-Ostenhof, T. Hoffman-Ostenhof, L 2-exponential lower bound of the solutions of the Schrödinger equation. Comm. Math. Phys. 87 (1982), 265–286.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    J.M. Jauch, Foundation of Quantum Mechanics, Addison-Wesley, London, 1973Google Scholar
  15. [15]
    M. Klein, A. Martinez, R. Seiler, X.P. Wang, On the Born-Oppenheimer expansion for polyatomic molecules. Comm. Math. Phys. 143 (1992), 3, 607–639.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Ya.A. Luckiy, V.S. Rabinovich, Pseudodifferential operators on spaces of functions of exponential behavior at infinity. Funct. Anal. Prilozh. 4 (1977), 79–80.Google Scholar
  17. [17]
    M. Mântoiu, Weighted estimations from a conjugate operator. Letter Math. Phys. 51 (2000), 17–35.zbMATHCrossRefGoogle Scholar
  18. [18]
    A. Martinez, Eigenvalues and resonances of polyatomic molecules in the Born-Oppenheimer approximation, in: Schrödinger Operators. The Quantum Mechanical Many-Body Problem, Lecture Notes in Physics 403, Springer-Verlag, Berlin, Heidelberg 1992, 145–152.CrossRefGoogle Scholar
  19. [19]
    A. Martinez, Microlocal exponential estimates and application to tunnelling. In: Microlocal Analysis and Spectral Theory, L. Rodino (Editor), NATO ASI Series, Series C: Mathematical and Physical Sciences 490, 1996, 349–376.Google Scholar
  20. [20]
    A. Martinez, An Introduction to Semiclassical and Microlocal Analysis. Springer, New York 2002.zbMATHGoogle Scholar
  21. [21]
    S. Nakamura, Agmon-type exponential decay estimates for pseudodifferential operators. J. Math. Sci. Univ. Tokyo 5 (1998), 693–712.zbMATHMathSciNetGoogle Scholar
  22. [22]
    L. Nedelec, Resonances for matrix Schrödinger operators. Duke Math. J. 106 (2001), 2, 209–236.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    V.S. Rabinovich, Pseudodifferential operators with analytic symbols and some of its applications. In: Linear Topological Spaces and Complex Analysis 2, Metu-Tübitak, Ankara 1995, 79–98.Google Scholar
  24. [24]
    V. Rabinovich, Pseudodifferential operators with analytic symbols and estimates for eigenfunctions of Schrödinger operators. Z. Anal. Anwend. (J. Anal. Appl.) 21 (2002), 2, 351–370.zbMATHMathSciNetGoogle Scholar
  25. [25]
    V.S. Rabinovich, S. Roch, Essential spectrum and exponential decay estimates of solutions of elliptic systems of partial differential equations. Applications to Schrödinger and Dirac operators. Georgian Math. J. 15 (2008), 2, 333–351.zbMATHMathSciNetGoogle Scholar
  26. [26]
    M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, Heidelberg, New York 2001.zbMATHGoogle Scholar
  27. [27]
    B. Thaller, The Dirac Equation, Springer Verlag, Berlin, Heidelberg, New York 1992.Google Scholar

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© Springer Basel AG 2010

Authors and Affiliations

  • V. Rabinovich
    • 1
  • S. Roch
    • 2
  1. 1.Instituto Politécnico NacionalESIME-ZacatencoMéxicoMéxico
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

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