Essential Spectra of Pseudodifferential Operators and Exponential Decay of Their Solutions. Applications to Schrödinger Operators

  • Vladimir S. Rabinovich
  • Steffen Roch
Part of the Operator Theory: Advances and Applications book series (OT, volume 181)


The aim of this paper is to study relations between the location of the essential spectrum and the exponential decay of eigenfunctions of pseudodifferential operators on L p (ℝ n ) perturbed by singular potentials.

Our approach to this problem is via the limit operators method. This method associates with each band-dominated operator A a family op(A) of so-called limit operators which reflect the properties of A at infinity. Consider the compactification of ℝn by the “infinitely distant” sphere S n−1. Then the set op(A) can be written as the union of its components op ηω (A) where ω runs through the points of S n−1 and where op ηω (A) collects all limit operators of A which reflect the properties of A if one tends to infinity “in the direction of ω”. Set \( sp_{n_\omega } A: = \cup _{A_h \in op_{\eta \omega } (A)} spA_h \).

We show that the distance of an eigenvalue λsp ess A to sp ηω A determines the exponential decay of the λ-eigenfunctions of A in the direction of ω. We apply these results to estimate the exponential decay of eigenfunctions of electro-magnetic Schrödinger operators for a large class of electric potentials, in particular, for multiparticle Schrödinger operators and periodic Schrödinger operators perturbed by slowly oscillating at infinity potentials.


Pseudodifferential operators limit operators essential spectra exponential decay Schrödinger operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Agmon, Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations. Princeton University Press, Princeton, 1982.zbMATHGoogle Scholar
  2. [2]
    S. Agmon, Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 4(1975), 151–218.MathSciNetGoogle Scholar
  3. [3]
    M.S. Agranovich, Elliptic Operators on Closed Manifolds. Modern Problems of Mathematics 63, Partial Differential Equations 6, VINITI, 1990 (Russian).Google Scholar
  4. [4]
    W. Amerin, M. MĂntoiu, R. Purice, Propagation properties for Schrödinger operators affiliated with certain C*-algebras. Ann. H. Poincaré In-t 6(2002), 3, 1215–1232.CrossRefGoogle Scholar
  5. [5]
    M.Sh. Birman, The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential. Algebra i Analiz 8(1996), 3–20. Russian, Engl. transl.: St. Petersburg Math. J. 8(1997), 1–14.MathSciNetzbMATHGoogle Scholar
  6. [6]
    M.Sh. Birman, The discrete spectrum in gaps of the perturbed periodic Schrödinger operator I. Regular perturbations. In: Boundary Value Problems, Schrödinger Operators, Deformation Quantization, Math. Top. 8, Akademie-Verlag, Berlin 1995, p. 334–352.Google Scholar
  7. [7]
    M.Sh. Birman, The discrete spectrum in gaps of the perturbed periodic Schrödinger operator II. Nonregular perturbations. Algebra i Analiz 9(1997), 62–89. Russian, Engl. transl.: St. Petersburg Math. J. 9(1998), 1073–1095.MathSciNetzbMATHGoogle Scholar
  8. [8]
    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
  9. [9]
    E.B. Davies, Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics 42, Cambridge University Press, Cambridge 1995.Google Scholar
  10. [10]
    R. Froese, I. Herbst, Exponential bound and absence of positive eigenvalue for N-body Schrödinger operators. Comm. Math. Phys. 87(1982), 429–447.CrossRefMathSciNetzbMATHGoogle Scholar
  11. [11]
    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.CrossRefMathSciNetzbMATHGoogle Scholar
  12. [12]
    I. Gohberg, I. Feldman, Convolution Equations and Projection Methods for Their Solution. Nauka, Moskva 1971. Russian, Engl. transl.: Amer. Math. Soc. Transl. Math. Monographs 41, Providence, R.I., 1974.Google Scholar
  13. [13]
    V. Georgescu, A. Iftimovici, Crossed Products of C*-Algebras and Spectral Analysis of Quantum Hamiltonians. Comm. Math. Phys. 228(2002), 519–560.CrossRefMathSciNetzbMATHGoogle Scholar
  14. [14]
    V. Georgescu, A. Iftimovici, Localization at infinity and essential spectrum of quantum Hamiltonians. arXiv:math-ph/0506051V1, June 20, 2005.Google Scholar
  15. [15]
    P. Kuchment, On some spectral problems of mathematical physics. In: Partial Differential Equations and Inverse Problems, C. Conca, R. Manasevich, G. Uhlmann, M. S. Vogelius (Editors), Contemp. Math. 362, Amer. Math. Soc. 2004.Google Scholar
  16. [16]
    S. Lang, Real and Functional Analysis. Graduate Texts inMathematics 142, Springer, New York 1993 (third ed.).Google Scholar
  17. [17]
    Y. Last, B. Simon, The essential spectrum of Schrödinger, Jacobi, and CMV operators. Preprint 304 at Scholar
  18. [18]
    P.D. Lax, Functional Analysis. Wiley-Interscience, 2002.Google Scholar
  19. [19]
    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
  20. [20]
    M. MĂntoiu, Weighted estimations from a conjugate operator. Letter in Math. Physics 51(2000), 17–35.CrossRefzbMATHGoogle Scholar
  21. [21]
    M. MĂntoiu, C*-algebras, dynamical systems at infinity and the essential spectrum of generalized Schrödinger operators. J. Reine Angew. Math. 550(2002), 211–229.MathSciNetzbMATHGoogle Scholar
  22. [22]
    M. MĂntoiu, R. Purice, A priori decay for eigenfunctions of perturbed Periodic Schrödinger operators. Preprint Université de Genève, UGVA-DPT 2000/02-1071.Google Scholar
  23. [23]
    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 Vol. 490, 1996, p. 349–376.Google Scholar
  24. [24]
    A. Martinez, An Introduction to Semiclassical and Microlocal Analysis. Springer, New York 2002.zbMATHGoogle Scholar
  25. [25]
    S. Nakamura, Agmon-type exponential decay estimates for pseudodifferential operators. J. Math. Sci. Univ. Tokyo 5(1998), 693–712.MathSciNetzbMATHGoogle Scholar
  26. [26]
    V.S. Rabinovich, Pseudodifferential operators with analytic symbols and some of its applications. Linear Topological Spaces and Complex Analysis 2, Metu-Tübitak, Ankara 1995, p. 79–98.Google Scholar
  27. [27]
    V. Rabinovich, Pseudodifferential operators with analytic symbols and estimates for eigenfunctions of Schrödinger operators. Z. f. Anal. Anwend. (J. Anal. Appl.) 21(2002), 2, 351–370.MathSciNetzbMATHGoogle Scholar
  28. [28]
    V.S. Rabinovich, On the essential spectrum of electromagnetic Schrödinger operators. In: Contemp. Math. 382, Amer. Math. Soc. 2005, p. 331–342.MathSciNetGoogle Scholar
  29. [29]
    V.S. Rabinovich, Essential spectrum of perturbed pseudodifferential operators. Applications to the Schrödinger, Klein-Gordon, and Dirac operators. Russian J. Math. Phys. 12(2005), 1, 62–80.MathSciNetzbMATHGoogle Scholar
  30. [30]
    V.S. Rabinovich, S. Roch, The essential spectrum of Schrödinger operators on lattices. J. Phys. A: Math. Gen. 39(2006), 8377–8394.CrossRefMathSciNetzbMATHGoogle Scholar
  31. [31]
    V.S. Rabinovich, S. Roch, B. Silbermann, Fredholm theory and finite section method for band-dominated operators. Integral Eq. Oper. Theory 30(1998), 4, 452–495.CrossRefMathSciNetzbMATHGoogle Scholar
  32. [32]
    V.S. Rabinovich, S. Roch, B. Silbermann, Band-dominated operators with operatorvalued coefficients, their Fredholm properties and finite sections. Integral Eq. Oper. Theory 40(2001), 3, 342–381.CrossRefMathSciNetzbMATHGoogle Scholar
  33. [33]
    V.S. Rabinovich, S. Roch, B. Silbermann, Limit Operators and Their Applications in Operator Theory. Operator Theory: Adv. and Appl. 150, Birkhäuser, Basel, Boston, Berlin 2004.Google Scholar
  34. [34]
    M. Reed, B. Simon, Methods of Modern Mathematical Physics II. Fourier Analysis, Selfadjointness. Academic Press, New York, San Francisco, London 1975.Google Scholar
  35. [35]
    M. Reed, B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, New York, San Francisco, London 1978.zbMATHGoogle Scholar
  36. [36]
    M.A. Shubin, Pseudodifferential Operators and Spectral Theory. Springer, New York 2001 (second ed.).zbMATHGoogle Scholar
  37. [37]
    B. Simon, Semiclassical analysis of low lying eigenvalues II. Tunnelling. Ann. Math. 120(1984), 89–118.CrossRefGoogle Scholar
  38. [38]
    E.M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, New Jersey, 1993.zbMATHGoogle Scholar
  39. [39]
    M.E. Taylor, Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey, 1981.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Vladimir S. Rabinovich
    • 1
  • Steffen Roch
    • 2
  1. 1.ESIME-ZacatencoInstituto Politechnico NationalMexico, D.F.Mexico
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations