Dislocation Problems for Periodic Schrödinger Operators and Mathematical Aspects of Small Angle Grain Boundaries

  • Rainer HempelEmail author
  • Martin Kohlmann
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)


We discuss two types of defects in two-dimensional lattices, namely (1) translational dislocations and (2) defects roduced by a rotation of the lattice in a half-space. For Lipschitz-continuous and ℤ2-periodic potentials, we first show that translational dislocations produce spectrum inside the gaps of the periodic problem; we also give estimates for the (integrated) density of the associated surface states.W e then study lattices with a small angle defect where we find that the gaps of the periodic problem fill with spectrum as the defect angle goes to zero.T o introduce our methods, we begin with the study of dislocation problems on the real line and on an infinite strip.F inally, we consider examples of muffin tin type.O ur overview refers to results in [HK1, HK2].


Schrödinger operators eigenvalues spectral gaps 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ADH.
    S. Alama, P.A. Deift, and R. Hempel, Eigenvalue branches of the Schrödinger operator H — λW in a gap of σ (H), Commun. Math. Phys. 121 (1989), 291–321.MathSciNetzbMATHCrossRefGoogle Scholar
  2. CFS.
    I.P. Cornfield, S.V. Fomin, and Y.G. Sinai, Ergodic theory, Springer, New York, 1982.CrossRefGoogle Scholar
  3. CFrKS.
    H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer, New York, 1987.Google Scholar
  4. DH.
    P.A. Deift and R. Hempel, On the existence of eigenvalues of the Schrödinger operator H — λW in a gap of σ (H), Commun. Math. Phys. 103 (1986), 461–490.MathSciNetzbMATHCrossRefGoogle Scholar
  5. DS.
    E.B. Davies and B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Commun. Math.Phys. 63 (1978), 277–301.MathSciNetzbMATHCrossRefGoogle Scholar
  6. E.
    M.S.P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, London, 1973.zbMATHGoogle Scholar
  7. EKSchrS.
    H. Englisch, W. Kirsch, M. Schröder, and B. Simon, Random Hamiltonians ergodic in all but one direction, Commun. Math. Phys. 128 (1990), 613–625.zbMATHCrossRefGoogle Scholar
  8. HK1.
    R. Hempel and M. Kohlmann, A variational approach to dislocation problems for periodic Schrödinger operators, J. Math. Anal. Appl., to appear.Google Scholar
  9. HK2.
    ____, Spectral properties of grain boundaries at small angles of rotation, J. Spect. Th., to appear.Google Scholar
  10. K1.
    E. Korotyaev, Lattice dislocations in a 1-dimensional model, Commun. Math. Phys. 213 (2000), 471–489.MathSciNetzbMATHCrossRefGoogle Scholar
  11. K2.
    ___, Schrödinger operators with a junction of two 1-dimensional periodic potentials, Asymptotic Anal. 45 (2005), 73–97.Google Scholar
  12. KS.
    V. Kostrykin and R. Schrader, Regularity of the surface density of states, J. Funct. Anal. 187 (2001), 227–246.MathSciNetzbMATHCrossRefGoogle Scholar
  13. PF.
    L. Pastur and A. Figotin, Spectra of Random and almost-periodic Operators, Springer, New York, 1991.Google Scholar
  14. RS-IV.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV, Analysis of Operators, Academic Press, New York, 1978.Google Scholar
  15. S.
    B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526.MathSciNetzbMATHCrossRefGoogle Scholar
  16. V.
    I. Veselic, Existence and regularity properties of the integrated density of states of random Schrödinger operators, Springer Lecture Notes in Mathematics, vol. 1917, Springer, New York, 2008.zbMATHGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Institute for Computational MathematicsTechnische Universität BraunschweigBraunschweigGermany
  2. 2.Institute for Applied MathematicsLeibniz Universität HannoverHannoverGermany

Personalised recommendations