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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)

Abstract

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].

Keywords

Schrödinger operators eigenvalues spectral gaps 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

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

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