Thursday, April 12, 2012

1101.5526 (Rainer Hempel et al.)

Dislocation problems for periodic Schrödinger operators and
mathematical aspects of small angle grain boundaries

Rainer Hempel, Martin Kohlmann
We discuss two types of defects in two-dimensional lattices, namely (1) translational dislocations and (2) defects produced by a rotation of the lattice in a half-space. For Lipschitz-continuous and $\Z^2$-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. We 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. To introduce our methods, we begin with the study of dislocation problems on the real line and on an infinite strip. Finally, we consider examples of muffin tin type. Our overview refers to results in [HK1, HK2].
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