Spectral properties of grain boundaries at small angles of rotation

  • Rainer Hempel

    Technische Universität Braunschweig, Germany
  • Martin Kohlmann

    Leibniz Universität Hannover, Germany


We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential V ⁣:R2RV \colon \Bbb R^2 \to \Bbb R, we let Vθ(x,y)=V(x,y)V_\theta(x,y) = V(x,y) in the right half-plane {x0}\{ x \ge 0\} and Vθ=VMθV_\theta = V \circ M_{-\theta} in the left half-plane {x<0}\{x < 0\}, where MθR2×2M_\theta \in \Bbb R^{2 \times 2} is the usual matrix describing rotation of the coordinates in R2\Bbb R^2 by an angle θ\theta. As a main result, it is shown that spectral gaps of the periodic Schrödinger operator H0=Δ+VH_0 = -\Delta + V fill with spectrum of Rθ=Δ+VθR_\theta = -\Delta + V_\theta as 0θ00 \ne \theta \to 0. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states.

Cite this article

Rainer Hempel, Martin Kohlmann, Spectral properties of grain boundaries at small angles of rotation. J. Spectr. Theory 1 (2011), no. 2, pp. 197–219

DOI 10.4171/JST/9