Spectral properties of grain boundaries at small angles of rotation
Rainer Hempel, Martin Kohlmann · Journal of Spectral Theory · 2011
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 \colon \Bbb R^2 \to \Bbb R , we let V_\theta(x,y) = V(x,y) in the right half-plane \{ x \ge 0\} and V_\theta = V \circ M_{-\theta} in the left half-plane \{x < 0\} , where M_\theta \in \Bbb R^{2 \times 2} is the usual matrix describing rotation of the coordinates in \Bbb R^2 by an angle \theta . As a main result, it is shown that spectral gaps of the periodic Schrödinger operator H_0 = -\Delta + V fill with spectrum of R_\theta = -\Delta + V_\theta as 0 e \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.