Spectral Properties of Grain Boundaries at Small Angles of Rotation

Details Spectral Properties of Grain Boundaries at Small Angles of Rotation Spectral Properties of Grain Boundaries at Small Angles of Rotation

2010-09-21

arxiv.org/abs/1009.4039v2

J. Spectr. Theory 1 (2011) 197-219

Physics

Mathematical Physics

22 pages, 3 figures

Scientific paper

10.4171/JST/9

We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential $V \colon \R^2 \to \R$, 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 $M_\theta \in \R^{2 \times 2}$ is the usual matrix describing rotation of the coordinates in $\R^2$ by an angle $\theta$. As a main result, it is shown that spectral gaps of the periodic Schr\"odinger operator $H_0 = -\Delta + V$ fill with spectrum of $R_\theta = -\Delta + V_\theta$ as $0 \ne \theta \to 0$. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states.

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