Abstract
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
, 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 \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.