  We consider the discrete eigenvalues of the operator
$H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x)$, where $V(\x)$ is periodic and $Q(\y)$
is localized on $\R^d,\ \ d\ge1$. For $\eps>0$ and sufficiently small, discrete
eigenvalues may bifurcate (emerge) from spectral band edges of the periodic
Schr\"odinger operator, $H_0 = -\Delta_\x+V(\x)$, into spectral gaps. The
nature of the bifurcation depends on the homogenized Schr\"odinger operator
$L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y)$. Here, $A$ denotes the inverse
effective mass matrix, associated with the spectral band edge, which is the
site of the bifurcation.
