Defect Modes and Homogenization of Periodic Schr\"odinger Operators

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.