Defect Modes and Homogenization of Periodic Schr\"odinger Operators

We consider the discrete eigenvalues of the operator $H_{\text{eps}}=-\mathrm{\Delta }+V(\text{x})+{\text{eps}}^{2}Q(\text{eps}\text{x})$, where $V(\text{x})$ is periodic and $Q(\text{y})$ is localized on ${\mathbb{R}}^d,d\ge 1$. For $\text{eps}>0$ and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schr"odinger operator, $H_0=-{\mathrm{\Delta }}_{\text{x}}+V(\text{x})$, into spectral gaps. The nature of the bifurcation depends on the homogenized Schr"odinger operator $L_{A,Q}=-{\mathrm{\nabla }}_{\text{y}}\cdot A{\mathrm{\nabla }}_{\text{y}}+Q(\text{y})$. Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of the bifurcation.