Defect Modes and Homogenization of Periodic Schr\"odinger Operators

Abstract

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.Comment: 26 pages, 3 figures, to appear SIAM J. Math. Ana