Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes

Abstract

Let $Q(x)$ denote a periodic function on the real line. The Schr\"odinger operator, $H_Q=-\partial_x^2+Q(x)$, has $L^2(\mathbb{R})-$ spectrum equal to the union of closed real intervals separated by open spectral gaps. In this article we study the bifurcation of discrete eigenvalues (point spectrum) into the spectral gaps for the operator $H_{Q+q_\epsilon}$, where $q_\epsilon$ is spatially localized and highly oscillatory in the sense that its Fourier transform, $\widehat{q}_\epsilon$ is concentrated at high frequencies. Our assumptions imply that $q_\epsilon$ may be pointwise large but $q_\epsilon$ is small in an average sense. For the special case where $q_\epsilon(x)=q(x,x/\epsilon)$ with $q(x,y)$ smooth, real-valued, localized in $x$, and periodic or almost periodic in $y$, the bifurcating eigenvalues are at a distance of order $\epsilon^4$ from the lower edge of the spectral gap. We obtain the leading order asymptotics of the bifurcating eigenvalues and eigenfunctions. Underlying this bifurcation is an effective Hamiltonian associated with the lower edge of the $(b_*)^{\rm th}$ spectral band: $H^\epsilon_{\rm eff}=-\partial_x A_{b_*,\rm eff}\partial_x - \epsilon^2 B_{b_*,\rm eff} \times \delta(x)$ where $\delta(x)$ is the Dirac distribution, and effective-medium parameters $A_{b_*,\rm eff},B_{b_*,\rm eff}>0$ are explicit and independent of $\epsilon$. The potentials we consider are a natural model for wave propagation in a medium with localized, high-contrast and rapid fluctuations in material parameters about a background periodic medium.