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A spatially localized initial condition for an energy-conserving waveequation with periodic coefficients disperses (spatially spreads) and decays inamplitude as time advances. This dispersion is associated with the continuousspectrum of the underlying differential operator and the absence of discreteeigenvalues. The introduction of spatially localized perturbations in aperiodic medium leads to defect modes, states in which energy remains trappedand spatially localized. In this paper we study weak, localized perturbationsof one-dimensional periodic Schr\"odinger operators. Such perturbations giverise to such defect modes, and are associated with the emergence of discreteeigenvalues from the continuous spectrum. Since these isolated eigenvalues arelocated near a spectral band edge, there is strong scale-separation between themedium period and the localization length of the defect mode. Bound statestherefore have a multi-scale structure: a "carrier Bloch wave" times a "waveenvelope", which is governed by a homogenized Schr\"odinger operator withassociated effective mass, depending on the spectral band edge which is thesite of the bifurcation. Our analysis is based on a reformulation of theeigenvalue problem in Bloch quasi-momentum space, using the Gelfand-Blochtransform and a Lyapunov-Schmidt reduction to a closed equation for thenear-band-edge frequency components of the bound state. A rescaling of thelatter equation yields the homogenized effective equation for the waveenvelope, and approximations to bifurcating eigenvalues and eigenfunctions.

Homogenized description of defect modes in periodic structures with localized defects