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