Bifurcation of localized eigenstates of perturbed periodic Schrödinger operators

A spatially localized initial condition for an energy-conserving wave equation with periodic coefficients disperses (spatially spreads) and decays 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 the wave is spatially localized and periodic in time. These modes are associated with eigenvalues which bifurcate from the continuous spectrum induced by the perturbation. This thesis investigates specific families of perturbations of one-dimensional periodic Schrödinger operators and studies the resulting bifurcating eigenvalues from the unperturbed continuous spectrum. For Q(x) a real-valued periodic function, the Schrödinger operator H_Q = -∂_x² + Q(x) has a continuous spectrum equal to the union of closed intervals, called spectral bands, separated by open spectral gaps. We find that upon the introduction of a bounded, ``small'', and sufficiently decaying perturbation W(x), the spectrum of H_{Q+W} has discrete eigenvalues (with corresponding eigenstates which are exponentially decaying in |x|) which lie in the open spectral gaps of H_Q. Our analysis covers two large classes of perturbations W(x): 1. W(x) = λ V(x), 0<λ ≪ 1, and V(x) sufficiently rapidly decaying as x → ± ∞; 2. W(x) = q(x, x/ε), 0<ε ≪ 1, where x ⟼ q(x,y) is spatially localized, q(x,y+1) = q(x,y) for x ∈ ℝ, and y ⟼ q(x,y) has mean zero. In Case 1. W(x) corresponds to a small and localized absolute change in the medium's material properties. In Case 2. W(x) corresponds to a high-contrast microstructure. Q(x) + W(x) may be pointwise very large, but on average it is a small perturbation of Q(x).