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 Schrödinger operator, HQ = −∂ 2 x + Q(x), has L 2 (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 HQ+q , where q is spatially localized and highly oscillatory in the sense that its Fourier transform, q is concentrated at high frequencies. Our assumptions imply that q may be pointwise large but q is small in an average sense. For the special case where q (x) = q(x, x/ ) 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 4 from the lower edge of the spectral gap. We obtain the leading order asymptotics of the bifurcating eigenvalues and eigenfunctions. Consider the (b * ) th spectral band (b * ≥ 1) of HQ. Underlying this bifurcation is an effective Hamiltonian associated with the lower spectral band edge: H eff = −∂xA b * ,eff ∂x − 2 B b * ,eff × δ(x) where δ(x) is the Dirac distribution, and effective-medium parameters A b * ,eff , B b * ,eff > 0 are explicit and independent of . 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.

Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures

“…In contrast to the edge state bifurcations obtained via Theorem 4.1, these bifurcations are not topologically protected; they may be destroyed by a localized perturbation of the edge. The formal multiplescale bifurcation analysis presented in Section 5.1 can be made rigorous along the lines of [31,32,33,34]; see also [35].…”

The global bifurcation picture for ground states in nonlinear Schrodinger equations

“…Hence, any curve approaching the E = 0 boundary bifurcates from the edge of the continuous spectrum of the linearization at its limit. Such bifurcations are notoriously difficult to handle, and, while some progress has been made in the case of periodic potentials, see [5], the problem remains open for our type of potentials.…”

Dynamics of Partial Differential Equations

“…However, while the analysis of bifurcation from discrete spectrum leads to a finite dimensional (nonlinear algebraic) bifurcation equation, bifurcation from the continuous spectrum leads to an infinite dimensional bifurcation equation, a nonlinear homogenized partial differential equation (3.37); see also [25,26]. Examples of recent applications of these and related ideas to other systems appear in [28,29,[31][32][33][34][35]61].…”