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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 3

### Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

### Homogenized description of defect modes in periodic structures with localized defects

Pages: 777 – 823

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n3.a9

#### Authors

#### Abstract

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, $\mathcal{O}(\lambda), 0 \lt \lambda \ll 1$, localized perturbations of one-dimensional periodic Schrödinger 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 $( \sim \textrm{order} \; 1$) and the localization length of the defect mode $( \sim \textrm{order} \; | \; \textrm{defect} \; \textrm{eigenvalue} \; |^{-\frac{1}{2}} = \lambda^{-1} \gg 1)$. Bound states therefore have a multi-scale structure: a “carrier Bloch wave” $\times$ a “wave envelope”, which is governed by a homogenized Schrödinger 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.

#### Keywords

Schrödinger operator, defect modes, effective operator, Floquet-Bloch states

#### 2010 Mathematics Subject Classification

34C23, 34E10, 34L15, 34L40, 35J10

Published 3 March 2015