Homogenized description of defect modes in periodic structures with localized defects

Vincent Duchêne (1) , Iva Vukicevic (2) , Michael I. Weinstein (3, 2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 APAM - Department of Applied Physics and Applied Mathematics [New York]
3 Dept of Mathematics, Columbia University

Waves in extended periodic structures are well-known to spatially disperse and decay 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 localized perturbations, leads to defect modes, states in which energy remains trapped and spatially localized. In this paper we present results on the bifurcation of such defect modes, associated with the emergence of discrete eigenvalues from the continuous spectrum.

Mots clés: Schrödinger operator; Floquet-Bloch states; defect modes; effective operator

Domaines: Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]; Physique mathématique [math-ph]

Vincent Duchêne, Iva Vukicevic, Michael I. Weinstein. Homogenized description of defect modes in periodic structures with localized defects. Communications in Mathematical Sciences, 2015, 13 (3), pp.777-823. ⟨10.4310/CMS.2015.v13.n3.a9⟩. ⟨hal-00773709⟩