Defect Modes and Homogenization of Periodic Schrödinger Operators

Abstract

We consider the discrete eigenvalues of the operator $H_\eps=-Δ+V(\x)+\eps^2Q(\eps\x)$, where $V(\x)$ is periodic and $Q(\y)$ is localized on ${\mathbb{R}}^d,d\ge 1$. For $\eps>0$ and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schrödinger operator, $H_0 = -Δ_\x+V(\x)$, into spectral gaps. The nature of the bifurcation depends on the homogenized Schrödinger operator $L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y)$. Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of the bifurcation.

26 pages, 3 figures, to appear SIAM J. Math. Anal

References in corpus (4)

• On the eigenvalues of operators with gaps. Application to Dirac operators
• Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential
• Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential
• Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps

Cited by in corpus (12)

• Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene
• Field Patterns: A New Mathematical Object
• High-frequency homogenization of zero frequency stop band photonic and phononic crystals
• Subwavelength localized modes for acoustic waves in bubbly crystals with a defect
• High frequency homogenization for travelling waves in periodic media
• Characterization of edge states in perturbed honeycomb structures
• Photonic realization of topologically protected bound states in domain-wall waveguide arrays
• Wavepackets in inhomogeneous periodic media: effective particle-field dynamics and Berry curvature
• Dirac operators and domain walls
• Defect resonances of truncated crystal structures
• Emergence of Periodic Structure from Maximizing the Lifetime of a Bound State Coupled to Radiation
• Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes