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Article Dans Une Revue SIAM Journal on Mathematical Analysis Année : 2015

Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes

Résumé

Let $Q(x)$ denote a periodic function on the real line. The Schrödinger operator, $H_Q=-\partial_x^2+Q(x)$, has $L^2(\mathbb R)-$ spectrum equal to the union of closed real intervals separated by open spectral gaps. It is known that a spatially localized and small perturbation of $H_Q$, say $H_{Q+\epsilon V}$, where $V\in L^1$, induces the bifurcation of discrete eigenvalues (point spectrum) into the spectral gaps at a distance of order $\epsilon^2$ from the spectral gap. In this article we study the bifurcation of a discrete spectrum for the operator $H_{Q+q_\epsilon}$, where $q_\epsilon$ is spatially localized and tends to zero weakly. For the special case where $q_\epsilon(x)=q(x,x/\epsilon)$ 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 $\epsilon^4$ from the lower edge of the spectral gap. We obtain detailed asymptotics of the bifurcating eigenvalues and eigenfunctions. Underlying this bifurcation is an effective Hamiltonian associated with the lower spectral and edge of the $(b_*)^{\rm th}$ spectral band: $H^\epsilon_{\rm eff}=-\partial_x A_{b_*,\rm eff}\partial_x - \epsilon^2 B_{b_*,\rm eff} \times \delta(x)$ where $\delta(x)$ is the Dirac distribution, and effective-medium parameters $A_{b_*,\rm eff},B_{b_*,\rm eff}>0$ are explicit and independent of $\epsilon$. The potentials we consider are a natural model of localized rapid fluctuations in material parameters about a background periodic medium.

Dates et versions

hal-01059211 , version 1 (29-08-2014)

Identifiants

Citer

Vincent Duchêne, Iva Vukicevic, Michael I. Weinstein. Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes. SIAM Journal on Mathematical Analysis, 2015, 47 (5), pp.3832-3883. ⟨10.1137/140980302⟩. ⟨hal-01059211⟩
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