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Vincent Duchêne

Publications and source records attributed to Vincent Duchêne.

23 records · Page 2Linked to original sources
arXiv2015

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

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. In this article we study the bifurcation of discrete eigenvalues (point spectrum) into the spectral gaps for the operator $H_{Q+q_ε}$, where $q_ε$ is spatially localized and highly oscillatory in the sense that its Fourier transform, $\widehat{q}_ε$ is concentrated at high frequencies. Our assumptions imply that $q_ε$ may be pointwise large but $q_ε$ is small in an average sense. For the special case where $q_ε(x)=q(x,x/ε)$ 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 $ε^4$ from the lower edge of the spectral gap. We obtain the leading order asymptotics of the bifurcating eigenvalues and eigenfunctions. Underlying this bifurcation is an effective Hamiltonian associated with the lower edge of the $(b_*)^{\rm th}$ spectral band: $H^ε_{\rm eff}=-\partial_x A_{b_*,\rm eff}\partial_x - ε^2 B_{b_*,\rm eff} \times δ(x)$ where $δ(x)$ is the Dirac distribution, and effective-medium parameters $A_{b_*,\rm eff},B_{b_*,\rm eff}>0$ are explicit and independent of $ε$. 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.

math-ph↗
arXiv2014

Homogenized description of defect modes in periodic structures with localized defects

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, 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 and the localization length of the defect mode. 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.

math-ph↗
arXiv2012

Scattering and Localization Properties of Highly Oscillatory Potentials

We investigate scattering, localization and dispersive time-decay properties for the one-dimensional Schrödinger equation with a rapidly oscillating and spatially localized potential, $q_ε=q(x,x/ε)$, where $q(x,y)$ is periodic and mean zero with respect to $y$. Such potentials model a microstructured medium. Homogenization theory fails to capture the correct low-energy ($k$ small) behavior of scattering quantities, e.g. the transmission coefficient, $t^{q_ε}(k)$, as $ε$ tends to zero. We derive an effective potential well, $σ^ε_{eff}(x)=-ε^2Λ_{eff}(x)$, such that $t^{q_ε}(k)-t^{σ^ε_{eff}}(k)$ is uniformly small on $\mathbb{R}$ and small in any bounded subset of a suitable complex strip. Within such a bounded subset, the scaled transmission coefficient has a universal form, depending on a single parameter, which is computable from the effective potential. A consequence is that if $ε$, the scale of oscillation of the microstructure potential, is sufficiently small, then there is a pole of the transmission coefficient (and hence of the resolvent) in the upper half plane, on the imaginary axis at a distance of order $ε^2$ from zero. It follows that the Schrödinger operator $H_{q_ε}=-\partial_x^2+q_ε(x)$ has an $L^2$ bound state with negative energy situated at a distance $O(ε^4)$ from the edge of the continuous spectrum. Finally, we use this detailed information to prove a local energy time-decay estimate of the time-dependent Schrödinger equation.

math.AP↗
arXiv2011

Scattering, homogenization and interface effects for oscillatory potentials with strong singularities

We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schrödinger equation \[ H_εψ:= (-\partial_x^2 + V_0(x) + q(x,x/ε)) ψ= k^2 ψ, \] for $k\in\RR$ and $ε<< 1$. Here, $q(.,y+1)=q(.,y)$, has mean zero and $|V_0(x)+q(x,.)|$ goes to zero as $|x|$ goes to infinity. The distorted plane waves of $H_ε$ are solutions of the form: $e^{\pm ikx}+u^s_\pm(x;k)$, $u^s_\pm$ outgoing as $|x|$ goes to infinity. We derive their $ε$ small asymptotic behavior, from which the asymptotic behavior of scattering quantities such as the transmission coefficient, $t^ε(k)$, follow. Let $t_0^{hom}(k)$ denote the homogenized transmission coefficient associated with the average potential $V_0$. If the potential is smooth, then classical homogenization theory gives asymptotic expansions of, for example, distorted plane waves, and transmission and reflection coefficients. Singularities of $V_0$ or discontinuities of $q_ε$, that our theory admits, are "interfaces" across which a solution must satisfy interface conditions (continuity or jump conditions). To satisfy these conditions it is necessary to introduce interface correctors, which are highly oscillatory in $ε$. A consequence of our main results is that $t^ε(k)-t_0^{hom}(k)$, the error in the homogenized transmission coefficient is (i) $O(ε^2)$ if $q_ε$ is continuous and (ii) $O(ε)$ if $q_ε$ has discontinuities. Moreover, in the discontinuous case the correctors are highly oscillatory in $ε$, so that a first order corrector is not well-defined. The analysis is based on a (pre-conditioned) Lippman-Schwinger equation, introduced in [SIAM J. Mult. Mod. Sim. (3), 3 (2005), pp. 477--521].

math-ph↗