Yves Colin de Verdiere

Yves Colin De Verdière

Description

Niveau: Supérieur, Licence, Bac+2
Scattering and correlations Yves Colin de Verdiere ? February 28, 2007 Introduction Let us consider the propagation of scalar waves with the speed v > 0 given by the wave equation utt?v2∆u = 0 outside a compact domain D in the Euclidean space Rd. Let us put ? = Rd \ D. We can assume for example Neumann boundary conditions. We will denote by ∆? the (self-adjoint) Laplace operator with the boundary conditions. So our stationary wave equation is the Helmholtz equation v2∆?f + ? 2f = 0 (1) with the boundary conditions. We consider a bounded interval I = [?2?, ? 2 +] ? ]0,+∞[ and the Hilbert subspace HI of L2(?) which is the image of the spectral projector PI of our operator ?v2∆?. Let us compute the integral kernel ?I(x, y) of PI defined by: PIf(x) = ∫ ? ?I(x, y)f(y)|dy| into 2 different ways: 1. From general spectral theory 2. From scattering theory. Taking the derivatives of ?I(x, y) w.r. to ?+, we get a simple general and exact relation between the correlation of scattered waves and the Green's function con- firming the calculations from Sanchez-Sesma and al.

• wave e0


• projector onto


• ?institut fourier


• scattered plane


• fourier transform


• using equation


• green's function


• full scattering

Sujets

• Fourier transform
• Green's function

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 9
• Langue: English

Extrait

Scattering and correlations

∗ YvesColindeVerdi`ere

February 28, 2007

Introduction Let us consider the propagation of scalar waves with the speedv >0 given by the 2 wave equationu tt −vΔu= 0 outside a compact domainDin the Euclidean space d d Rus put Ω =. LetR\D. Wecan assume for example Neumann boundary conditions. Wewill denote by Δ Ω the (self-adjoint) Laplace operator with the boundary conditions.So our stationary wave equation is the Helmholtz equation 2 2 vΔ Ω f+ω f(1)= 0 2 2 with the boundary conditions.We consider a bounded intervalI= [ωω ,]⊂ −+ 2 ]0,+∞[ and the Hilbert subspaceH I ofL(Ω) which is the image of the spectral 2 projectorP I of our operator−vΔ Ω . Let us compute the integral kernel Π I (x, y) ofP I defined by: Z P I f(x) =Π I (x, y)f(y)|dy| Ω into 2 different ways: 1. Fromgeneral spectral theory 2. Fromscattering theory. Taking the derivatives of Π I (x, y) w.r.toω + , we get a simple general and exact relation between the correlation of scattered waves and the Green’s function con- firming the calculations fromSanchez-Sesma and al.[March 2006]in the case whereDis a disk. ∗ InstitutFourier,Unit´emixtederechercheCNRS-UJF5582,BP74,38402-SaintMartin d’H`eresCedex(France);[e-mail]

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