Yves Colin de Verdiere

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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


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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 equationuttvΔu= 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 ΔΩthe (self-adjoint) Laplace operator with the boundary conditions.So our stationary wave equation is the Helmholtz equation 2 2 vΔΩf+ω f(1)= 0 2 2 with the boundary conditions.We consider a bounded intervalI= [ωω ,]+ 2 ]0,+[ and the Hilbert subspaceHIofL(Ω) which is the image of the spectral 2 projectorPIof our operatorvΔΩ. Let us compute the integral kernel ΠI(x, y) ofPIdefined by: Z PIf(x) =ΠI(x, y)f(y)|dy| Ω into 2 different ways: 1. Fromgeneral spectral theory 2. Fromscattering theory. Taking the derivatives of ΠI(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 dH`eresCedex(France);yves.colin-de-verdiere@ujf-grenoble.fr
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