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Localization of Classical Waves I: Acoustic Waves.
 Commun. Math. Phys
, 1996
"... We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existe ..."
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Cited by 37 (0 self)
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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the selfadjoint operators A = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almost the wh...
Localization of Classical Waves II: Electromagnetic Waves.
 Commun. Math. Phys
, 1997
"... We consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r \Theta 1 " 0 (x) r \Theta has a gap in the spectrum, were r \Thet ..."
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We consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r \Theta 1 " 0 (x) r \Theta has a gap in the spectrum, were r \Theta \Psi = r\Theta\Psi. We prove the existence of localized waves, i.e., finite energy solutions of Maxwell's equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times. Localization of electromagnetic waves is a consequence of Anderson localization for the selfadjoint operators M = r \Theta 1 "(x) r \Theta . We prove that, in the random medium described by "(x), the random operator M exhibits Anderson localization inside the gap in the spectrum of M 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almo...
Generalized Eigenfunctions for Waves in Inhomogeneous Media
, 2000
"... Many wave propagation phenomena in classical physics are governed by equations that can be recast in Schrodinger form. In this approach the classical wave equation (e.g., Maxwell's equations, acoustic equation, elastic equation) is rewritten in Schrodinger form, leading to the study of the spectral ..."
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Cited by 11 (5 self)
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Many wave propagation phenomena in classical physics are governed by equations that can be recast in Schrodinger form. In this approach the classical wave equation (e.g., Maxwell's equations, acoustic equation, elastic equation) is rewritten in Schrodinger form, leading to the study of the spectral theory of its classical wave operator, a selfadjoint, partial differential operator on a Hilbert space of vectorvalued, square integrable functions. Physically interesting inhomogeneous media give rise to nonsmooth coefficients. We construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients. Our construction yields polynomially bounded generalized eigenfunctions, the set of generalized eigenvalues forming a subset of the operator's spectrum with full spectral measure. 1 Introduction A selfadjoint operator in a finite dimensional Hilbert spaces can always be diagonalized in an orthonormal basis of eigenvectors (i.e., it has a complete set of eig...