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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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Cited by 20 (0 self)
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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...
Spectral Properties of Classical Waves in High Contrast Periodic Media
 SIAM J. Appl. Math
, 1998
"... We introduce and investigate the band gap sturcture of the frequency spectrum for classical electromagnetic and acoustic waves in a high contrast twocomponent periodic medium. The asymptotics with respect to the high contrast is considered. The limit medium is described in terms of appropriate sefl ..."
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Cited by 15 (5 self)
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We introduce and investigate the band gap sturcture of the frequency spectrum for classical electromagnetic and acoustic waves in a high contrast twocomponent periodic medium. The asymptotics with respect to the high contrast is considered. The limit medium is described in terms of appropriate sefladjoint operators and the convergence to the limit is proven. These limit operators give an idea of the spectral structure and suggest new numerical approaches as well. The results are obtained in arbitrary dimension and for rather general geometry of the medium. In particular, 2D photonic band gap structures and their acoustic analogs are covered. Key words: propagation of electromagnetic and acoustic waves, band gap structure of the spectrum, periodic dielectrics, periodic acoustic media, high contrast. AMS subject classiifcation: 35B27, 73D25, 78A45. To appear in SIAM Journal on Applied Mathematics. y This work was supported by the U.S. Air Force Grant F496209410172DEF z This ...
Localized Classical Waves Created by Defects
 J. Stat. Phys
, 1997
"... We study acoustic and electromagnetic waves in a periodic medium (or any other background medium with a spectral gap) disturbed by a single defect, i.e., a local disturbance analogous to a well potential in solid state physics. We show that defects do not change the essential spectrum of the associa ..."
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Cited by 11 (2 self)
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We study acoustic and electromagnetic waves in a periodic medium (or any other background medium with a spectral gap) disturbed by a single defect, i.e., a local disturbance analogous to a well potential in solid state physics. We show that defects do not change the essential spectrum of the associated nonnegative operators and can only create isolated eigenvalues of finite multiplicity in a gap of the periodic medium, with the eigenmodes decaying exponentially. We give a constructive and simple description of defects in acoustic and dielectric media, including a simple condition on the parameters of the medium and of the defect, which ensures the rise of a localized eigenmode with the corresponding eigenvalue in a specified subinterval of the given gap of the periodic medium. KEY WORDS: Electromagnetic waves, acoustic waves, localization, photonic localization, periodic medium, spectral gap, photonic crystals, photonic band gaps, defects. 1 Introduction Localization of classical wav...