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 self-adjoint 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...

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 self-adjoint 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...

We consider lattice versions of Maxwell's equations and of the equation that governs the propagation of acoustic waves in a random medium. The vector nature of electromagnetic waves is fully taken into account. The medium is assumed to be a small perturbation of a periodic one. We prove rigorously that localized eigenstates arise in a vicinity of the edges of the gaps in the spectrum. A key ingredient is a new Wegner-type estimate for a class of lattice operators with off-diagonal disorder. Key words: localization, random media, electromagnetic waves, acoustic waves, lattice model. The work is supported by U. S. Air Force grant AFOSR-91-0243 y The work is partially supported by NSF grant DMS 9208029 1 Introduction Decades after P. W. Anderson [1] described the remarkable phenomenon of the localization in space of electron wave functions in disordered solids, physicists have begun to ask whether other waves, say electromagnetic or acoustic, can be localized if the propagating m...

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...

. We consider three dimensional lossless periodic dielectric (photonic crystals) and acoustic media having a gap in the spectrum. If such a periodic medium is perturbed by a strong enough defect, defect eigenmodes arise, localized exponentially around the defect, with the corresponding eigenvalues in the gap. We use a modified Birman-Schwinger method to derive equations for these eigenmodes and corresponding eigenvalues in the gap, in terms of the spectral attributes of an auxiliary Hilbert-Schmidt operator. We prove that in three dimensions, under some natural conditions on the periodic background, the number of eigenvalues generated in a gap of the periodic operator is finite, and give an estimate on the number of these midgap eigenvalues. In particular, we show that if the defect is weak there are no midgap eigenvalues. Key words. photonic crystal, photonic bandgap, periodic acoustic medium, periodic dielectric medium, midgap states, defect modes, localization of light AMS subject...

We consider the problems of existence and structure of gaps (pseudogaps) in the spectra associated with Maxwell equations and equations that governs the propagation of acoustic waves in periodic two component media. The dielectric constant " is assumed to be real and positive, and the value of " = " b on the background is supposed to be essentially larger than the value of " = " a on the embedded component. We prove the existence of pseudogaps in the spectra of the relevant operators. In particular, we give an accurate treatment of the term "pseudogap". We also show that if the contrast " b =" a approaches infinity then the bands of the spectrum shrink to a discrete set which can be identified with the set of eigenvalues of a Neumann type boundary value problem and, thus, can be effectively calculated. Key words: waves, periodic dielectrics, periodic acoustic media, pseudogaps in the spectrum. Introduction The idea of finding and designing periodic and disordered dielectric materials...

this paper, experimental results are presented for what we believe to be the first experimental realization and characterization of stationary light sources of this kind

Abstract. We extend the Bloch-decomposition based time-splitting spectral method introduced in an ear-lier paper [13] to the case of (non-)linear Klein-Gordon equations. This provides us with an unconditionally stable numerical method which achieves spectral convergence in space, even in the case where the peri-odic coefficients are highly oscillatory and/or discontinuous. A comparison to a traditional pseudo-spectral method and to a finite difference/volume scheme shows the superiority of our method. We further estimate the stability of our scheme in the presence of random perturbations and give numerical evidence for the well-known phenomenon of Anderson’s localization. version: February 15, 2008 1.

Abstract: Melosh [1996] has suggested that acoustic fluidization could provide an alternative to theories that are invoked as explanations for why some faults appear to be weak. We show that there is a subtle but profound inconsistency in the theory that unfortunately invalidates the results. We propose possible remedies but must In the standard rebound theory of earthquakes, deformation elastic energy is progressively stored in the crust and is suddenly released in an earthquake when a threshold is reached. The Ruina-Dieterich friction laws [Dieterich, 1972; 1978; Ruina, 1983] constitute the basic ingredient used to describe the interaction between the two sides