## Localization of Classical Waves I: Acoustic Waves. (1996)

Venue: | Commun. Math. Phys |

Citations: | 40 - 1 self |

### BibTeX

@ARTICLE{Figotin96localizationof,

author = {Alexander Figotin and Abel Klein},

title = {Localization of Classical Waves I: Acoustic Waves.},

journal = {Commun. Math. Phys},

year = {1996},

volume = {180},

pages = {439--482}

}

### Years of Citing Articles

### OpenURL

### Abstract

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...

### Citations

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Citation Context ... addition, the decomposition of oe(A g;! ) into pure point spectrum, absolutely continuous spectrum and singular continuous spectrum is also independent of the choice of ! with probability one (e.g., =-=[PF]-=-). In this article we are interested in the phenomenon of localization. According to the philosophy of Anderson localization we will assume that the operator A 0 has at least one gap in the spectrum. ... |

164 | Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-body Schrödinger - Agmon - 1982 |

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The spectral theory of periodic differential equations. Scottish Acad
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24 |
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- Figotin, Klein
- 1994
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21 |
Absence of Diffusion
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11 | applications of materials exhibiting photonic band gaps - Development - 1993 |

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4 |
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3 |
Localization for Random and Quasiperiodic
- Spencer
- 1988
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Citation Context ...;L0s1 L 2 0 9 = ;s1 \Gamma 1 L p 0 : (194) Theorem 32 suffices to prove Theorem 6 with the stronger hypothesis j ? 2d in (23). To deal with the weaker hypothesis j ? d we adapt an argument of Spencer =-=[Sp]-=- to obtain the starting hypothesis (Q1) from a weaker (and easier to verify) hypothesis. Definition 34 Let the operator A be as in (10) with (11). Givens? 0, E ? 0, x 2 qZ d and L 2 2qN , Ls4q, we say... |

2 | Band-Gap Structure of Spectra of Periodic and Acoustic - Figotin, Kuchment |

2 | Acoustic Anderson Localization, in "Random Media and Composites - Maynard - 1988 |

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1 |
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Citation Context ...ustic and electromagnetic waves. Acoustic waves are treated in this paper; electromagnetic waves will be discussed in a sequel [FK3]. Localized classical waves created by local defects are studied in =-=[FK4]-=-. We assume that the underlying periodic medium has a gap in the spectrum. The existence of periodic media exhibiting gaps in the spectrum is proved for acoustic and 2Dperiodic dielectric structures [... |