Physique de la Matière Condensée, Ecole Polytechnique, Centre National de la Recherche Scientifique, 91128 Palaiseau, France.
Proc Natl Acad Sci U S A. 2012 Sep 11;109(37):14761-6. doi: 10.1073/pnas.1120432109. Epub 2012 Aug 27.
Localization of stationary waves occurs in a large variety of vibrating systems, whether mechanical, acoustical, optical, or quantum. It is induced by the presence of an inhomogeneous medium, a complex geometry, or a quenched disorder. One of its most striking and famous manifestations is Anderson localization, responsible for instance for the metal-insulator transition in disordered alloys. Yet, despite an enormous body of related literature, a clear and unified picture of localization is still to be found, as well as the exact relationship between its many manifestations. In this paper, we demonstrate that both Anderson and weak localizations originate from the same universal mechanism, acting on any type of vibration, in any dimension, and for any domain shape. This mechanism partitions the system into weakly coupled subregions. The boundaries of these subregions correspond to the valleys of a hidden landscape that emerges from the interplay between the wave operator and the system geometry. The height of the landscape along its valleys determines the strength of the coupling between the subregions. The landscape and its impact on localization can be determined rigorously by solving one special boundary problem. This theory allows one to predict the localization properties, the confining regions, and to estimate the energy of the vibrational eigenmodes through the properties of one geometrical object. In particular, Anderson localization can be understood as a special case of weak localization in a very rough landscape.
驻波的局域化现象存在于各种振动系统中，无论是机械的、声学的、光学的还是量子的。它是由不均匀介质、复杂的几何形状或淬火无序引起的。它最显著和著名的表现形式之一是安德森局域化，例如，它负责无序合金中的金属-绝缘体转变。然而，尽管有大量相关文献，但仍未找到局域化的清晰统一图像，以及其多种表现形式之间的确切关系。在本文中，我们证明了安德森局域化和弱局域化都源于相同的普遍机制，该机制作用于任何类型的振动、任何维度和任何域形状。该机制将系统划分为弱耦合的子区域。这些子区域的边界对应于隐藏景观的山谷，该景观是由波算子和系统几何形状的相互作用产生的。景观沿着其山谷的高度决定了子区域之间的耦合强度。通过求解一个特殊的边界问题，可以严格确定景观及其对局域化的影响。该理论允许通过一个几何对象的特性来预测局域化性质、限制区域，并估计振动本征模式的能量。特别是，安德森局域化可以被理解为非常粗糙景观中的弱局域化的一个特例。
