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.vibrations | eigenfunctions | elliptic operator | confinement A puzzling feature exhibited by complex, irregular, or inhomogeneous systems is their ability to maintain standing waves or vibrations in a very restricted subregion of their domain even in the absence of confining force or potential. The most striking manifestation of this phenomenon is the famous Anderson localization which has fascinated scientists and spurred an extraordinarily abundant wealth of literature in the past 50 years (1-6). Since Anderson's seminal work in 1958 it is known that a sufficiently large structural disorder can lead to strongly localized quantum states, which are standing waves of the Schrödinger equation. This phenomenon has now been experimentally demonstrated in optic or electromagnetic systems (7-9).Another well-known example of vibration confinement is the weak localization occurring in domains of irregular geometry and characterized by a slow decay of the vibration amplitude away from its main existence subregion (10-13), as opposed to the exponential decay of Anderson localization.Considerable progress has been made to understand the onset of weak and Anderson localization, as well as the possible link between these two types of localization (14). Even though a few approaches statistically relate the energy levels to the potential in disordered solids (15,16), there has been no general theory able to directly determine for any domain and any type of inhomogeneity the precise relationship between the g...
First, we show that the known ranges of boundedness in L p for the heat semigroup and Riesz transform of L, are sharp. In particular, the heat semigroup e −tL need not be bounded inThen we provide a complete description of all Sobolev spaces in which L admits a bounded functional calculus, in particular, where e −tL is bounded.Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces associated to L, that serves the range of p beyond [2n/(n + 2), 2n/(n − 2)]. It includes, in particular, characterizations by the sharp maximal function and the Riesz transform (for certain ranges of p), as well as the molecular decomposition and duality and interpolation theorems.R. -Soit L un opérateur elliptique du second ordre de formes de divergence, à coefficients complexes bornés et mesurables. Les opérateurs associés à L tels que le semi-groupe de la chaleur ou la transformée de Riesz ne sont en général pas de type Calderón-Zygmund et présentent des comportements différents de leurs analogues construits à partir du laplacien. Cet article a pour objectif de décrire de manière exhaustive les propriétés de ces opérateurs dans L p , dans les espaces de Sobolev ainsi que dans certains nouveaux espaces de Hardy naturellement associés à L. Tout d'abord, nous montrons que les plages de valeurs connues pour lesquelles ces opérateurs sont bornés en norme L p sont strictes. En particulier, le semi-groupe de la chaleur et la transformée de Riesz ne sont pas obligatoirement bornés si p ∈ [2n/(n + 2), 2n/(n − 2)]. Nous fournissons ensuite une description complète de tous les espaces de Sobolev pour lesquels L admet un calcul fonctionnel borné, en particulier, pour lesquels e −tL est borné.Puis, nous développons une théorie extensive des espaces de Hardy et de Lipschitz associés à L, pour les valeurs de p hors de [2n/(n + 2), 2n/(n − 2)]. Cette théorie comprend, en particulier, des caractérisations par la fonction maximale « dièse » et par la transformée de Riesz (pour certaines plages de p), ainsi que leur décomposition moléculaire, leur dualité et les théorèmes d'interpolation.ANNALES SCIENTIFIQUES DE L'ÉCOLE NORMALE SUPÉRIEURE 0012-9593/05
The amplitude of localized quantum states in random or disordered media may exhibit long-range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long-range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of one-dimensional systems, periodic or random.
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