The landscape and its impact on localization can be determined rigorously by solving one special boundary problem. The height of the landscape along its valleys determines the strength of the coupling between the 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. This mechanism partitions the system into weakly coupled subregions. 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. 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. One of its most striking and famous manifestations is Anderson localization, responsible for instance for the metal-insulator transition in disordered alloys.
It is induced by the presence of an inhomogeneous medium, a complex geometry, or a quenched disorder. Localization of stationary waves occurs in a large variety of vibrating systems, whether mechanical, acoustical, optical, or quantum.
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