Localization landscape and localization potential in disordered or complex structures
Marcel Filoche (Laboratoire de Physique de la Matière Condensée, Ecole Polytechnique)
Standing waves in disordered or complex systems can be subject to a strange and intriguing phenomenon which has puzzled the physics and mathematical communities for more than 60 years, namely wave localization. This phenomenon consists of a concentration (or a focusing) of the wave energy in a very restricted sub-region of the entire domain. It has been evidenced experimentally in mechanics, acoustics and quantum physics. Determining the conditions for the onset of localization, depending on the disorder amplitude, the energy, or the wave type, is the aim of many theoretical studies. We will present a theory that unifies different types of localization within a single mathematical framework . To that end, we will introduce the notion of « localization landscape », solution to an associated Dirichlet problem. Going further, this will enable us to define an « effective localization potential », providing a new insight into the confinement of the waves in disordered media. This potential allows us to predict the localization region, the energies of the localized modes, the density of states, and the long range decay of the wave functions. We will present experimental and numerical examples of this theory in mechanics, in semiconductor physics, and in molecular systems, as well as theoretical perspectives with cold atom systems.
 M. Filoche and S. Mayboroda, Universal mechanism for Anderson and weak localization, PNAS 109, 14761–14766 (2012).