title: Effective Confining Potential of Quantum States in Disordered Media.
abstract: 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.
authors.affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA.
authors.forename: Douglas N
authors.initials: DN
authors.lastname: Arnold
authors.affiliation: Université Paris-Sud, Laboratoire de Mathématiques, CNRS, UMR 8658, Orsay F-91405, France.
authors.forename: Guy
authors.initials: G
authors.lastname: David
authors.affiliation: Mathematics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
authors.forename: David
authors.initials: D
authors.lastname: Jerison
authors.affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA.
authors.forename: Svitlana
authors.initials: S
authors.lastname: Mayboroda
authors.affiliation: Physique de la Matière Condensée, Ecole Polytechnique, CNRS, Palaiseau F-91128, France.
authors.forename: Marcel
authors.initials: M
authors.lastname: Filoche