The effective confining potential of quantum states in disordered media

Douglas L. Arnold (1) , Guy David (2) , David Jerison (3) , Svitlana Mayboroda (1) , Marcel Filoche (4)

1 UMN-MATH - School of Mathematics
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 MIT - Massachusetts Institute of Technology
4 LPMC - Laboratoire de physique de la matière condensée

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 the Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of 1D systems, periodic or random.

Mots clés: Anderson localization; Agmon distance; quantum confinement; density of states

Domaines: Systèmes désordonnés et réseaux de neurones [cond-mat.dis-nn]; Matière Molle [cond-mat.soft]; Pneumologie et système respiratoire; Optique / photonique

Douglas L. Arnold, Guy David, David Jerison, Svitlana Mayboroda, Marcel Filoche. The effective confining potential of quantum states in disordered media. 2015. ⟨hal-01150766⟩