Some mathematical aspects of Anderson localization: boundary effect, multimodality, and bifurcation

TL;DR

This paper extends the landscape theory of Anderson localization to complex boundary conditions, revealing boundary localization, multimodality, and bifurcation phenomena through probabilistic and analytical methods.

Contribution

It generalizes Anderson localization theory to elliptic operators with complex boundaries and analytically characterizes boundary effects, multimodality, and bifurcation phenomena.

Findings

Low energy states localize on the boundary under Neumann conditions

Quantum states can localize in multiple subregions with high probability

Bifurcation of localization regions occurs as disorder strength varies

Abstract

Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered medium. Here we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach, and further investigate some mathematical aspects of Anderson localization that are rarely discussed before. First, we observe that under the Neumann boundary condition, the low energy quantum states are localized on the boundary of the domain with high probability. We provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional case. Second, we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary conditions. Finally, we examine a bifurcation phenomenon of the localization subregion as the strength of disorder varies. The critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed.