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Anderson localization is a famous wave phenomenon that describes the absenceof diffusion of waves in a disordered medium. Here we generalize the landscapetheory of Anderson localization to general elliptic operators and complexboundary conditions using a probabilistic approach, and further investigatesome mathematical aspects of Anderson localization that are rarely discussedbefore. First, we observe that under the Neumann boundary condition, the lowenergy quantum states are localized on the boundary of the domain with highprobability. We provide a detailed explanation of this phenomenon using theconcept of extended subregions and obtain an analytical expression of thisprobability in the one-dimensional case. Second, we find that the quantumstates may be localized in multiple different subregions with high probabilityin the one-dimensional case and we derive an explicit expression of thisprobability for various boundary conditions. Finally, we examine a bifurcationphenomenon of the localization subregion as the strength of disorder varies.The critical threshold of bifurcation is analytically computed based on a toymodel and the dependence of the critical threshold on model parameters isanalyzed.

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