Poisson statistics in higher-dimensional localized Anderson–Bernoulli regimes

Establish Poisson local eigenvalue statistics for higher-dimensional lattice Anderson–Bernoulli operators in the localized regimes where localization near the spectral edge is known, including dimensions two and higher.

Background

The paper proves Poisson local eigenvalue statistics for one-dimensional Anderson–Bernoulli Schrödinger operators at every fixed nonzero coupling and every Bernoulli bulk energy. In its comparison with higher-dimensional work, it notes that localization near spectral edges has been established in dimensions two, three, and higher, while the corresponding Poisson eigenvalue-statistics problem has not been resolved.

The unresolved problem is to extend the one-dimensional Poisson-statistics conclusion to these higher-dimensional localized regimes, where eigenfunction localization is available but the requisite local eigenvalue statistics remain unknown.