Metal–insulator transition and extended states in d ≥ 3 for the Anderson model

Establish for the discrete Anderson Hamiltonian H = Δ + λV on Zd with d ≥ 3 and small coupling λ > 0 the existence of a metal–insulator transition: prove that the spectrum exhibits pure point spectrum with localized eigenfunctions near the spectral edges and continuous spectrum consisting of delocalized ("extended") eigenstates in the bulk energy regime.

Background

The notes study quantum diffusion for the random Schrödinger (Anderson) model H = Δ + λV on Zd at weak disorder. While localization near spectral edges is established in many settings, the behavior of the bulk in dimensions d ≥ 2 remains a central unresolved issue. Physically, one expects a transition between insulating (localized) and conducting (extended) regimes as energy varies, known as a metal–insulator transition, with a mobility edge separating them.

The authors highlight that this transition is conjectured in d ≥ 3 but remains out of reach of current mathematical techniques. This problem is tied to the long-standing extended states/mobility edge conjectures and is listed among Simon’s problems.

References

It is conjectured that in d≥3 there is a “metal-insulator” transition in the spectrum between localized eigenfunctions and pure point spectrum near the edges and a continuous spectrum consisting of “extended” or delocalized eigenstates in the bulk.

— Lecture notes on Quantum Diffusion and Random Matrix Theory  (2511.04380 - Hernández, 6 Nov 2025) in Introduction, Section 1

Beyond the accuracy of the electronic structure itself, a related open issue is how the degree of structural disorder controls the crossover between extended and localized electronic states, and how this crossover is reflected in measurable transport properties.

— Electron transport in amorphous materials: from localization to predictive transport modeling  (2608.17814 - Lee, 18 Aug 2026) in Section 5, Summary and outlook