OP-3090OpenMathematical physics

Delocalization in the 3D Anderson model

For $H_\lambda=-\Delta+\lambda V_\omega$ on $\ell^2(\mathbb Z^3)$ , with independent potentials uniform on $[-1,1]$ , prove that all sufficiently small $\lambda>0$ admit a nonempty interval in the free band with almost-sure nonzero absolutely continuous spectrum

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Background

The Anderson Hamiltonian $H=-\Delta+\lambda V_\omega$ on $\ell^2(\mathbb{Z}^d)$ , with i.i.d. random potential $V_\omega$ , models a quantum particle in a disordered medium. Physics (the scaling theory of localization) predicts that for $d\ge3$ and weak disorder $\lambda$ there is a mobility edge separating a low-energy/strong-disorder localized regime from a delocalized regime with absolutely continuous (a.c.) spectrum and diffusive transport.

What is known: Multiscale analysis and fractional-moment methods establish localization in one dimension and, under appropriate distribution hypotheses, at strong disorder or spectral edges in higher dimensions. They do not prove full-spectrum localization for arbitrarily weak disorder in the standard scalar two-dimensional model; that separate question is M12. Nonzero a.c. spectrum at weak disorder for the standard scalar i.i.d. cubic-lattice model remains open. This statement does not exclude results for other random operators, internal degrees of freedom or non-Euclidean graphs.

An important solved comparison is the Bethe lattice (regular tree), where Klein proved persistence of a.c. spectrum at weak disorder, followed by further results of Aizenman, Warzel, Sims and others. A tree is not the cubic lattice. Diffusive transport is a stronger, separate dynamical target; a.c. spectrum alone does not supply it.

Mobility-edge subproblem (formerly M3): for this same fixed model, establish a boundary between localized and a.c. energy regions, then bound its location or regularity. Do not assume there is exactly one such boundary or identify spectral localization with every possible transport criterion.

Progress & source assessment

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

References

[1]Aizenman & Warzel, 'Random Operators: Disorder Effects on Quantum Spectra and Dynamics' (AMS, 2015) link
[2]Klein, 'Extended states in the Anderson model on the Bethe lattice', Adv. Math. 133 (1998) link
[3]Fröhlich & Spencer, 'Absence of diffusion in the Anderson tight binding model for large disorder or low energy', Comm. Math. Phys. 88 (1983) link
[4]Aizenman & Molchanov, 'Localization at large disorder and at extreme energies: An elementary derivation', Comm. Math. Phys. 157 (1993) link
[5]Bucaj et al., localization and the unresolved higher-dimensional regimes link

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