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OP-3098OpenMathematical physics

Two-dimensional Anderson localization at arbitrary disorder

For Hλ=−Δ+λVωH_\lambda=-\Delta+\lambda V_\omega on ℓ2(Z2)\ell^2(\mathbb Z^2) with independent potentials uniform on [−1,1][-1,1], prove that for every λ>0\lambda>0 the almost-sure spectrum is pure point with exponentially localized eigenfunctions throughout the spectrum.

Source-reported statusAs of 2026-09-04Source & evidence ↓
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Problem statement

For Hλ=−Δ+λVωH_\lambda=-\Delta+\lambda V_\omega on ℓ2(Z2)\ell^2(\mathbb Z^2) with independent potentials uniform on [−1,1][-1,1], prove that for every λ>0\lambda>0 the almost-sure spectrum is pure point with exponentially localized eigenfunctions throughout the spectrum.

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Background

This is the full-spectrum two-dimensional localization question, not the three-dimensional delocalization problem M2. The fixed scalar, real i.i.d. model excludes magnetic, correlated and multi-orbital variants with different physics.

Established baseline: localization is proved in one dimension and at strong disorder or suitable spectral edges in higher dimensions. These results do not settle all energies at arbitrarily weak disorder on the square lattice. Hurtado's 2026 nonstationary-potential advance concerns the bottom of the spectrum, not this universal energy range.

The stated target is spectral localization. Uniform-in-time dynamical localization with quantified moment or eigenfunction-correlator bounds is a stronger related target and should be recorded separately.

Progress & source assessment

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

References
  1. [1]Bucaj et al., primary account of localization and the open 2D problem link
  2. [2]Hurtado, bottom-of-spectrum localization for nonstationary potentials, CMP (2026) link

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