Extended states in random Schrodinger operators

6 problems

• Localization at all energies in the weakly disordered 2D Anderson model

  Open

  Scaling theory predicts that in two dimensions every electron state is trapped by any amount of disorder, however weak. A proof exists only near the edges of the energy band or at strong disorder.

• Absolutely continuous spectrum of the 3D Anderson model at weak disorder

  Open

  Prove that at least some electron states in a weakly disordered 3D lattice spread over the whole sample. This is the half of the mobility-edge problem that has no proof at all.

• Delocalization of random band matrices in two and three dimensions

  Partially resolved

  Random band matrices are simplified models of disordered conductors in which each site couples randomly to all sites within a distance W. The transition between trapped and spreading states in one dimension was settled in 2025, but in 2D and 3D at fixed band width it is open.

• Existence of a mobility edge in the 3D Anderson model

  Open

  Show that in a weakly disordered 3D crystal, electrons at energies near the band edges are trapped while those in the middle of the band move freely. The boundary energy between the two regimes is the mobility edge.

• Quantum diffusion for all times in the weakly disordered 3D lattice

  Open

  An electron in a weakly disordered 3D crystal should wander like a random walker, with its mean squared distance growing linearly in time forever. This is proven only up to a long but finite time.

• Random-matrix bulk statistics for random regular graphs of fixed degree

  Open

  A random network in which every node has exactly d links is a discrete model of a chaotic system, and its eigenvalues should be spaced like those of random matrices. This is proven at the spectrum's edge and for slowly growing d, but not in the bulk at fixed d.