Dynamical delocalization in random Landau Hamiltonians

Abstract

We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disordered-broadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field goes to infinity or the disorder goes to zero.

Delocalization for random Landau Hamiltonians with unbounded random variables

“…In this note we prove the existence of a dynamical localization/delocalization transition for Landau Hamiltonian randomly perturbed by an electric potential with unbounded amplitude, extending results from [GKS1,GKS2]. In [GKS1] the perturbation had to be sufficiently small compared to the strength of the magnetic field: the amplitude of the random potential was such that the Landau gaps survived after adding the perturbation.…”

“…In [GKS1] the perturbation had to be sufficiently small compared to the strength of the magnetic field: the amplitude of the random potential was such that the Landau gaps survived after adding the perturbation. In [GKS2] the Landau gaps where allowed to close, but the random potentials were bounded.…”

An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

“…However, it is easier to pursue a purely local result and also obtain a Wegner estimate. Motivated by transport questions for random Landau Hamiltonians (1.8), Germinet, Klein, and Schenker [17] used the result (4.1) to prove a purely local version of the quantitative unique continuation principle. This allowed them to prove a Wegner estimate for Landau Hamiltonians at any energy, including the Landau levels.…”

“…As in [17], given a magnetic field strength B > 0, we define a number The spectrum of these local operators is discrete and consists of finite multiplicity eigenvalues at the Landau levels E n (B). We denote by Π n,L the finite rank projection onto the eigenspace corresponding to the n th Landau level E n (B).…”

Approach to the Extended States Conjecture

“…Although many physicists consider the problem solved, many mathematical questions with striking physical relevance remain open. The field has grown into a rich mathematical theory (see [Germinet-Klein-Schenker 2007] and [Ghribi-Hislop-Klopp 2007] for the study of different Anderson models; for refined notions of Anderson localization see [del Rio-Jitomirskaya-Last-Simon 1986] and [Last 2007]).…”