Space–Adiabatic Theory for Random–Landau Hamiltonian: Results and Prospects

Abstract

The study of the (integer) Quantum Hall Effect (QHE) requires a careful analysis of the spectral properties of the 2D, single-electron Hamiltonian

$$ {H}_{\Gamma ,B} :={ \left (-\mathrm{i}{\partial }_{x} - B\ y\right )}^{2} +{ \left (-\mathrm{i}{\partial }_{y} + B\ x\right )}^{2} + {V }_{\Gamma }(x,y)$$

(1)

where ${H}_{B} := {H}_{\Gamma ,B} - {V }_{\Gamma }$ is the usual Landau Hamiltonian (in symmetric gauge) with magnetic field B and ${V }_{\Gamma }$ is a $\Gamma \equiv {\mathbb{Z}}^{2}$ periodic potential which models the electronic interaction with a crystalline structure. Under usual conditions (e.g., ${V }_{\Gamma } \in {L}_{\mathrm{loc}}^{2}({\mathbb{R}}^{2})$) the Hamiltonian 1 is self-adjoint on a suitable domain of ${L}^{2}({\mathbb{R}}^{2})$. A direct analysis of the fine spectral properties of 1 is extremely difficult and one needs resorting to simpler effective models hoping to capture (some of) the main physical features in suitable physical regimes.