Absolute continuity of the spectrum of a Landau Hamiltonian perturbed by a generic periodic potential

Consider Γ, a non-degenerate lattice in R and a constant magnetic field B with a flux though a cell of Γ that is a rational multiple of 2π. We prove that for a generic Γ-periodic potential V , the spectrum of the Landau Hamiltonian with magnetic field B and periodic potential V is purely absolutely continuous. Résumé. On considère Γ, un réseau non-dégénéré dans R et un champ magnétique constant B dont le flux à travers une cellule du réseau est un multiple rationnel de 2π. On démontre que, pour un potentiel Γpériodique V continu générique, le spectre du hamiltonien de Landau de champ magnétique constant B perturbé par le potentiel périodique V est purement absolument continu. Written in the Coulomb gauge, on L2(R2), the Landau Hamiltonian is defined by (1) H = (−i∇−A), where A(x1, x2) = B 2 (−x2, x1), Let Γ = ⊕i=1Zei be a non-degenerate lattice such that (2) B e1 ∧ e2 ∈ 2πQ. Define the set of real valued, continuous, Γ-periodic functions (3) CΓ = {V ∈ C(R ,R); ∀x ∈ R, ∀γ ∈ Γ, V (x+ γ) = V (x)}. The space CΓ is endowed with the uniform topology, the associated norm being denoted by ‖ · ‖. Our main result is Theorem 1. There exists a Gδ-dense subset of CΓ such that, for V in this set, the spectrum of H(V ) := H + V is purely absolutely continuous. The absence of singular continuous spectrum can be obtained from the sole analytic direct integral representation of H(V ) that we use below ([2]). Our result is optimal in the sense that there are examples of periodic V for which the spectrum of H contains eigenvalues such as V constant. Of Part of this work was done during the conference “Spectral analysis of differential operators” held at the MFO, Oberwolfach (29/11-03/12/2004); it is a pleasure to thank T. Weidl and A. Sobolev, the organizers of the conference, for their invitation to participate to the meeting as well as D. Elton for stimulating discussions. It is also a pleasure to thank the Institute of Mathematics of Hanoi, where this work was completed, for its kind hospitality. 1