Limiting absorption principle for the Magnetic Dirichlet Laplacian in a half-plane.

Nicolas Popoff (1) , Eric Soccorsi (2, 3)

1 Équipe EDP et Physique Mathématique
2 CPT - Centre de Physique Théorique - UMR 7332
3 CPT - E8 Dynamique quantique et analyse spectrale

We consider the Dirichlet Laplacian in the half-plane with constant magnetic field. Due to the translational invariance this operator admits a fiber decomposition and a family of dis- persion curves, that are real analytic functions. Each of them is simple and monotically decreasing from positive infinity to a finite value, which is the corresponding Landau level. These finite limits are thresholds in the purely absolutely continuous spectrum of the magnetic Laplacian. We prove a limiting absorption principle for this operator both outside and at the thresholds. Finally, we establish analytic and decay properties for functions lying in the absorption spaces. We point out that the analysis carried out in this paper is rather general and can be adapted to a wide class of fibered magnetic Laplacians with thresholds in their spectrum that are finite limits of their band functions.

Mots clés: limit absorption; constant magnetic field; thresholds; Two-dimensional Schrödinger operators; thresholds.

Domaines: Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]

Nicolas Popoff, Eric Soccorsi. Limiting absorption principle for the Magnetic Dirichlet Laplacian in a half-plane.. Communications in Partial Differential Equations, 2016, 41 (6), pp.879-893. ⟨10.1080/03605302.2016.1167081⟩. ⟨hal-01066569v2⟩