LIMITING ABSORPTION PRINCIPLE FOR SCHRÖDINGER OPERATORS WITH OSCILLATING POTENTIAL.

Théorème d'absorption limite pour des opérateurs de Schrödinger avec potentiel oscillant.

Thierry Jecko (1) , Aiman Mbarek (1)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

Making use of the weighted Mourre theory developed in [GJ1], we show the limiting absorption principle for Schrödinger operators with perturbed oscillating potential on appropriate energy intervals. We focus on a certain class of oscillating potentials (larger than the one in [GJ2]) that was already studied in [BD, MU, ReT1, ReT2]. We allow long-range and short-range components and local singularities in the perturbation. We improve known results, the main novelty being the presence of a long-range perturbation. A subclass of the considered potentials actually cannot be treated by the usual Mourre commutator method. Inspired by [FH], we also show, in some cases, the absence of positive eigenvalues for our Schrödinger operators.

Mots clés: oscillating potential; Wigner-Von Neumann potential.; continuous spectrum; Schrödinger operators; Weighted Mourre commutator theory; Mourre estimate; limiting absorption principle

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Thierry Jecko, Aiman Mbarek. LIMITING ABSORPTION PRINCIPLE FOR SCHRÖDINGER OPERATORS WITH OSCILLATING POTENTIAL. . Documenta Mathematica, 2017. ⟨hal-01381212v2⟩