Limiting Absorption Principle for Schrödinger Operators with Oscillating Potentials
Thierry JeckoAGM, UMR 8088 du CNRS, Université de Cergy Pontoise, Site de Saint Martin 2, rue Adolphe Chauvin F-95302 Cergy-Pontoise cedex, France
Aiman MbarekAGM, UMR 8088 du CNRS, Université de Cergy Pontoise, Site de Saint Martin, 2, rue Adolphe Chauvin, F-95302 Cergy-Pontoise cedex, France
Making use of the localised Putnam theory developed in [gj], 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, DMR, DR1, DR2, MU, ReT1, ReT2]. Allowing long-range and short-range components and local singularities in the perturbation, we improve known results. A subclass of the considered potentials actually cannot be treated by the Mourre commutator method with the generator of dilations as conjugate operator. Inspired by [FH], we also show, in some cases, the absence of positive eigenvalues for our Schrödinger operators.
Cite this article
Thierry Jecko, Aiman Mbarek, Limiting Absorption Principle for Schrödinger Operators with Oscillating Potentials. Doc. Math. 22 (2017), pp. 727–776DOI 10.4171/DM/577