DOCUMENTA MATHEMATICA, Vol. 22 (2017), 727-776

Thierry Jecko, Aiman Mbarek

Limiting Absorption Principle for Schrödinger Operators with Oscillating Potentials

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.

2010 Mathematics Subject Classification: 35J10, 35P25, 35Q40, 35S05, 47B15, 47B25, 47F05.

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