Limiting Absorption Principle for Schrödinger Operators with Oscillating Potential

2016/10/14 by Thierry Jecko, Jecko, Thierry, Aiman Mbarek +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1610.04369

openalex publication_date 2016/10/14 · openalex created_date 2016/10/28 · openalex updated_date 2026/07/28

Abstract

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.

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