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Peer-reviewedCC BY-NC-ND 4.0Documenta Mathematica

LIMITING ABSORPTION PRINCIPLE FOR SCHRÖDINGER OPERATORS WITH OSCILLATING POTENTIAL.

  • Thierry Jecko
  • Aiman Mbarek
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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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Cite

BibTeXRIS
BibTeX
@article{jecko2026limiting,
  title = {{LIMITING ABSORPTION PRINCIPLE FOR SCHRÖDINGER OPERATORS WITH OSCILLATING POTENTIAL.}},
  author = {Thierry Jecko and Aiman Mbarek},
  journal = {Documenta Mathematica},
  year = {2026},
  month = oct,
  doi = {10.4171/dm/577},
  eprint = {1610.04369},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-01381212v3},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - JOUR
TI  - LIMITING ABSORPTION PRINCIPLE FOR SCHRÖDINGER OPERATORS WITH OSCILLATING POTENTIAL.
AU  - Thierry Jecko
AU  - Aiman Mbarek
PY  - 2026
DA  - 2026/10/28
JO  - Documenta Mathematica
DO  - 10.4171/dm/577
UR  - https://hal.science/hal-01381212v3
AB  - 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.
SN  - 1431-0643
SN  - 1431-0635
ER  -