{
  "annotation_id": "6791d58f-7970-4e91-a786-929546bb5122",
  "date_created": "2026-03-02T18:02:44.530000Z",
  "date_modified": "2026-03-02T18:02:44.530000Z",
  "file_hash": "806cf376af9a6b18999ead2257a5c04ed9b430772e459f1e0436032d3c79c339",
  "private": false,
  "record": {
    "abstract": "We study a complex intertwining relation of second order for Schroedinger\noperators and construct third order symmetry operators for them. A modification\nof this approach leads to a higher order shape invariance. We analyze with\nparticular attention irreducible second order Darboux transformations which\ntogether with the first order act as building blocks. For the third order\nshape-invariance irreducible Darboux transformations entail only one sequence\nof equidistant levels while for the reducible case the structure consists of up\nto three infinite sequences of equidistant levels and, in some cases, singlets\nor doublets of isolated levels.",
    "arxiv_id": "quant-ph/9902057",
    "authors": [
      "A. Andrianov",
      "F. Cannata",
      "M. Ioffe",
      "D. Nishnianidze"
    ],
    "categories": [
      "quant-ph",
      "math-ph",
      "math.MP",
      "nlin.SI",
      "solv-int"
    ],
    "doi": "10.1016/S0375-9601(00)00031-1",
    "journal_ref": "Phys.Lett. A266 (2000) 341-349",
    "title": "Systems with Higher-Order Shape Invariance: Spectral and Algebraic   Properties",
    "url": "https://arxiv.org/abs/quant-ph/9902057"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "e03fbe96-338a-4983-8c8d-4ea639a87071",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}