{
  "annotation_id": "672ba603-2672-46d2-8575-479a1d13b60d",
  "date_created": "2026-03-02T18:02:45.189000Z",
  "date_modified": "2026-03-02T18:02:45.189000Z",
  "file_hash": "dfb6bb796bbcbae49b4bd39a4bd976291f7942484d977207d128f2e5b6552e36",
  "private": false,
  "record": {
    "abstract": "Levinson\u0027s theorem for the Schr\\\"{o}dinger equation with a cylindrically\nsymmetric potential in two dimensions is re-established by the Sturm-Liouville\ntheorem. The critical case, where the Schr\\\"{o}dinger equation has a finite\nzero-energy solution, is analyzed in detail. It is shown that, in comparison\nwith Levinson\u0027s theorem in non-critical case, the half bound state for $P$\nwave, in which the wave function for the zero-energy solution does not decay\nfast enough at infinity to be square integrable, will cause the phase shift of\n$P$ wave at zero energy to increase an additional $\\pi$.",
    "arxiv_id": "quant-ph/9806004",
    "authors": [
      "Shi-Hai dong",
      "Xi-Wen Hou",
      "Zhong-Qi Ma"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1103/PhysRevA.58.2790",
    "title": "Levinson\u0027s theorem for the Schr\\\"{o}dinger equation in two dimensions",
    "url": "https://arxiv.org/abs/quant-ph/9806004"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "e747c94b-0b65-4cdf-977d-ab4b7289885b",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}