{
  "annotation_id": "2a000532-b3f3-4140-a77c-51fd126209e0",
  "date_created": "2026-02-17T05:53:24.043000Z",
  "date_modified": "2026-02-17T05:53:24.043000Z",
  "file_hash": "ba090dc437364fae90e69f605d4fee86317d61fc5a0e255dabce084f7e45c57b",
  "private": false,
  "record": {
    "abstract": "For a broad class of polynomial potentials $V$, with an important and instructive representative being $V(x) = x^{2a} + i x^b$, $x \\in \\mathbb R$, $a, b \\in \\mathbb N$, we show that the system of spectral projections $\\{P_n\\}_n$ of an anharmonic operator $L = - (\\mathrm{d}/ \\mathrm{d}x)^2 + V(x)$ does not generate a (Riesz) basis in $L^2(\\mathbb R)$ if $a - 1 \u003c b \u003c 2a$. Moreover, for $\\sigma = [b - (a - 1)]/(1 + a)$ and $\\gamma \u003e 0$ small enough, $\\limsup_n \\|P_n\\|/ \\exp(\\gamma n^\\sigma) = \\infty$. Proofs are based on two groups of results which are of great interest on their own: (a) relationship between behavior (growth) of the norms of projections $\\|P_n\\|$ and of the resolvent $\\|(z - L)^{-1}\\|$ outside of the spectrum $\\sigma(L)$; (b) partial fraction decompositions of special meromorphic functions $1/F$ where $F(w) = \\prod_{k=1}^\\infty \\left( 1 + \\frac{w}{a_k} \\right)$, $a_{k+1} \\geq a_k\u003e0$, $k \\in \\mathbb N$, and the generalization of the first resolvent identity.",
    "arxiv_id": "2601.09800",
    "authors": [
      "Boris Mityagin",
      "Petr Siegl"
    ],
    "categories": [
      "math.SP",
      "math-ph",
      "math.FA",
      "math.MP"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "title": "Spectral projections of an anharmonic oscillator with complex polynomial potential",
    "url": "https://arxiv.org/abs/2601.09800",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "97f4d55c-8c2d-4c32-9b8e-0079a7fdb43b",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}