{
  "abstract": "We study eigenvalues of discrete Schrödinger operators $H=Δ+V$ on $\\mathbb{Z}^2$, where $Δ$ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of $\\mathbb{Z}^2$ and $V$ decays at infinity. By Weyl's theorem, the essential spectrum of $H$ is $[-4,4]$. We determine the sharp decay thresholds for existence of eigenvalues in three distinct spectral regimes. While it is natural to expect different behavior at the spectral edge $λ=\\pm4$, and the bulk, there is a further distinction between the regular energies $0<|λ|<4$ and the interior critical energy $λ=0$ stemming from the reducibility of the corresponding Fermi surface in the latter case. For every $0<|λ|<4$, we construct potentials $V$ satisfying $|V(n)|\\leq C|n|^{-1}$ for which $λ$ is an eigenvalue of $Δ+V$, and prove absence of eigenvalues when $|V(n)|\\leq C|n|^{-1-\\varepsilon}$ for some $\\varepsilon>0$. At $λ=0$, the critical power changes and we construct potentials $V$ satisfying $|V(n)|\\leq C|n| ^{-2}$ for which $0$ is an eigenvalue of $Δ+V$, as well as prove absence when $|V(n)|\\leq C|n|^{-2-\\varepsilon}$ for any $\\varepsilon>0$. Finally, at each spectral edge, we show that, for every $K\\geq3$, an eigenvalue can be created by potentials supported on exactly $K$ sites, whereas a potential supported on at most two sites cannot create an edge eigenvalue. The proofs combine Green-function expansions and moment cancellation, Hilbert-space-valued iterations, discrete Carleman estimates, and a uniform Green-kernel estimate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15908",
  "author_total": 2,
  "authors": [
    {
      "name": "Shirshendu Ganguly"
    },
    {
      "name": "Wencai Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ck15s15t3n1vm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15908",
  "fulltext_url": null,
  "id": "ck15s15t3n1vm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15908",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7213365053",
  "pdf_url": "https://arxiv.org/pdf/2609.15908",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ck15s15t3n1vm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2609.15908",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15908"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Sharp decay thresholds for eigenvalues of discrete Schrödinger operators on $\\mathbb{Z}^2$",
  "title_text": "Sharp decay thresholds for eigenvalues of discrete Schrödinger operators on ℤ^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/ck15s15t3n1vm",
  "venue": "arXiv",
  "versions": []
}