{"paper":{"title":"Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Ari Laptev, Leon A. Takhtajan, Lukas Schimmer","submitted_at":"2015-09-30T21:34:50Z","abstract_excerpt":"We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are $H(\\zeta)=U+U^{-1}+V+\\zeta V^{-1}$ and $H_{m,n}=U+V+q^{-mn}U^{-m}V^{-n}$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{i\\pi b^{2}}$, $b>0$ and $\\zeta>0$, $m,n\\in\\mathbb{N}$. We prove that $H(\\zeta)$ and $H_{m,n}$ are self-adjoint operators with purely discrete spectrum on $L^{2}(\\mathbb{R})$. Using the coherent state transform we find the asymptotical behaviour for the Riesz mean $\\sum_{j\\ge 1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00045","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}