{"paper":{"title":"On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"Konstantin Pankrashkin","submitted_at":"2014-11-07T15:49:05Z","abstract_excerpt":"Let $\\Omega\\subset \\mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\\ell_1,...,\\ell_M$. For $\\alpha>0$, let $H^\\Omega_\\alpha$ denote the Laplacian in $\\Omega$, $u\\mapsto -\\Delta u$, with the Robin boundary conditions $\\partial u/\\partial\\nu =\\alpha u$, where $\\nu$ is the exterior unit normal at the boundary of $\\Omega$. We show that, for any fixed $m\\in\\mathbb{N}$, the $m$th eigenvalue $E^\\Omega_m(\\alpha)$ of $H^\\Omega_\\alpha$ behaves as \\[ E^\\Omega_m(\\alpha)=-\\alpha^2+\\mu^D_m +\\mathcal{O}\\Big(\\dfrac{1}{\\sqrt\\alpha}\\Big) \\quad {as $\\alpha$ tends to $+\\infty$}, \\] where"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1956","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"}