{"paper":{"title":"Counting Nodal Lines Which Touch the Boundary of an Analytic Domain","license":"","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"John A. Toth, Steve Zelditch","submitted_at":"2007-09-30T02:59:46Z","abstract_excerpt":"We consider the zeros on the boundary $\\partial \\Omega$ of a Neumann eigenfunction $\\phi_{\\lambda}$ of a real analytic plane domain $\\Omega$. We prove that the number of its boundary zeros is $O (\\lambda)$ where $-\\Delta \\phi_{\\lambda} = \\lambda^2 \\phi_{\\lambda}$. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is $O(\\lambda)$. It follows that the number of nodal lines of $\\phi_{\\lambda}$ (components of the nodal set) which touch the boundary is of order $\\lambda$. This upper bound is of the same order of magnitude as the length of the t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.0101","kind":"arxiv","version":4},"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"}