Counting nodal lines which touch the boundary of an analytic domain

John A. Toth*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

47 Scopus citations


We consider the zeros on the boundary ∂Ω of a Neumann eigen- function φλj of a real analytic plane domain Ω. We prove that the number of its boundary zeros is O(Ωj) where −Δφλj = λ2jφλj. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(Ωj). It follows that the number of nodal lines of φλj (components of the nodal set) which touch the boundary is of order Ωj . This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains.

Original languageEnglish (US)
Pages (from-to)649-686
Number of pages38
JournalJournal of Differential Geometry
Issue number3
StatePublished - 2009

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology


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