Logarithmic lower bound on the number of nodal domains

Steve Zelditch*

*Corresponding author for this work

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

We prove that the number of nodal domains of eigenfunctions grows at least logarithmically with the eigenvalue (for almost the entire sequence of eigenvalues) on certain negatively curved surfaces. The geometric model is the same as in prior joint work with J. Jung, where the number of nodal domainswas shown to tend to infinity. The surfaces are assumed to be "real Riemann surfaces," i.e. Riemann surfaceswith an anti-holomorphic involution σ with non-empty fixed point set. The eigenfunctions are assumed to be even or odd, which is automatically the case for generic invariant metrics. The logarithmic growth rate gives a quantitative refinement of the prior results.

Original languageEnglish (US)
Pages (from-to)1047-1086
Number of pages40
JournalJournal of Spectral Theory
Volume6
Issue number4
DOIs
StatePublished - Jan 1 2016

Keywords

  • Eigenfunctions
  • Negative curvature
  • Nodal domains
  • Restriction

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Geometry and Topology

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