Nodal sets and growth exponents of Laplace eigenfunctions on surfaces

Guillaume Roy-Fortin*

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

We prove a result, announced by F. Nazarov, L. Polterovich and M. Sodin, that exhibits a relation between the average local growth of a Laplace eigenfunction on a closed surface and the global size of its nodal set. More precisely, we provide a lower and an upper bound to the Hausdorff measure of the nodal set in terms of the expected value of the growth exponent of an eigenfunction on disks of wavelength-like radius. Combined with Yau's conjecture, the result implies that the average local growth of an eigenfunction on such disks is bounded by constants in the semiclassical limit. We also obtain results that link the size of the nodal set to the growth of solutions of planar Schrödinger equations with small potential.

Original languageEnglish (US)
Pages (from-to)223-255
Number of pages33
JournalAnalysis and PDE
Volume8
Issue number1
DOIs
StatePublished - Jan 1 2015

Keywords

  • Growth of eigenfunctions
  • Laplace eigenfunctions
  • Nodal sets
  • Spectral geometry

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

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