Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 1 (2015), 223-255.

Nodal sets and growth exponents of Laplace eigenfunctions on surfaces

Guillaume Roy-Fortin

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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.

Article information

Source
Anal. PDE, Volume 8, Number 1 (2015), 223-255.

Dates
Received: 6 September 2014
Accepted: 26 November 2014
First available in Project Euclid: 28 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1511895962

Digital Object Identifier
doi:10.2140/apde.2015.8.223

Mathematical Reviews number (MathSciNet)
MR3336925

Zentralblatt MATH identifier
1316.58025

Subjects
Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]

Keywords
spectral geometry Laplace eigenfunctions nodal sets growth of eigenfunctions

Citation

Roy-Fortin, Guillaume. Nodal sets and growth exponents of Laplace eigenfunctions on surfaces. Anal. PDE 8 (2015), no. 1, 223--255. doi:10.2140/apde.2015.8.223. https://projecteuclid.org/euclid.apde/1511895962


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