The Leray measure of nodal sets for random eigenfunctions on the torus

2006/09/26 by Ferenc Oravecz, Zeév Rudnick, Oravecz, Ferenc +3 · 1 citation
Mathematics · Computer Science · #Geometry and complex manifolds #Numerical methods in inverse problems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.math-ph/0609072

Abstract

We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in 2 or more dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with multiplicity N tending to infinity. The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant. Our main result is that the variance of Leray measure is asymptotically 1/(4 pi N), as N tends to infinity, at least in dimensions 2 and at least 5.

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