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

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

2006-09-26

arxiv.org/abs/math-ph/0609072v2

Physics

Mathematical Physics

change of title and other very minor changes

Scientific paper

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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