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On the volume of nodal sets for eigenfunctions of the Laplacian on the torus

Zeev Rudnick·Igor Wigman
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Abstract

We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues $4π^2\eigenvalue$ with growing multiplicity $\Ndim\to\infty$, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is $const \sqrt{\eigenvalue}$. Our main result is that the variance of the volume normalized by $\sqrt{\eigenvalue}$ is bounded by $O(1/\sqrt{\Ndim})$, so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.

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Zeev Rudnick, Igor Wigman. 2007-09-24. On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. https://doi.org/10.1007/s00023-007-0352-6

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