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

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

2007-02-23

arxiv.org/abs/math-ph/0702081v2

Physics

Mathematical Physics

20 pages, Was accepted for publication in the Annales Henri Poincare

Scientific paper

10.1007/s00023-007-0352-6

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 $4\pi^2\eigenvalue$ 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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