On the Volume of nodal sets for Eigenfunctions of the Laplacian on the Torus

Rudnick, Zeév and Wigman, Igor 2008. On the Volume of nodal sets for Eigenfunctions of the Laplacian on the Torus. Annales Henri Poincaré 9 (1) , pp. 109-130. 10.1007/s00023-007-0352-6

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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 4π2 E with growing multiplicity N→∞, 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 √E. Our main result is that the variance of the volume normalized by √E is bounded by O(1/√N), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Springer
ISSN:	1424-0637
Last Modified:	26 Jun 2019 01:57
URI:	https://orca.cardiff.ac.uk/id/eprint/12407

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