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

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On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. / Rudnick, Zeev; Wigman, Igor.

In: Annales Henri Poincare, Vol. 9, No. 1, 02.2008, p. 109-130.

Research output: Contribution to journalArticlepeer-review

Harvard

Rudnick, Z & Wigman, I 2008, 'On the volume of nodal sets for eigenfunctions of the Laplacian on the torus', Annales Henri Poincare, vol. 9, no. 1, pp. 109-130. https://doi.org/10.1007/s00023-007-0352-6

APA

Rudnick, Z., & Wigman, I. (2008). On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Annales Henri Poincare, 9(1), 109-130. https://doi.org/10.1007/s00023-007-0352-6

Vancouver

Rudnick Z, Wigman I. On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Annales Henri Poincare. 2008 Feb;9(1):109-130. https://doi.org/10.1007/s00023-007-0352-6

Author

Rudnick, Zeev ; Wigman, Igor. / On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. In: Annales Henri Poincare. 2008 ; Vol. 9, No. 1. pp. 109-130.

Bibtex Download

@article{b2859bf5f5fc482c95ee0168c42577a5,
title = "On the volume of nodal sets for eigenfunctions of the Laplacian on the torus",
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 4 pi(2) E with growing multiplicity N -> infinity, 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 root E. Our main result is that the variance of the volume normalized by root E is bounded by O(1/root N), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.",
author = "Zeev Rudnick and Igor Wigman",
year = "2008",
month = feb,
doi = "10.1007/s00023-007-0352-6",
language = "English",
volume = "9",
pages = "109--130",
journal = "Annales Henri Poincare",
issn = "1424-0637",
publisher = "Birkhauser Verlag Basel",
number = "1",

}

RIS (suitable for import to EndNote) Download

TY - JOUR

T1 - On the volume of nodal sets for eigenfunctions of the Laplacian on the torus

AU - Rudnick, Zeev

AU - Wigman, Igor

PY - 2008/2

Y1 - 2008/2

N2 - 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) E with growing multiplicity N -> infinity, 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 root E. Our main result is that the variance of the volume normalized by root E is bounded by O(1/root N), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.

AB - 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) E with growing multiplicity N -> infinity, 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 root E. Our main result is that the variance of the volume normalized by root E is bounded by O(1/root N), so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.

U2 - 10.1007/s00023-007-0352-6

DO - 10.1007/s00023-007-0352-6

M3 - Article

VL - 9

SP - 109

EP - 130

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 1

ER -

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