TY - JOUR
T1 - On the volume of nodal sets for eigenfunctions of the Laplacian on the torus
AU - Rudnick, Zeév
AU - Wigman, Igor
N1 - Funding Information:
Z. Rudnick was supported by the Israel Science Foundation (grant No. 925/06). I. Wigman was supported by CRM analysis laboratory fellowship.
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 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 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.
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 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 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.
UR - http://www.scopus.com/inward/record.url?scp=40949105097&partnerID=8YFLogxK
U2 - 10.1007/s00023-007-0352-6
DO - 10.1007/s00023-007-0352-6
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AN - SCOPUS:40949105097
SN - 1424-0637
VL - 9
SP - 109
EP - 130
JO - Annales Henri Poincare
JF - Annales Henri Poincare
IS - 1
ER -