The Leray measure of nodal sets for random eigenfunctions on the torus

Ferenc Oravecz[1]; Zeév Rudnick[2]; Igor Wigman[3]

  • [1] Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Reáltanoda utca 13-15 1053 Budapest (Hungary)
  • [2] Tel Aviv University School of Mathematical Sciences 69978 Tel Aviv (Israel)
  • [3] Université de Montréal Centre de recherches mathématiques (CRM) C.P. 6128, succ. centre-ville Montréal, Québec H3C 3J7 (Canada)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 1, page 299-335
  • ISSN: 0373-0956

Abstract

top
We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in d ≥ 2 dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity 𝒩 → ∞ .The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal to 1 / 2 π . Our main result is that the variance of Leray measure is asymptotically 1 / 4 π 𝒩 , as 𝒩 → ∞ , at least in dimensions d = 2 and d ≥ 5

How to cite

top

Oravecz, Ferenc, Rudnick, Zeév, and Wigman, Igor. "The Leray measure of nodal sets for random eigenfunctions on the torus." Annales de l’institut Fourier 58.1 (2008): 299-335. <http://eudml.org/doc/10312>.

@article{Oravecz2008,
abstract = {We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in $d\ge 2$ dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity $\mathcal\{N\}\rightarrow \infty $.The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal to $1/\sqrt\{2\pi \}$. Our main result is that the variance of Leray measure is asymptotically $1/4\pi \mathcal\{N\}$, as $\mathcal\{N\}\rightarrow \infty $, at least in dimensions $d=2$ and $d\ge 5$},
affiliation = {Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Reáltanoda utca 13-15 1053 Budapest (Hungary); Tel Aviv University School of Mathematical Sciences 69978 Tel Aviv (Israel); Université de Montréal Centre de recherches mathématiques (CRM) C.P. 6128, succ. centre-ville Montréal, Québec H3C 3J7 (Canada)},
author = {Oravecz, Ferenc, Rudnick, Zeév, Wigman, Igor},
journal = {Annales de l’institut Fourier},
keywords = {Nodal sets; Leray measure; eigenfunctions of the Laplacian; trigonometric polynomials; nodal set; random eigenfunction; torus},
language = {eng},
number = {1},
pages = {299-335},
publisher = {Association des Annales de l’institut Fourier},
title = {The Leray measure of nodal sets for random eigenfunctions on the torus},
url = {http://eudml.org/doc/10312},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Oravecz, Ferenc
AU - Rudnick, Zeév
AU - Wigman, Igor
TI - The Leray measure of nodal sets for random eigenfunctions on the torus
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 1
SP - 299
EP - 335
AB - We study nodal sets for typical eigenfunctions of the Laplacian on the standard torus in $d\ge 2$ dimensions. Making use of the multiplicities in the spectrum of the Laplacian, we put a Gaussian measure on the eigenspaces and use it to average over the eigenspace. We consider a sequence of eigenvalues with growing multiplicity $\mathcal{N}\rightarrow \infty $.The quantity that we study is the Leray, or microcanonical, measure of the nodal set. We show that the expected value of the Leray measure of an eigenfunction is constant, equal to $1/\sqrt{2\pi }$. Our main result is that the variance of Leray measure is asymptotically $1/4\pi \mathcal{N}$, as $\mathcal{N}\rightarrow \infty $, at least in dimensions $d=2$ and $d\ge 5$
LA - eng
KW - Nodal sets; Leray measure; eigenfunctions of the Laplacian; trigonometric polynomials; nodal set; random eigenfunction; torus
UR - http://eudml.org/doc/10312
ER -

References

top
  1. P. Bérard, Volume des ensembles nodaux des fonctions propres du laplacien, Bony-Sjostrand-Meyer seminar, 1984–1985 12 (1962), 591-613 Zbl0589.58033MR1046051
  2. M. V. Berry, Regular and irregular semiclassical wavefunctions, J.Phys.A 10 (1977), 2083-2091 Zbl0377.70014MR489542
  3. M. V. Berry, Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature, J.Phys.A 35 (2002), 3025-3038 Zbl1044.81047MR1913853
  4. M. Borovoi, Z. Rudnick, Hardy-Littlewood varieties and semisimple groups, Inventiones Math 119 (1995), 37-66 Zbl0917.11025MR1309971
  5. J. Bourgain, Eigenfunction bounds for the Laplacian on the n -torus, Internat. Math. Res. Notices 3 (1993), 61-66 Zbl0779.58039MR1208826
  6. J. Cilleruelo, The distribution of the lattice points on circles, J. Number Theory 43 (1993), 198-202 Zbl0777.11036MR1207499
  7. H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities. Second edition. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman. Edited and prepared for publication by T. D. Browning., (2005), Cambridge Mathematical Library. Cambridge University Press,, Cambridge Zbl1125.11018MR2152164
  8. P. Erdös, R. R. Hall, On the angular distribution of Gaussian integers with fixed norm, Discrete Math. 200, Paul Erdös memorial collection (1999), 87-94 Zbl1044.11073MR1692282
  9. L. Fainsilber, P. Kurlberg, B. Wennberg, Lattice points on circles and discrete velocity models for the Boltzmann equation, SIAM J. Math. Anal. 37 (2006), 1903-1922 Zbl1141.11046MR2213399
  10. I. M. Gelfand, G. E. Shilov, Generalized functions. Vol. 1. Properties and operations. Translated from the Russian by Eugene Saletan., (1964 [1977]), Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London Zbl0115.33101MR435831
  11. M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 87-91, 938 Zbl0060.28602MR7812
  12. I. Kátai, I. Környei, On the distribution of lattice points on circles, Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 19 (1977), 87-91 Zbl0348.10036MR506102
  13. M. S. Longuet-Higgins, The statistical analysis of a random, moving surface, Philos. Trans. Roy. Soc. London. Ser. A. 249 (1957), 321-387 Zbl0077.12707MR87257
  14. M. S. Longuet-Higgins, Statistical properties of an isotropic random surface, Philos. Trans. Roy. Soc. London. Ser. A. 250 (1957), 157-174 Zbl0078.32701MR91569
  15. J. Neuheisel, The asymptotic distribution of nodal sets on spheres, (2000) 
  16. V. P. Palamodov, Distributions and harmonic analysis, in Commutative harmonic analysis, Encyclopaedia Math. Sci., III (Havin and N.K. Nikol’skij, eds.) 72 (1995), 1-127, 261–266 Zbl0826.46025
  17. C. Pommerenke, Über die Gleichverteilung von Gitterpunkten auf m -dimensionalen Ellipsoiden, erratum, Acta Arith. 5,7 (1959,1961/1962), 115-137,279 Zbl0089.26802MR122788
  18. Z. Rudnick, I. Wigman, On the volume of nodal sets for eigenfunctions of the Laplacian on the torus Zbl1142.60029
  19. Christopher D. Sogge, Fourier integrals in classical analysis, 105 (1993), Cambridge University Press, Cambridge Zbl0783.35001MR1205579
  20. S. Zelditch, A random matrix model for quantum mixing, Cambridge Tracts in Mathematics 3 (1996), 115-137 Zbl0858.58048MR1383753
  21. A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 12 (1974), 189-201 Zbl0278.42005MR387950

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.