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

Oravecz, Ferenc; Rudnick, Zeév; Wigman, Igor

Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 299-335.

Nous étudions les ensembles nodaux des fonctions propres du Laplacien sur le tore standard de dimension $d\ge 2$. En utilisant la multiplicité du spectre du Laplacien et en introduisant une mesure gaussienne sur l’espace propre, nous nous servons de cette dernière afin d’évaluer des moyennes dans l’espace. Nous considérons une suite de valeurs propres ayant une multiplicité croissante $𝒩\to \infty$.

La quantité que nous étudions est la mesure de Leray (mesure microcanonique). Nous montrons que la moyenne de la mesure de Leray pour une fonction propre est constante et qu’elle vaut $1/\sqrt{2\pi }$. Notre résultat principal précise que la variance de la mesure de Leray est asymptotiquement $1/4\pi 𝒩$ lorsque $𝒩\to \infty$ pour $d=2$ et $d\ge 5$.

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 $𝒩\to \infty$.

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 $1/4\pi 𝒩$, as $𝒩\to \infty$, at least in dimensions $d=2$ and $d\ge 5$

[1] Bérard, P. Volume des ensembles nodaux des fonctions propres du laplacien, Bony-Sjostrand-Meyer seminar, 1984–1985, Volume 12 (1962), pp. 591-613 | Numdam | MR | Zbl

[2] Berry, M. V. Regular and irregular semiclassical wavefunctions, J.Phys.A, Volume 10 (1977), pp. 2083-2091 | DOI | MR | Zbl

[3] Berry, M. V. Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature, J.Phys.A, Volume 35 (2002), pp. 3025-3038 | DOI | MR | Zbl

[4] Borovoi, M.; Rudnick, Z. Hardy-Littlewood varieties and semisimple groups, Inventiones Math, Volume 119 (1995), pp. 37-66 | DOI | MR | Zbl

[5] Bourgain, J. Eigenfunction bounds for the Laplacian on the $n$-torus, Internat. Math. Res. Notices, Volume 3 (1993), pp. 61-66 | DOI | MR | Zbl

[6] Cilleruelo, J. The distribution of the lattice points on circles, J. Number Theory, Volume 43 (1993) no. 2, pp. 198-202 | DOI | MR | Zbl

[7] Davenport, H. 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., Cambridge Mathematical Library. Cambridge University Press,, Cambridge, 2005 | MR | Zbl

[8] Erdös, P.; Hall, R. R. On the angular distribution of Gaussian integers with fixed norm, Discrete Math., Volume 200, Paul Erdös memorial collection (1999), pp. 87-94 | DOI | MR | Zbl

[9] Fainsilber, L.; Kurlberg, P.; Wennberg, B. Lattice points on circles and discrete velocity models for the Boltzmann equation, SIAM J. Math. Anal., Volume 37 (2006) no. 6, pp. 1903-1922 | DOI | MR | Zbl

[10] Gelfand, I. M.; Shilov, G. E. Generalized functions. Vol. 1. Properties and operations. Translated from the Russian by Eugene Saletan., Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977] | MR | Zbl

[11] Kac, M. On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc., Volume 49 (1943), p. 87-91, 938 | MR | Zbl

[12] Kátai, I.; Környei, I. On the distribution of lattice points on circles, Ann. Univ. Sci. Budapest. Eotvos Sect. Math., Volume 19 (1977), pp. 87-91 | MR | Zbl

[13] Longuet-Higgins, M. S. The statistical analysis of a random, moving surface, Philos. Trans. Roy. Soc. London. Ser. A., Volume 249 (1957), pp. 321-387 | DOI | MR | Zbl

[14] Longuet-Higgins, M. S. Statistical properties of an isotropic random surface, Philos. Trans. Roy. Soc. London. Ser. A., Volume 250 (1957), pp. 157-174 | DOI | MR | Zbl

[15] Neuheisel, J. The asymptotic distribution of nodal sets on spheres, Johns Hopkins Ph.D. thesis (2000) (Ph. D. Thesis)

[16] Palamodov, V. P. Distributions and harmonic analysis, in Commutative harmonic analysis, Encyclopaedia Math. Sci., III (Havin and N.K. Nikol’skij, eds.), Volume 72 (1995), p. 1-127, 261–266 | Zbl

[17] Pommerenke, C. Über die Gleichverteilung von Gitterpunkten auf $m$-dimensionalen Ellipsoiden, erratum, Acta Arith., Volume 5,7 (1959,1961/1962), p. 115-137,279 | MR | Zbl

[18] Rudnick, Z.; Wigman, I. On the volume of nodal sets for eigenfunctions of the Laplacian on the torus (In preparation)

[19] Sogge, Christopher D. Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, 105, Cambridge University Press, Cambridge, 1993 | MR | Zbl

[20] Zelditch, S. A random matrix model for quantum mixing, Cambridge Tracts in Mathematics, Volume 3 (1996), pp. 115-137 | MR | Zbl

[21] Zygmund, A. On Fourier coefficients and transforms of functions of two variables, Studia Math., Volume 12 (1974), pp. 189-201 | MR | Zbl