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The Leray measure of nodal sets for random eigenfunctions on the torus. (English) Zbl 1153.35058
For \(E>0\), let \(N(E)\) be the dimension of the eigenspace of the Laplacian on the \(d\)-dimensional torus, i.e. \(N(E)\) is the number of the set of all \(\lambda\in \mathbb Z^d\) such that \(| \lambda| ^2 =E.\) Let \(b_\lambda, c_\lambda\) be independent standard Gaussian random variables such that \(b_{-\lambda}=b_\lambda\) and \(c_{-\lambda} =-c_\lambda\), and let \(L(f)\) be the Leray measure of the nodal set of the random eigenfunction
\[ f(x):= (2N(E))^{-1/2} \sum_{\lambda\in \mathbb Z^d: | \lambda | ^2=E} b_\lambda \cos 2\pi I \lambda\cdot x -c_\lambda \sin 2 \pi i \lambda\cdot x. \] The introduced Leray measure has the property that \(\mathbb E L(f)= (2\pi)^{-1/2}\) and \(\text{Var}(L(f))\) behaves as \((4\pi N(E))^{-1} \) as \(N\to \infty\).

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:
[1] Bérard, P., Volume des ensembles nodaux des fonctions propres du laplacien, Bony-Sjostrand-Meyer seminar, 1984-1985, 12, 591-613, (1962) · Zbl 0589.58033
[2] Berry, M. V., Regular and irregular semiclassical wavefunctions, J.Phys.A, 10, 2083-2091, (1977) · Zbl 0377.70014
[3] Berry, M. V., Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature, J.Phys.A, 35, 3025-3038, (2002) · Zbl 1044.81047
[4] Borovoi, M.; Rudnick, Z., Hardy-Littlewood varieties and semisimple groups, Inventiones Math, 119, 37-66, (1995) · Zbl 0917.11025
[5] Bourgain, J., Eigenfunction bounds for the Laplacian on the \(n\)-torus, Internat. Math. Res. Notices, 3, 61-66, (1993) · Zbl 0779.58039
[6] Cilleruelo, J., The distribution of the lattice points on circles, J. Number Theory, 43, 2, 198-202, (1993) · Zbl 0777.11036
[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., (2005), Cambridge Mathematical Library. Cambridge University Press,, Cambridge · Zbl 1125.11018
[8] Erdös, P.; Hall, R. R., On the angular distribution of Gaussian integers with fixed norm, Discrete Math., 200, Paul Erdös memorial collection, 87-94, (1999) · Zbl 1044.11073
[9] Fainsilber, L.; Kurlberg, P.; Wennberg, B., Lattice points on circles and discrete velocity models for the Boltzmann equation, SIAM J. Math. Anal., 37, 6, 1903-1922, (2006) · Zbl 1141.11046
[10] Gelfand, I. M.; Shilov, G. E., Generalized functions. Vol. 1. Properties and operations. Translated from the Russian by Eugene Saletan., (19641977), Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London · Zbl 0115.33101
[11] Kac, M., On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc., 49, 87-91, 938, (1943) · Zbl 0060.28602
[12] Kátai, I.; Környei, I., On the distribution of lattice points on circles, Ann. Univ. Sci. Budapest. Eotvos Sect. Math., 19, 87-91, (1977) · Zbl 0348.10036
[13] Longuet-Higgins, M. S., The statistical analysis of a random, moving surface, Philos. Trans. Roy. Soc. London. Ser. A., 249, 321-387, (1957) · Zbl 0077.12707
[14] Longuet-Higgins, M. S., Statistical properties of an isotropic random surface, Philos. Trans. Roy. Soc. London. Ser. A., 250, 157-174, (1957) · Zbl 0078.32701
[15] Neuheisel, J., The asymptotic distribution of nodal sets on spheres, (2000)
[16] Palamodov, V. P., Distributions and harmonic analysis, in commutative harmonic analysis, Encyclopaedia Math. Sci., III (Havin and N.K. Nikol’skij, eds.), 72, 1-127, 261-266, (1995) · Zbl 0826.46025
[17] Pommerenke, C., Über die gleichverteilung von gitterpunkten auf \(m\)-dimensionalen ellipsoiden, erratum, Acta Arith., 5,7, 115-137,279, (195919611962) · Zbl 0089.26802
[18] Rudnick, Z.; Wigman, I., On the volume of nodal sets for eigenfunctions of the Laplacian on the torus · Zbl 1142.60029
[19] Sogge, Christopher D., Fourier integrals in classical analysis, 105, (1993), Cambridge University Press, Cambridge · Zbl 0783.35001
[20] Zelditch, S., A random matrix model for quantum mixing, Cambridge Tracts in Mathematics, 3, 115-137, (1996) · Zbl 0858.58048
[21] Zygmund, A., On Fourier coefficients and transforms of functions of two variables, Studia Math., 12, 189-201, (1974) · Zbl 0278.42005
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