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CENTRAL LIMIT THEOREM FOR PLANCK-SCALE MASS DISTRIBUTION OF TORAL LAPLACE EIGENFUNCTIONS

Part of: Miscellaneous applications of number theory Geometry of numbers Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  12 April 2019

Igor Wigman  and
Nadav Yesha
Igor Wigman
Affiliation:
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, U.K. email igor.wigman@kcl.ac.uk
Nadav Yesha
Affiliation:
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, U.K.
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Abstract

We study the fine-scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

MSC classification

Primary: 11H06: Lattices and convex bodies 11Z05: Miscellaneous applications of number theory 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Type
Research Article
Information
Mathematika , Volume 65 , Issue 3 , 2019 , pp. 643 - 676
Copyright
Copyright © University College London 2019 

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Footnotes

1

Current address: Department of Mathematics, University of Haifa, 3498838 Haifa, Israel email nyesha@univ.haifa.ac.il

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