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Small Scale Equidistribution of Eigenfunctions on the Torus

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Abstract

We study the small scale distribution of the L 2 mass of eigenfunctions of the Laplacian on the flat torus \({\mathbb{T}^{d}}\). Given an orthonormal basis of eigenfunctions, we show the existence of a density one subsequence whose L 2 mass equidistributes at small scales. In dimension two our result holds all the way down to the Planck scale. For dimensions d = 3, 4 we can restrict to individual eigenspaces and show small scale equidistribution in that context. We also study irregularities of quantum equidistribution: We construct eigenfunctions whose L 2 mass does not equidistribute at all scales above the Planck scale. Additionally, in dimension d = 4 we show the existence of eigenfunctions for which the proportion of L 2 mass in small balls blows up at certain scales.

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Authors and Affiliations

  1. Department of Mathematics, KTH, 100 44, Stockholm, Sweden

    Stephen Lester

  2. Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Zeév Rudnick

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  1. Stephen Lester
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  2. Zeév Rudnick
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Correspondence to Stephen Lester.

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Communicated by S. Zelditch

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Lester, S., Rudnick, Z. Small Scale Equidistribution of Eigenfunctions on the Torus. Commun. Math. Phys. 350, 279–300 (2017). https://doi.org/10.1007/s00220-016-2734-4

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