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Semiclassical Limits of Eigenfunctions on Flat n-Dimensional Tori

  • Tayeb Aϊssiou (a1)
Abstract

We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an n-dimensional flat torus 𝕋n, and the Fourier transform of squares of the eigenfunctions |φ λ|2 of the Laplacian have uniform ln bounds that do not depend on the eigenvalue λ. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on 𝕋n+2. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an n-dimensional sphere Sn(λ) of radius √λ, and we use it in the proof.

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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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