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Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori

  Published:2011-08-03
 Printed: Mar 2013
  • Tayeb Aïssiou,
    Department of Mathematics, McGill University, Montréal, Qc.
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Abstract

We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. 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 $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in the proof.
Keywords: semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits
MSC Classifications: 58G25, 81Q50, 35P20, 42B05 show english descriptions unknown classification 58G25
Quantum chaos [See also 37Dxx]
Asymptotic distribution of eigenvalues and eigenfunctions
Fourier series and coefficients
58G25 - unknown classification 58G25
81Q50 - Quantum chaos [See also 37Dxx]
35P20 - Asymptotic distribution of eigenvalues and eigenfunctions
42B05 - Fourier series and coefficients
 

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