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Semiclassical Limits of Eigenfunctions on Flat n-Dimensional Tori

Published online by Cambridge University Press:  20 November 2018

Tayeb Aϊssiou
Tayeb Aϊssiou*
Affiliation:
Department of Mathematics, McGill University, Montréal, QC e-mail: aissiou@math.mcgill.ca
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Abstract

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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 ${{\left| \varphi \lambda \right|}^{2}}$ 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 ${{S}^{n}}\,\left( \text{ }\!\!\lambda\!\!\text{ } \right)$ of radius $\sqrt{\lambda }$, and we use it in the proof.

Keywords

58G2581Q5035P2042B05semiclassical limitseigenfunctions of Laplacian on a torusquantum limits
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 1 , 01 March 2013 , pp. 3 - 12
Copyright
Copyright © Canadian Mathematical Society 2013

References

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