Semiclassical Limits of Eigenfunctions on Flat n-Dimensional Tori

Aϊssiou, Tayeb

Canadian mathematical bulletin, Tome 56 (2013) no. 1, pp. 3-12

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-152-9

Mots-clés : 58G25, 81Q50, 35P20, 42B05, semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits