2 results
Semiclassical Limits of Eigenfunctions on Flat n-Dimensional Tori
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- Journal:
- Canadian Mathematical Bulletin / Volume 56 / Issue 1 / 01 March 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 3-12
- Print publication:
- 01 March 2013
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- Article
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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.
Semiclassical Limits of Heat Kernels of Laplacians on theh-Heisenberg Group
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- Journal:
- Mathematical Modelling of Natural Phenomena / Volume 8 / Issue 1 / 2013
- Published online by Cambridge University Press:
- 28 January 2013, pp. 132-142
- Print publication:
- 2013
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