Semiclassical limits of eigenfunctions on flat $n$-dimensional tori

We provide a proof of the conjecture formulated in \cite{Jak97,JNT01} which states that on a $n$-dimensional flat torus $\T^{n}$, the Fourier transform of squares of the eigenfunctions $|ϕ_λ|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $λ$. The proof is a generalization of the argument by Jakobson, {\it et al}. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on $\TT^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices which satisfy a certain restriction on an $n$-dimensional sphere $S^n(λ)$ of radius $\sqrtλ$ and use it in the proof.

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