594 research outputs found
Problems related to the concentration of eigenfunctions
We survey recent results related to the concentration of eigenfunctions. We
also prove some new results concerning ball-concentration, as well as showing
that eigenfunctions saturating lower bounds for -norms must also, in a
measure theoretical sense, have extreme concentration near a geodesic.Comment: 13 page
Concerning Nikodym-type sets in 3-dimensional curved spaces
We obtain estimates for maximal functions that arise when one studies
Nikodym-type sets. We also formulate a curvature condition that allows
favorable estimates for these maximal functions
Semiclassical limits of eigenfunctions on flat -dimensional tori
We provide a proof of the conjecture formulated in \cite{Jak97,JNT01} which
states that on a -dimensional flat torus \T^{n}, the Fourier transform of
squares of the eigenfunctions of the Laplacian have uniform
bounds that do not depend on the eigenvalue . 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
-dimensional sphere of radius and use it in
the proof.Comment: 10 pages; Canadian Mathematical Bulletin, 201
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