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Quantum Limits on Flat Tori

Dmitry Jakobson
Annals of Mathematics
Second Series, Vol. 145, No. 2 (Mar., 1997), pp. 235-266
Published by: Annals of Mathematics
DOI: 10.2307/2951815
Stable URL: http://www.jstor.org/stable/2951815
Page Count: 32
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Quantum Limits on Flat Tori
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Abstract

We classify all weak * limits of squares of normalized eigenfunctions of the Laplacian on two-dimensional flat tori (called quantum limits). We also obtain several results about such limits in dimensions three and higher. Many of the results are a consequence of a geometric lemma which describes a property of simplices of codimension one in Rn whose vertices are lattice points on spheres. The lemma follows from the finiteness of the number of solutions of a system of two Pell equations. A consequence of the lemma is a generalization of the result of B. Connes. We also indicate a proof (communicated to us by J. Bourgain) of the absolute continuity of the quantum limits on a flat torus in any dimension. After generalizing a two-dimensional result of Zygmund to three dimensions, we discuss various possible generalizations of that result to higher dimensions and the relation to Lp norms of densities of quantum limits and their Fourier series.

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