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On quantum limits on flat tori

Author: Dmitry Jakobson
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 80-86
MSC (1991): Primary 42B05, 81Q50, 58C40, 52B20, 11D09, 11J86
MathSciNet review: 1350683
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Abstract: We classify all weak $*$ limits of squares of normalized eigenfunctions of the Laplacian on two-dimensional flat tori (we call these limits 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 $\mathbb{R}^n$ 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. We generalize a two-dimensional result of Zygmund to three dimensions; we discuss various possible generalizations of that result to higher dimensions and the relation to $L^p$ norms of the densities of quantum limits and their Fourier series.

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Additional Information

Dmitry Jakobson
Affiliation: address Department of Mathematics, Princeton University, Princeton, NJ 08544

Received by editor(s): April 20, 1995
Received by editor(s) in revised form: July 19, 1995
Communicated by: Yitzhak Katznelson
Article copyright: © Copyright 1995 American Mathematical Society