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We classify all weak $*$ limits of squares of normalized eigenfunctions of the Laplacian on two-dimensional flat tori (we call these limits \it 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 $\RR^n$ 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.

Copyright American Mathematical Society 1995

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Article Info

• ERA Amer. Math. Soc. 01 (1995), pp. 80-86
• Publisher Identifier: S 1079-6762(95)02004-6
• 1991 Mathematics Subject Classification. 42B05, 81Q50, 58C40, 52B20, 11D09, 11J86
• Received by the editors April 20, 1995, and, in revised form, July 19, 1995
• Key words and phrases. Flat tori, Laplacian, resonances, Pell equation, weak* limits, Fourier series
• Communicated by Yitzhak Katznelson
• Comments

Dmitry Jakobson

Department of Mathematics, Princeton University, Princeton, NJ 08544

E-mail address: diy@math.princeton.edu