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On Quantum Limits on Flat Tori
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## On quantum limits on flat tori

### Dmitry Jakobson

**Abstract.**
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 `

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