Quantum limits on flat tori
- Creators
- Jakobson, Dmitry
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 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. 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 L^p norms of densities of quantum limits and their Fourier series.
Additional Information
© 1997 Annals of Mathematics. Received August 9, 1995. Dedicated to the memory of Anya Pogosyants and Igor Slobodkin. This research was partially supported by an NSF postdoctoral fellowship.Additional details
- Eprint ID
- 28879
- Resolver ID
- CaltechAUTHORS:20120120-085616919
- NSF Postdoctoral Fellowship
- Created
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2012-01-20Created from EPrint's datestamp field
- Updated
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2021-11-09Created from EPrint's last_modified field