Jakobson, Dmitry
(1997)
*Quantum limits on flat tori.*
Annals of Mathematics, 145
(2).
pp. 235-266.
ISSN 0003-486X.
http://resolver.caltech.edu/CaltechAUTHORS:20120120-085616919

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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 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.

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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. | |||||||||

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Record Number: | CaltechAUTHORS:20120120-085616919 | |||||||||

Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20120120-085616919 | |||||||||

Official Citation: | Quantum Limits on Flat Tori Dmitry Jakobson The Annals of Mathematics Second Series, Vol. 145, No. 2 (Mar., 1997) (pp. 235-266) | |||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||

ID Code: | 28879 | |||||||||

Collection: | CaltechAUTHORS | |||||||||

Deposited By: | Ruth Sustaita | |||||||||

Deposited On: | 20 Jan 2012 17:34 | |||||||||

Last Modified: | 20 Jan 2012 17:34 |

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