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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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.|
|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.|
|Deposited By:||Ruth Sustaita|
|Deposited On:||20 Jan 2012 17:34|
|Last Modified:||20 Jan 2012 17:34|
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