A Caltech Library Service

Quantum limits on flat tori

Jakobson, Dmitry (1997) Quantum limits on flat tori. Annals of Mathematics, 145 (2). pp. 235-266. ISSN 0003-486X.

Full text is not posted in this repository. Consult Related URLs below.

Use this Persistent URL to link to this item:


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.

Item Type:Article
Related URLs:
URLURL TypeDescription
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.
Funding AgencyGrant Number
NSF Postdoctoral FellowshipUNSPECIFIED
Record Number:CaltechAUTHORS:20120120-085616919
Persistent URL:
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
Deposited By: Ruth Sustaita
Deposited On:20 Jan 2012 17:34
Last Modified:20 Jan 2012 17:34

Repository Staff Only: item control page