Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Electronic Research Announcements
Electronic Research Announcements
ISSN 1079-6762

 

On quantum limits on flat tori


Author: Dmitry Jakobson
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 80-86
MSC (1991): Primary 42B05, 81Q50, 58C40, 52B20, 11D09, 11J86
MathSciNet review: 1350683
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We classify all weak $*$ limits of squares of normalized eigenfunctions of the Laplacian on two-dimensional flat tori (we call these limits 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 $\mathbb{R}^n$ 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. 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.


References [Enhancements On Off] (What's this?)

  • Bak1 Alan Baker, Transcendental number theory, Cambridge University Press, London-New York, 1975. MR 0422171 (54 #10163)
  • Bak2 A. Baker, A sharpening of the bounds for linear forms in logarithms. II, Acta Arith. 24 (1973), 33–36. (errata insert). Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, I. MR 0376549 (51 #12724)
  • Bou1 J. Bourgain, Eigenfunction bounds for the Laplacian on the 𝑛-torus, Internat. Math. Res. Notices 3 (1993), 61–66. MR 1208826 (94f:58127), http://dx.doi.org/10.1155/S1073792893000066
  • Bou2 J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Shroödinger equations, preprint (1994).
  • Con Bernard Connes, Sur les coefficients des séries trigonométriques convergentes sphériquement, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 4, Aii, A159–A161 (French, with English summary). MR 0422991 (54 #10975)
  • HL G. H. Hardy, Collected papers of G. H. Hardy (Including Joint papers with J. E. Littlewood and others). Vol. I, Edited by a committee appointed by the London Mathematical Society, Clarendon Press, Oxford, 1966. MR 0201267 (34 #1151)
  • Jak D. Jakobson, Ph. D. Thesis, Princeton University (1995).
  • Mi M. Mignotte, Intersection des images de certaines suites récurrentes linéaires, Theoret. Comput. Sci. 7 (1978), no. 1, 117–122 (French). MR 0498356 (58 #16486)
  • Sar P. Sarnak, Arithmetic Quantum Chaos, Blyth Lectures, Toronto (1993).
  • Zyg A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189–201. MR 0387950 (52 #8788)

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (1991): 42B05, 81Q50, 58C40, 52B20, 11D09, 11J86

Retrieve articles in all journals with MSC (1991): 42B05, 81Q50, 58C40, 52B20, 11D09, 11J86


Additional Information

Dmitry Jakobson
Affiliation: address Department of Mathematics, Princeton University, Princeton, NJ 08544
Email: diy@math.princeton.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-95-02004-X
PII: S 1079-6762(95)02004-X
Received by editor(s): April 20, 1995
Received by editor(s) in revised form: July 19, 1995
Communicated by: Yitzhak Katznelson
Article copyright: © Copyright 1995 American Mathematical Society