On quantum limits on flat tori
Author:
Dmitry Jakobson
Journal:
Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 8086
MSC (1991):
Primary 42B05, 81Q50, 58C40, 52B20, 11D09, 11J86
MathSciNet review:
1350683
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Abstract: We classify all weak limits of squares of normalized eigenfunctions of the Laplacian on twodimensional 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 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 twodimensional result of Zygmund to three dimensions; we discuss various possible generalizations of that result to higher dimensions and the relation to norms of the densities of quantum limits and their Fourier series.
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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/S107967629502004X
PII:
S 10796762(95)02004X
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
