Mathematics – Number Theory
Scientific paper
1999-01-28
Mathematics
Number Theory
37 pages. New title. Spelling mistake in bibliography corrected. To appear in Duke Math. J
Scientific paper
We study semi-classical limits of eigenfunctions of a quantized linear hyperbolic automorphism of the torus ("cat map"). For some values of Planck's constant, the spectrum of the quantized map has large degeneracies. Our first goal in this paper is to show that these degeneracies are coupled to the existence of quantum symmetries. There is a commutative group of unitary operators on the state-space which commute with the quantized map and therefore act on its eigenspaces. We call these "Hecke operators", in analogy with the setting of the modular surface. We call the eigenstates of both the quantized map and of all the Hecke operators "Hecke eigenfunctions". Our second goal is to study the semiclassical limit of the Hecke eigenfunctions. We will show that they become equidistributed with respect to Liouville measure, that is the expectation values of quantum observables in these eigenstates converge to the classical phase-space average of the observable.
Kurlberg Par
Rudnick Zeev
No associations
LandOfFree
Hecke theory and equidistribution for the quantization of linear maps of the torus does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Hecke theory and equidistribution for the quantization of linear maps of the torus, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hecke theory and equidistribution for the quantization of linear maps of the torus will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-598637