On Matrix Elements for the Quantized Cat Map Modulo Prime Powers
Abstract
The quantum cat map is a model for a quantum system with underlying chaotic dynamics. In this paper we study the matrix elements of smooth observables in this model, when taking arithmetic symmetries into account. We give explicit formulas for the matrix elements as certain exponential sums. With these formulas we can show that there are sequences of eigenfunctions for which the matrix elements decay significantly slower then was previously expected. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average.
- Publication:
-
Annales Henri Poincaré
- Pub Date:
- December 2008
- DOI:
- 10.1007/s00023-008-0394-4
- arXiv:
- arXiv:0802.3237
- Bibcode:
- 2008AnHP....9.1479K
- Keywords:
-
- Matrix Element;
- Chaotic Dynamic;
- Prime Power;
- Toral Automorphism;
- Matrix Element Decay;
- Mathematics - Number Theory;
- Mathematical Physics;
- 81Q50;
- 11L03
- E-Print:
- 26 pages, final version, to appear in AHP