On matrix elements for the quantized cat map modulo prime powers

Details On matrix elements for the quantized cat map modulo prime powers On matrix elements for the quantized cat map modulo prime powers

2008-02-21

arxiv.org/abs/0802.3237v2

Mathematics

Number Theory

26 pages, final version, to appear in AHP

Scientific paper

10.1007/s00023-008-0394-4

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 conjectured. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average.

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