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On the fluctuations of matrix elements of the quantum cat map. (English) Zbl 1229.81108

Summary: We study the fluctuations of the diagonal matrix elements of the quantum cat map about their limit. We show that after suitable normalization, the fifth centered moment for the Hecke basis vanishes in the semiclassical limit, confirming in part a conjecture of P. Kurlberg and Z. Rudnick [Ann. Math. (2) 161, No. 1, 489–507 (2005; Zbl 1082.81054)].
We also study sums of matrix elements lying in short windows. For observables with zero mean, the first moment of these sums is zero, and the variance was determined by P. Kurlberg, the author and Z. Rudnick [Nonlinearity 20, No. 10, 2289–2304 (2007; Zbl 1187.81131)]. We show that if the window is sufficiently small in terms of Planck’s constant, then the third moment vanishes if we normalize so that the variance is of order 1.

MSC:

81Q50 Quantum chaos
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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