Alex Gamburd et al 2003 J. Phys. A: Math. Gen. 36 3487 doi:10.1088/0305-4470/36/12/336
Alex Gamburd1, John Lafferty2 and Dan Rockmore3
Show affiliationsAccording to one of the basic conjectures in quantum chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms—'cat maps'. Our numerical experiments indicate that for 'generic' choices of cat maps, the unfolded consecutive spacing distribution in the irreducible components of the Nth quantization (given by the N-dimensional Weil representation) approaches the GOE/GSE law of random matrix theory. For certain special 'arithmetic' transformations, related to the Ramanujan graphs of Lubotzky, Phillips and Sarnak, the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.
05.45.Mt Quantum chaos; semiclassical methods
11F27 Theta series; Weil representation
Issue 12 (28 March 2003)
Received 7 August 2002, in final form 6 November 2002
Published 12 March 2003
Alex Gamburd et al 2003 J. Phys. A: Math. Gen. 36 3487
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