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Prequantum chaos: resonsances of the prequantum cat map. (English) Zbl 1145.81034

Prequantum dynamics was introduced in the 1970s by Kostant, Souriau and Kirillov as an intermediate between classical and quantum dynamics. In this paper the author presents the prequantum quantization of a linear hyperbolic map on the torus and its relation to the corresponding quantum system.
The classical dynamical system considered is a simple model for chaotic dynamics given by the iteration of a linear hyperbolic map \(M\) on the torus \(\mathbb{T}^2\). The corresponding quantum dynamics is given by the iteration of a unitary operator \(\widehat{M}\) on a finite dimensional Hilbert space of dimension \(N=\frac{1}{2\pi\hbar}\). The prequantization is a lift of the classical map to a Hermitian complex line bundle \(L\) over \(\mathbb{T}^2\) with curvature \(2\pi iN \omega\) where \(\omega\) the symplectic two form on \(\mathbb{T}^2\). This map defines a unitary operator \(\widetilde M\) on the infinite dimensional prequantum Hilbert space \(L^2(L)\).
The main result of this paper is the following relation between the quantum spectrum and the resonances for the prequantum dynamics. The author shows that the Ruelle-Pollicott resonances for the prequantum dynamics are given by \(r_{n,k}=e^{i\varphi_k-\lambda(n+1/2)}\) where \(e^{i\varphi_k},\;k=1,\ldots,N\) is the spectrum of the quantum map (i.e., the eigenvalues of \(\widehat{M}\)) and \(\lambda\) is the Lyapunov exponent of the classical map. He further shows that the prequantum operator satisfies an exact trace formula which can be used to derive the trace formula for the quantum map (which is also an exact formula in this case).
The author suggests that this method could be used also for nonlinear maps, in order to derive a semi-classical trace formula valid beyond the Ehrenfest time.

MSC:

81Q50 Quantum chaos
81S10 Geometry and quantization, symplectic methods
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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