Prequantum transfer operator for symplectic Anosov diffeomorphism

We define the prequantization of a symplectic Anosov diffeomorphism f : M → M , which is a U(1) extension of the diffeomorphism f preserving an associated specific connection, and study the spectral properties of the associated transfer operator, called prequantum transfer operator. This is a model for the transfer operators asso10 ciated to geodesic flows on negatively curved manifolds (or contact Anosov flows). We restrict the prequantum transfer operator to the N -th Fourier mode with respect to the U(1) action and investigate the spectral property in the limit N → ∞, regarding the transfer operator as a Fourier integral operator and using semi-classical analysis. In the main result, we show a “ band structure ” of the spectrum, that is, the 15 spectrum is contained in a few separated annuli and a disk concentric at the origin. We show that, with the special (Hölder continuous) potential V0 = 1 2 log |detDfx|Eu |, the outermost annulus is the unit circle and separated from the other parts. For this, we use an extension of the transfer operator to the Grassmanian bundle. Using Atiyah-Bott trace formula, we establish the Gutzwiller trace formula with exponen20 tially small reminder for large time. We show also that, for a potential V such that the outermost annulus is separated from the other parts, most of the eigenvalues 1 ha l-0 07 03 23 4, v er si on 2 8 M ay 2 01 3 in the outermost annulus concentrate on a circle of radius exp (〈V − V0〉) where 〈.〉 denotes the spatial average on M . The number of these eigenvalues is given by the “Weyl law”, that is, NdVolM with d = 12dimM in the leading order. 25 We develop a semiclassical calculus associated to the prequantum operator by defining quantization of observables Op~ (ψ) in an intrinsic way. We obtain that the semiclassical Egorov formula of quantum transport is exact. We interpret all these results from a physical point of view as the emergence of quantum dynamics in the classical correlation functions for large time. We compare these results with standard 30 quantization (geometric quantization) in quantum chaos. 2 ha l-0 07 03 23 4, v er si on 2 8 M ay 2 01 3