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We define the prequantization of a symplectic Anosov diffeomorphism f:M-> M,which is a U(1) extension of the diffeomorphism f preserving an associatedspecific connection, and study the spectral properties of the associatedtransfer operator, called prequantum transfer operator. This is a model for thetransfer operators associated to geodesic flows on negatively curved manifolds(or contact Anosov flows). We restrict the prequantum transfer operator to theN-th Fourier mode with respect to the U(1) action and investigate the spectralproperty in the limit N->infinity, regarding the transfer operator as a Fourierintegral operator and using semi-classical analysis. In the main result, weshow a " band structure " of the spectrum, that is, the spectrum is containedin a few separated annuli and a disk concentric at the origin. We show that,with the special (H\"older continuous) potential V0=1/2 log |det Df_x|_{E_u}|,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 Grassmanianbundle. Using Atiyah-Bott trace formula, we establish the Gutzwiller traceformula with exponentially small reminder for large time. We show also that,for a potential V such that the outermost annulus is separated from the otherparts, most of the eigenvalues in the outermost annulus concentrate on a circleof radius exp where <.> denotes the spatial average on M. The number of theseeigenvalues is given by the "Weyl law", that is, N^d.Vol(M) with d=1/2. dim(M)in the leading order. We develop a semiclassical calculus associated to theprequantum operator by defining quantization of observables Op(psi) in anintrinsic way. We obtain that the semiclassical Egorov formula of quantumtransport is exact. We interpret all these results from a physical point ofview as the emergence of quantum dynamics in the classical correlationfunctions for large time. We compare these results with standard quantization(geometric quantization) in quantum chaos.

Prequantum transfer operator for symplectic Anosov diffeomorphism