We define the prequantization of a symplectic Anosov diffeomorphism

f : M -> M as a U(1) extension of the diffeomorphism f preserving

a connection related to the symplectic structure on M. We study the

spectral properties of the associated transfer operator with a given

potential V (...) C ^Infini ( M ), called prequantum transfer operator. This is

a model of transfer operators for 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 -> Infini, regarding the transfer operator as a

Fourier integral operator and using semi-classical analysis. In the main

result, under some pinching conditions, we show a " band structure " of

the spectrum, that is, the spectrum is contained in a few separated

annuli and a disk concentric at the origin.

We show that, with the special (Hölder continuous) potential

V ₀ = 1/2 log |det D f | E_u |, where E_u is the unstable subspace, 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 exponentially 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 in the outermost

annulus concentrate on a circle of radius exp ( <V - V ₀>) where

<.> denotes the spatial average on M. The number of the eigenvalues

in the outermost annulus satisfies a Weyl law, that is, N^d Vol ( M ) in

the leading order with d = 1/2dim M .

We develop a semiclassical calculus associated to the prequantum operator

by defining quantization of observables Op_N ( Psi ) as the spectral

projection of multiplication operator by Psi to this outer annulus. We

obtain that the semiclassical Egorov formula of quantum transport

is exact. The correlation functions defined by the classical transfer

operator are governed for large time by the restriction to the outer

annulus that we call the quantum operator. We interpret 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 quantization (geometric quantization) in

quantum chaos.