PREQUANTUM TRANSFER OPERATOR FOR SYMPLEC

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
V₀ = 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 - V₀>) where
<.> denotes the spatial average on M. The number of the eigenvalues
in the outermost annulus satisfies a Weyl law, that is, N^dVol (M) in
the leading order with d = 1/2dimM.

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.