By Frederic Faure, Masato Tsujii
We outline the prequantization of a symplectic Anosov diffeomorphism f:MM as a U(1) extension of the diffeomorphism f holding a connection on the topic of the symplectic constitution on M. We learn the spectral homes of the linked move operator with a given strength VC^(M), referred to as prequantum move operator. it is a version of move operators for geodesic flows on negatively curved manifolds (or touch Anosov flows). We limit the prequantum move operator to the N-th Fourier mode with appreciate to the U(1) motion and examine the spectral estate within the restrict N, concerning the move operator as a Fourier critical operator and utilizing semi-classical research. more often than not consequence, less than a few pinching stipulations, we exhibit a ``band structure'' of the spectrum, that's, the spectrum is contained in a number of separated annuli and a disk concentric on the beginning. We express that, with the distinct (Hölder non-stop) strength V_0=12|Df|_E_u|, the place E_u is the volatile subspace, the outermost annulus is the unit circle and separated from the opposite components. For this, we use an extension of the move operator to the Grassmanian package deal. utilizing Atiyah-Bott hint formulation, we identify the Gutzwiller hint formulation with exponentially small reminder for big time. We exhibit additionally that, for a possible V such that the outermost annulus is separated from the opposite components, lots of the eigenvalues within the outermost annulus be aware of a circle of radius (V-V_0) the place . denotes the spatial ordinary on M. The variety of the eigenvalues within the outermost annulus satisfies a Weyl legislation, that's, N^dVol(M) within the major order with d=12dimM. We improve a semiclassical calculus linked to the prequantum operator by way of defining quantization of observables Op_N() because the spectral projection of multiplication operator through to this outer annulus. We receive that the semiclassical Egorov formulation of quantum shipping is detailed. The correlation features outlined via the classical move operator are ruled for big time through the restrict to the outer annulus that we name the quantum operator. We interpret those effects from a actual viewpoint because the emergence of quantum dynamics within the classical correlation capabilities for giant time. We examine those effects with general quantization (geometric quantization) in quantum chaos.
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DYNAMICAL CORRELATION FUNCTIONS AND QUANTUM DYNAMICS 25 ˆ and Fˆ have the same spectrum on |z| > θ. 3) their spectrum is in |z| > 1 − ε > θ. Therefore all their spectrum coincides. 4. 6. 3 above. , the Weyl quantization or geometric quantization, one obtains a family of unitary operators Fˆ : H → H acting in some finite dimensional (family of) Hilbert spaces. 3). 4) but with an error term on the right hand side of the form O ( θn ) with θ = eh0 /2 > 1 where h0 > 0 is the topological entropy which represents the exponential growing number of periodic orbits (See  and the references therein).
12 shows that the external spectrum of the transfer operator concentrates uniformly on the unit circle as N = 1/ (2π ) → ∞. (We have not represented here the structure of the internal bands inside the disc of radius r1+ < 1). 12. 6) with the special choice of the smooth potential V˜0 (l) = 12 log detDf |f −1 (l) on G. Then the operator G FN χ ˆ extends to a bounded operator on H rN (PG ). The Ruelle-Pollicott resonances Res FN of FN χ ˆ concentrates on the unit circle as N = 1/ (2π ) → ∞ and is separated from the internal resonances by a non vanishing asymptotic spectral gap (r1+ < r0− = r0+ = 1).
K1 is a proper subset of K0 . 1(b). For any n ≥ 1, let n Kn := fG (K0 ) . 11) Kn . n=1 Let χ ∈ C ∞ (G) be a function such that χ (l) = 0 for l ∈ / K0 and χ (l) = 1 for l ∈ K1 . 12) χ ˆ : C ∞ (PG ) → C ∞ (PG ) for the multiplication operator by the function χ ◦ πG , where πG : PG → G is the projection. 13) = F n ◦ χ. 7). By Also χ ˆ preserves the space of equivariant functions CN duality the operator FN ◦ χ ˆ extends to the space of equivariant distributions DN (PG ). 14) supp FN ◦ χ ˆ −1 u ⊂ πG (Kn ) .