辛Anosov同态的预量子转移算子
Prequantum transfer operator for symplectic Anosov diffeomorphism
摘要：
本文定义了辛Anosov同态f：M-> M的预量子化，它是保相关特定联络的保相关同态f的U（1）扩张，并研究了相关转移算子的谱性质，称之为预量子转移算子.这是一个与负弯曲流形上的测地线流（或接触Anosov流）相关的转移算子的模型。我们将前量子转移算符限制在关于U（1）作用量的N阶Fourier模，并将转移算符看作Fourier积分算符，利用半经典分析方法研究了在极限N →无穷大时的谱性质.在主要结果中，我们给出了光谱的“能带结构“，即光谱包含在几个分离的环和一个与原点同心的圆盘中。我们表明，与特殊（Hölder连续）势V0=1/2 log| det Df_x|_{E_u}|最外面的环是单位圆，与其他部分分开。为此，我们使用的转移算子的格拉斯曼束的扩展。利用Atiyah-Bott迹公式，建立了大时间下具有指数小余项的Gutzwiller迹公式。我们还表明，对于一个潜在的V，使最外层的环是从其他部分分离，在最外层的环的特征值集中在一个圆的半径exp其中<.>表示的空间平均M。这些本征值的个数由“外尔定律”给出，即N^d. N（M），其中d=1/2。dim（M）在最前面的顺序。我们开发了一个半经典演算相关的前量子运营商通过定义量子化的观测Op（psi）的内在方式。我们得到了量子输运的半经典Egorov公式是精确的。我们从物理学的角度解释了这些结果，即在大时间内经典关联函数中量子动力学的出现。我们将这些结果与量子混沌中的标准量子化（几何量子化）进行比较。
英文摘要：
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 associated 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->infinity, 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 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 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 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 where <.> denotes the spatial average on M. The number of these eigenvalues 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 the prequantum operator by defining quantization of observables Op(psi) 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 quantization (geometric quantization) in quantum chaos.
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