Quantization of multidimensional cat maps
Quantization of multidimensional cat maps
复制标题
多维猫图的量化
DOI：
10.1088/0951-7715/13/2/302
复制
发表时间：
1999
期刊：
影响因子：
1.7
通讯作者：
A. O. D. Almeida
中科院分区：
文献类型：
--
作者：
A. Rivas;M. Saraceno;A. O. D. Almeida
在这项工作中，我们研究了具有多个自由度的猫地图。经典猫映射分类使用辛矩阵的凯莱参数化和密切相关的中心和弦生成函数。特别注意的是专门斜驶和椭圆双曲的行为，这是新的功能，四维地图：我们构建一个地图，是不是Anosov，但遍历和混合。然后，使用外尔表示的环面上的映射进行量化，并推导出一个特定的映射的Floquet角度的一般条件是量化。半经典近似是精确的，与维数或不动点的性质无关。我们挑出量子周期函数（QPF）的研究，即作为有限希尔伯特空间维数的函数的量子映射的周期。结果表明，QPF基本上依赖于猫映射的遍历性和退化Lyapunov指数的存在性。
In this work we study cat maps with many degrees of freedom. Classical cat maps are classified using the Cayley parametrization of symplectic matrices and the closely associated centre and chord generating functions. Particular attention is dedicated to loxodromic and elliptic-hyperbolic behaviour, which are new features of four-dimensional maps: we construct a map that is not Anosov, but is ergodic and mixing. The maps are then quantized using a Weyl representation on the torus and the general condition on the Floquet angles is derived for a particular map to be quantizable. The semiclassical approximation is exact, regardless of the dimensionality or of the nature of the fixed points. We single out the study of the quantum period function (QPF), that is the period of the quantum map as a function of the finite Hilbert space dimension. It is found that the QPF depends basically on the ergodicity and on the existence of degenerate Lyapunov exponents for some power of the cat map.