论文标题
对二进制量子图的特征多项式的完整动态评估
Complete Dynamical Evaluation of the Characteristic Polynomial of Binary Quantum Graphs
论文作者
论文摘要
我们使用动力学方法评估了特征多项式的特征多项式系数的方差。这是第一个示例，可以根据具有混乱的经典动力学的系统的周期性轨道来评估光谱统计量，而无需采用半经典限制，这是大图的极限。该方差取决于特定原始伪轨道（独特的原始周期轨道的集合）的大小：没有自我干扰的原始伪轨道和原始伪齿轨道的集合，它们具有固定数量的自身交叉，所有这些都由两个Arc组成，由两个pseud or Bits Crosss组成。为了显示其他伪轨道没有贡献，我们给出了两个论点。第一个是基于从对原始伪轨道对的总和到一个伪轨道的总和的降低，而没有重复键。第二个对单词的lyndon分解采用了平等论点。对于二进制图的家族，在半经典限制下，我们显示伪轨道公式接近通用常数，独立于多项式的系数。这是通过计算给定长度的原始伪轨道总数获得的。
We evaluate the variance of coefficients of the characteristic polynomial for binary quantum graphs using a dynamical approach. This is the first example where a spectral statistic can be evaluated in terms of periodic orbits for a system with chaotic classical dynamics without taking the semiclassical limit, which here is the limit of large graphs. The variance depends on the sizes of particular sets of primitive pseudo orbits (sets of distinct primitive periodic orbits): the set of primitive pseudo orbits without self-intersections and the sets of primitive pseudo orbits with a fixed number of self-intersections, all of which consist of two arcs of the pseudo orbit crossing at a single vertex. To show other pseudo orbits do not contribute we give two arguments. The first is based on a reduction of the variance formula from a sum over pairs of primitive pseudo orbits to a sum over pseudo orbits where no bonds are repeated. The second employs a parity argument for the Lyndon decomposition of words. For families of binary graphs, in the semiclassical limit, we show the pseudo orbit formula approaches a universal constant independent of the coefficient of the polynomial. This is obtained by counting the total number of primitive pseudo orbits of a given length.
