论文标题
统计核谱与$ Q $ - 正常和双变量$ Q $ - 正常分布和$ Q $ - hermite多项式
Statistical Nuclear Spectroscopy with $q$-normal and bivariate $q$-normal distributions and $q$-Hermite polynomials
论文作者
论文摘要
J.B. French在60年代后期引入的统计核谱法（也称为光谱分布方法），并在后来由他的小组和许多其他组在后来的几年进行了详细开发，它基于该州的高斯形式（特征值）（特征值）和壳模型中的过渡强度密度，并将其扩展到针对壳模型的部分密度，以延伸到壳模型模型下。高斯形式在嵌入的随机矩阵集合中具有其基础，其核汉密尔顿人由平均场一体部分和残留的两体部分组成。 However, following the recent random matrix results for the so called Sachdev-Ye-Kitaev model due to Verbaaarschot et al, embedded random matrix ensembles with $k$-body interactions are re-examined and it is shown that the density of states, transition strength densities and strength functions (partial densities) in fact follow more closely the $q$-normal distribution (the parameter $q$ is related to the fourth这些分布的力矩$ q = 1 $给出高斯，$ q = 0 $给出半圆形的表格）。 $ q $ - 正常具有重要的属性，其限制为$ 0 \ le q <1 $。 Bryc，Szabowski等人研究了$ Q $ - 纳米广告（也是其双变量和一般的多变量扩展）和相关的$ Q $ - hermite多项式[P.J. Szabowski，电子概率杂志{\ bf 15}，1296（2010）]。遵循这些内容，在本文中开发的是统计核谱，基于$ q $ - normal（单变量和双变量）分布和相关的$ q $ hermite多项式。特别是，提出了针对核水平密度，壳模型轨道占用，过渡强度（用于电磁和$β$以及双$β$ - decay型运算符）和强度总和的公式。
Statistical nuclear spectroscopy (also called spectral distribution method), introduced by J.B. French in late 60's and developed in detail in the later years by his group and many other groups, is based on the Gaussian forms for the state (eigenvalue) and transition strength densities in shell model spaces with their extension to partial densities defined over shell model subspaces. The Gaussian forms have their basis in embedded random matrix ensembles with nuclear Hamiltonians consisting of a mean-field one-body part and a residual two-body part. However, following the recent random matrix results for the so called Sachdev-Ye-Kitaev model due to Verbaaarschot et al, embedded random matrix ensembles with $k$-body interactions are re-examined and it is shown that the density of states, transition strength densities and strength functions (partial densities) in fact follow more closely the $q$-normal distribution (the parameter $q$ is related to the fourth moment of these distributions with $q=1$ giving Gaussian and $q=0$ giving semi-circle form). The $q$-normal has the important property that it is bounded for $0 \le q < 1$. The $q$-normal (also its bivariate and general multi-variate extensions) and the associated $q$-Hermite polynomials are studied for their properties by Bryc, Szabowski and others [P.J. Szabowski, Electronic Journal of Probability {\bf 15}, 1296 (2010)]. Following these, in the present article developed is statistical nuclear spectroscopy based on $q$-normal (univariate and bivariate) distributions and the associated $q$-Hermite polynomials. In particular, formulation is presented for nuclear level densities, shell model orbit occupancies, transition strengths (for electromagnetic and $β$ and double $β$-decay type operators) and strength sums.
