具有$q$ -正态和二元$q$ -正态分布以及$q$ -Hermite多项式的统计核谱学
Statistical Nuclear Spectroscopy with $q$-normal and bivariate $q$-normal distributions and $q$-Hermite polynomials
摘要：
统计核谱学（也称为谱分布法）是由J. B. French在60年代后期提出的，并在后来由他的小组和许多其他小组详细发展的，是基于壳模型空间中状态（本征值）和跃迁强度密度的高斯形式，并将其扩展到壳模型子空间上定义的部分密度。高斯形式的基础是嵌入随机矩阵系综，核哈密顿量由平均场的单体部分和剩余的两体部分组成。然而，在Verbaaarschot等人最近对所谓的Sachdev-Ye-Kitaev模型的随机矩阵结果之后，重新检查了具有$k$ -体相互作用的嵌入随机矩阵系综，并且表明态密度，过渡强度密度和强度函数（部分密度）实际上更接近于$q$ -正态分布（参数$q$与这些分布的四阶矩有关，其中$q=1$给出高斯分布，而$q=0$给出半圆形）。$q$ -正规有一个重要的性质，即它对于$0 \le q < 1$是有界的. $q$ -正态（以及其双变量和一般多变量扩展）和相关联的$q$ -Hermite多项式的性质由Bryc、Szabowski等人研究[P. J. Szabowski，Electronic Journal of Probability { \bf 15}，1296（2010）]。在此基础上，本文发展了基于$q$ -正态（单变量和双变量）分布和相关的$q$ -Hermite多项式的统计核谱学。特别是，制定的核能级密度，壳模型轨道占领，过渡强度（电磁和$β$和双$β$衰变型运营商）和强度总和。
英文摘要：
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.
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