实随机矩阵的高斯伪正交系综
Gaussian pseudo-Orthogonal Ensemble of Real Random Matrices
AI总结：
本文构建了实伪对称矩阵的高斯伪正交系综并研究其特征值统计，发现参数正负决定特征值为实数或共轭复数对且实特征值呈中间统计，该矩阵可表示 PT 对称量子系统。
摘要：
本文使用两个非零实参数 $λ$ 和 $μ$，在度量 $η$ 下，利用从高斯随机总体中独立抽取的 $\altmathcal{N}=n(n+1)/2$ 个元素，构建了大量 $N$ 个 $n \times n$（$n$ 为偶数且较大）实伪对称矩阵的高斯伪正交系综，并研究了其特征值的统计特性。当 $λμ>0$ 时，我们证明该伪对称矩阵相似于实对称矩阵，因此所有特征值均为实数，其谱分布满足 Wigner 统计。但当 $λμ<0$ 时，特征值为实数或共轭复数对。我们发现这些实特征值表现出中间统计特性。我们证明这些伪对称矩阵的对角化矩阵 ${\cal D}$ 在常数度量 $ζ$ 下满足 $\altmathcal{D}^t ζ\altmathcal{D}= ζ$ 因而是伪正交的，故属于伪正交群。这些伪对称矩阵可用于表示具有精确（未破缺）或破缺 PT 对称性的宇称-时间（PT）对称量子系统。
英文摘要：
Here, using two real non-zero parameters $λ$ and $μ$, we construct Gaussian pseudo-orthogonal ensembles of a large number $N$ of $n \times n$ ($n$ even and large) real pseudo-symmetric matrices under the metric $η$ using $ \altmathcal {N}=n(n+1)/2$ elements independently drawn from a Gaussian random population and investigate the statistical properties of the eigenvalues. When $λμ>0$, we show that the pseudo-symmetric matrix is similar to a real symmetric matrix, consequently, all the eigenvalues are real and so the spectral distributions satisfy Wigner's statistics. But when $λμ<0$ the eigenvalues are either real or complex conjugate pairs. We find that these real eigenvalues exhibit intermediate statistics. We show that the diagonalizing matrices ${ \cal D}$ of these pseudo-symmetric matrices are pseudo-orthogonal under a constant metric $ζ$ as $ \altmathcal{D}^t ζ\altmathcal{D}= ζ$, and hence they belong to a pseudo-orthogonal group. These pseudo-symmetric matrices serve to represent the parity-time (PT)-symmetric quantum systems having exact (un-broken) or broken PT-symmetry.
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