Let ${\left\{{\epsilon }_i\right\}}_{i=1}^{\infty }$ represent the set of unfolded energy eigenvalues of a quantized billiard system. Then, the set $\{{\delta }_n={\epsilon }_{n+1}-{\epsilon }_1-n\}$ may be considered as a finite time series of size $M=n_m-n_0$, in which $n$ plays the role of a discrete time and ${\delta }_n$ fluctuates around zero. These random fluctuations are known to exhibit the $1/f^{\alpha }$ noise characteristic, where $\alpha \in [1,2]$, the limit $\alpha =1\left(2\right)$ corresponding to a fully chaotic (regular) classical dynamics. Previously, the presence of $1/f^{\alpha }$ noise in the ${\delta }_n$ statistics has been demonstrated in a family of billiards for which the classical phase space is divided (Kolmogorov-Arnold-Moser systems). Here, the ${\delta }_n$ statistics is further explored, so as to include two new features, namely, (i) nonchaotic full ergodicity and (ii) symmetry. Four billiard families, classified as type $P$ and type $S$, are considered. The type-$P$ families are comprised of polygonal billiards, which are never chaotic, but may exhibit a mixing classical dynamics characterized quantitatively through the position autocorrelation function decay exponent $\sigma$. A type-$S$ billiard has a smooth boundary and, correspondingly, a divided phase space with normalized chaotic volume ${\rho }_{\text{c}}$. Possible correlations between the spectral noise parameter $\alpha$ and the classical quantities, $\sigma$ for type-$P$ and ${\rho }_{\text{c}}$ for type-$S$ domains, are numerically investigated. Spearman's rank correlation coefficients support that the quantum spectral exponent $\alpha$ scales with the classical quantities. In addition, one family in each class is $C_3$ symmetric. In this case, singlets and doublets have different spectral statistics in the chaotic limit, namely, the singlet (doublet) subspectrum follows the Gaussian orthogonal ensemble (Gaussian unitary ensemble) of random matrices. Let ${\alpha }_1\left({\alpha }_2\right)$ be the noise exponent in a singlet (doublet) time series. Numerical calculations of Pearson's correlation coefficients show that ${\alpha }_1$ and ${\alpha }_2$ are compatible with each other for both type-$S$ and type-$P$ within symmetrical families in the regime $1<\alpha <2$.

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• Chaos
• Quantum chaos
• Hamiltonian systems
• Quantum billiards
• Quantum chaotic systems
• Phase space methods
• Time series analysis