It is shown that the energy spectrum fluctuations of quantum systems can be formally considered as a discrete time series. The power spectrum behavior of such a signal for different systems suggests the following conjecture: The energy spectra of chaotic quantum systems are characterized by $1/f$ noise.

It was recently conjectured that $1/f$ noise is a fundamental characteristic of spectral fluctuations in chaotic quantum systems. This conjecture is based on the power spectrum behavior of the excitation energy fluctuations, which is different for chaotic and integrable systems. Using random matrix theory, we derive theoretical expressions that explain without free parameters the universal behavior of the excitation energy fluctuations power spectrum. The theory gives excellent agreement with numerical calculations and reproduces to a good approximation the $1/f$ ($1/f^2$) power law characteristic of chaotic (integrable) systems. Moreover, the theoretical results are valid for semiclassical systems as well.

The power law $1/f^{\alpha }$ in the power spectrum characterizes the fluctuating observables of many complex natural systems. Considering the energy levels of a quantum system as a discrete time series where the energy plays the role of time, the level fluctuations can be characterized by the power spectrum. Using a family of quantum billiards, we analyze the order-to-chaos transition in terms of this power spectrum. A power law $1/f^{\alpha }$ is found at all the transition stages, and it is shown that the exponent $\alpha$ is related to the chaotic component of the classical phase space of the quantum system.

The continuous Gaussian ensemble, also known as the $\nu$-Gaussian or $\nu$-Hermite ensemble, is a natural extension of the classical Gaussian ensembles of real $\left(\nu =1\right)$, complex $\left(\nu =2\right)$, or quaternion $\left(\nu =4\right)$ matrices, where $\nu$ is allowed to take any positive value. From a physical point of view, this ensemble may be useful to describe transitions between different symmetries or to describe the terrace-width distributions of vicinal surfaces. Moreover, its simple form allows one to speed up and increase the efficiency of numerical simulations dealing with large matrix dimensions. We analyze the long-range spectral correlations of this ensemble by means of the ${\delta }_n$ statistic. We derive an analytical expression for the average power spectrum of this statistic, $\overline{P_k^{\delta }}$, based on approximated forms for the two-point cluster function and the spectral form factor. We find that the power spectrum of ${\delta }_n$ evolves from $\overline{P_k^{\delta }}\propto 1/k$ at $\nu =1$ to $\overline{P_k^{\delta }}\propto 1/k^2$ at $\nu =0$. Relevantly, the transition is not homogeneous with a $1/f^{\alpha }$ noise at all scales, but heterogeneous with coexisting $1/f$ and $1/f^2$ noises. There exists a critical frequency $k_c\propto \nu$ that separates both behaviors: below $k_c$, $\overline{P_k^{\delta }}$ follows a $1/f$ power law, while beyond $k_c$, it transits abruptly to a $1/f^2$ power law. For $\nu >1$ the $1/f$ noise dominates through the whole frequency range, unveiling that the $1/f$ correlation structure remains constant as we increase the level repulsion and reduce to zero the amplitude of the spectral fluctuations. All these results are confirmed by stringent numerical calculations involving matrices with dimensions up to ${10}^5$.