Related papers: Fractal Weyl Law for Open Chaotic Maps
We consider compact Lie groups extensions of expanding maps of the circle, essentially restricting to U(1) and SU(2) extensions. The central object of the paper is the associated Ruelle transfer (or pull-back) operator $\hat{F}$. Harmonic…
We study relevant features of the spectrum of the quantum open baker map. The opening consists of a cut along the momentum $p$ direction of the 2-torus phase space, modelling an open chaotic cavity. We study briefly the classical forward…
A clear signature of classical chaoticity in the quantum regime is the fractal Weyl law, which connects the density of eigenstates to the dimension $D_0$ of the classical invariant set of open systems. Quantum systems of interest are often…
Under general assumptions, the numbers of semiclassical resonances is known to be bounded from above by a negative power of $h$ which is given by the fractal dimension of the trapped set. In this paper we provide examples of operators with…
The semiclassical structure of resonance states of classically chaotic scattering systems with partial escape is investigated. We introduce a local randomization on phase space for the baker map with escape, which separates the smallest…
We prove a new fractal Weyl upper bound for the high-energy distribution of resonances of convex co-compact hyperbolic surfaces which matches the improved spectral gap given by Fourier decay. This improves upon the fractal Weyl bound of…
Wave scattering in chaotic systems can be characterized by its spectrum of resonances, $z_n=E_n-i\frac{\Gamma_n}{2}$, where $E_n$ is related to the energy and $\Gamma_n$ is the decay rate or width of the resonance. If the corresponding ray…
Previous studies have shown that fractal scatterings in weak interactions of solitary waves in the generalized nonlinear Schr\"odinger equations are described by a universal second-order separatrix map. In this paper, this separatrix map is…
Resonance states in quantum chaotic scattering systems have a multifractal structure that depends on their decay rate. We show how classical dynamics describes this structure for all decay rates in the semiclassical limit. This result for…
This review article will present some recent results and methods in the study of 1-particle quantum or wave scattering systems, in the semiclassical/high frequency limit, in cases where the corresponding classical/ray dynamics is chaotic.…
Fractal scatterings in weak solitary wave interactions is analyzed for generalized nonlinear Schr\"odiger equations (GNLS). Using asymptotic methods, these weak interactions are reduced to a universal second-order map. This map gives the…
We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show that the spectrum is characterized by the fractal Weyl law recently established…
We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is…
In recent years fractal Weyl laws and related quantum eigenfunction hypothesis have been studied in a plethora of numerical model systems, called quantum maps. In some models studied there one can easily prove uniform hyperbolicity. Yet, a…
We prove an analogue of the pointwise Weyl law for families of unitary matrices obtained from quantization of one-dimensional interval maps. This quantization for interval maps was introduced by Pako\'nski et al. [J. Phys. A: Math. Gen. 34…
On a closed manifold $M$, we consider a smooth vector field $X$ that generates an Anosov flow. Let $V\in C^{\infty}\left(M;\mathbb{R}\right)$ be a smooth function called potential. It is known that for any $C>0$, there exists some…
For a class of quantized open chaotic systems satisfying a natural dynamical assumption, we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is…
We find Weyl upper bounds for the quantum open baker's map in the semiclassical limit. For the number of eigenvalues in an annulus, we derive the asymptotic upper bound $\mathcal O(N^\delta)$ where $\delta$ is the dimension of the trapped…
A fractal method to detect, locate and quantify chaos in multi-dimensional-conservative-closed systems, based on the creation of artificial exits, is presented. The method is invariant under space-time changes of coordinates and can be used…
We consider quantum maps induced by periodically-kicked scattering systems and discuss the computation of their resonance spectra in terms of complex scaling and sufficiently weak absorbing potentials. We also show that strong absorptive…