Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems

Körber, Martin J.; Michler, Matthias; Bäcker, Arnd; Ketzmerick, Roland

doi:10.1103/physrevlett.111.114102

Cited by 22 publications

(29 citation statements)

References 67 publications

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“…Mathematical work established that in consequence the resonances follow a modified, fractal Weyl law [9][10][11][12][13][14]. This fractal Weyl law has been confirmed numerically for a wide range of quantum maps [15][16][17][18][19][20][21][22][23][24], and numerical [25][26][27][28] as well as first experimental [29] work shows that the fractal Weyl law also holds for autonomous systems. In a simple physical picture, this law originates from the increasing phase-space resolution as one approaches the classical limit, which results in a proliferation of short-lived resonances that follow ballistical classical escape routes [15].…”

Section: Introductionmentioning

confidence: 84%

Formation and interaction of resonance chains in the open three-disk system

et al. 2014

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In ballistic open quantum systems, one often observes that the resonances in the complex-energy plane form a clear chain structure. Taking the open three-disk system as a paradigmatic model system, we investigate how this chain structure is reflected in the resonance states and how it is connected to the underlying classical dynamics. Using an efficient scattering approach, we observe that resonance states along one chain are clearly correlated, while resonance states of different chains show an anticorrelation. Studying the phase-space representations of the resonance states, we find that their localization in phase space oscillates between different regions of the classical trapped set as one moves along the chains, and that these oscillations are connected to a modulation of the resonance spacing. A single resonance chain is thus not a WKB quantization of a single periodic orbit, but the structure of several oscillating chains arises from the interaction of several periodic orbits. We illuminate the physical mechanism behind these findings by combining the semiclassical cycle expansion with a quantum graph model.

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Section: Introductionmentioning

confidence: 84%

Formation and interaction of resonance chains in the open three-disk system

et al. 2014

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“…The work of Sjöstrand [52] on semiclassical bounds for resonance counting has led to a general expectation for chaotic scattering systems that the number of resonances near the continuous spectrum should satisfy a power law with exponent equal to half of the dimension of the classical trapped set. Recently a large number of theoretical [56,15,37,53,36,38,35,9], numerical [48,24,25,27,49] and experimental [26,28,46,1,23] studies have appeared in support of this conjectural 'fractal Weyl law'.…”

Section: Fractal Weyl Conjecturementioning

confidence: 99%

Distribution of Resonances for Hyperbolic Surfaces

Borthwick

2014

Experimental Mathematics

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We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by counting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspects of the resonance distribution that have attracted attention recently: the fractal Weyl law, the spectral gap, and the concentration of decay rates.

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“…In open quantum system s the fractality o f the classical dynam ics appears in the distribution o f eigenstates [4] and in the Weyl law [5]. R ecent studies considered the effect of periodic orbits [6] and w eak chaos [7][8][9] and w ere conducted on four-dim ensional H am iltonian system s [7] and different area-preserving m aps [6,[8][9][10]. In these system s the fractal Weyl law can be w ritten as N (|v ,| > |v[Cutoff) ~ M d°)/2, (1) w here M is the dim ension o f the H ilbert space, DgS) is the fractal dim ension o f the chaotic saddle, and |v |cutoff is a cu to ff separating long-lived from short-lived states with eigenvalues u,-.…”

Section: Introductionmentioning

confidence: 99%

Quantum signatures of classical multifractal measures

Schönwetter

Altmann

2015

Phys. Rev. E

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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 partially open (e.g., cavities in which trajectories are partially reflected or absorbed). In the corresponding classical systems D(0) is trivial (equal to the phase-space dimension), and the fractality is manifested in the (multifractal) spectrum of Rényi dimensions D(q). In this paper we investigate the effect of such multifractality on the Weyl law. Our numerical simulations in area-preserving maps show for a wide range of configurations and system sizes M that (i) the Weyl law is governed by a dimension different from D(0)=2, and (ii) the observed dimension oscillates as a function of M and other relevant parameters. We propose a classical model that considers an undersampled measure of the chaotic invariant set, explains our two observations, and predicts that the Weyl law is governed by a nontrivial dimension D(asymptotic)

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Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems

Cited by 22 publications

References 67 publications

Formation and interaction of resonance chains in the open three-disk system

Formation and interaction of resonance chains in the open three-disk system

Distribution of Resonances for Hyperbolic Surfaces

Quantum signatures of classical multifractal measures

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