Scars in Groups of Eigenstates in a Classically Chaotic System

Polavieja, Gonzalo G. de; Borondo, F.; Benito, R. M.

doi:10.1103/physrevlett.73.1613

Cited by 90 publications

(96 citation statements)

References 26 publications

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“…In this section we briefly describe the method [9] that will be used to construct states initially located on a given PO and the corresponding manifolds. The procedure has been previously applied to the analysis of the scarring effect of unstable POs in chaotic systems [14], as well as to the study of the quantum dynamics that takes place beyond the first recurrence of the corresponding motion [7].…”

Section: Constructing Localized Wave Functionsmentioning

confidence: 99%

“…In Ref. [9] the necessary information is obtained by Fourier transforming the short-time exact quantum dynamics, so that wave functions averaged over the PO path are obtained. Kaplan and Heller [10] used a sum of coherent Gaussian wave packets centered along the PO to obtain the desired function.…”

Section: Introductionmentioning

confidence: 99%

“…[9], applying them to the study of the dynamics in the Bunimovich stadium billiard [13], a system which is known to behave ergodically from the classical point of view.…”

Section: Introductionmentioning

confidence: 99%

“…II we describe in detail how the method of Ref. [9] can be used to construct the desired wave functions, highly localized on a given unstable PO and their associated manifolds. Numerical results concerning the topological characteristics, both in configuration and phase space, of these functions and the corresponding time evolution will be presented and analyzed in Sects.…”

Section: Introductionmentioning

confidence: 99%

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Localization properties of groups of eigenstates in chaotic systems

et al. 2001

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In this paper we study in detail the localized wave functions defined in Phys. Rev. Lett. 76, 1613Lett. 76, (1994, in connection with the scarring effect of unstable periodic orbits in highly chaotic Hamiltonian system. These functions appear highly localized not only along periodic orbits but also on the associated manifolds. Moreover, they show in phase space the hyperbolic structure in the vicinity of the orbit, something which translates in configuration space into the structure induced by the corresponding self-focal points. On the other hand, the quantum dynamics of these functions are also studied. Our results indicate that the probability density first evolves along the unstable manifold emanating from the periodic orbit, and localizes temporarily afterwards on only a few, short related periodic orbits. We believe that this type of studies can provide some keys to disentangle the complexity associated to the quantum mechanics of these kind of systems, which permits the construction of a simple explanation in terms of the dynamics of a few classical structures.

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Section: Constructing Localized Wave Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[9], applying them to the study of the dynamics in the Bunimovich stadium billiard [13], a system which is known to behave ergodically from the classical point of view.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Localization properties of groups of eigenstates in chaotic systems

et al. 2001

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“…1). This can be done in different ways [11,12]; in particular, in Ref. 12 a definition of scar functions, based on transversally excited resonances along POs at given BS quantized energies with minimum dispersion, is provided.…”

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confidence: 99%

Signatures of Homoclinic Motion in Quantum Chaos

et al. 2005

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Homoclinic motion plays a key role in the organization of classical chaos in Hamiltonian systems. In this Letter, we show that it also imprints a clear signature in the corresponding quantum spectra. By numerically studying the fluctuations of the widths of wavefunctions localized along periodic orbits we reveal the existence of an oscillatory behavior, that is explained solely in terms of the primary homoclinic motion. Furthermore, our results indicate that it survives the semiclassical limit. , an enhanced localization of quantum density along unstable short POs in certain individual eigenfunctions. This phenomenon was first noticed in the Bunimovich stadium billiard [3], and subsequently studied systematically by Heller, who constructed a theory of scarring based on wave packet propagation [4]. Another important contribution to scar theory is due to Bogomolny [5], who derived an explicit expression for the PO contributions to the smoothed quantum probability density over small ranges of space and energy (i.e. average over a large number of eigenfunctions). A corresponding theory for Wigner functions was developed by Berry [6]. Recently, there has been a flurry of activity focusing on the influence of bifurcations (mixed systems) on scarring [7].The existence of a scar implies a clear regularity in the corresponding quantum spectrum, related to the period of the PO. In time domain, the dynamics of a packet running along a PO induce recurrences in the autocorrelation function, that when Fourier transformed define an envelope in the spectrum, giving rise to peaks of width proportional to the Lyapunov exponent, λ, at energies given by a Bohr-Sommerfeld (BS) quantization condition [1,8].In this Letter, we demonstrate the existence of an additional superimposed spectral regularity also related to scarring. It is originated by the associated homoclinic motion and is given by the area enclosed by the stable and unstable manifolds up to the first crossing. To unveil this regularity we consider the fluctuations of the spectral widths corresponding to localized wavefunctions along unstable POs. Our data demonstrate that these fluctuations have a surprisingly simple oscillatory behavior essentially governed by the quantization of only the primary homoclinic dynamics. We provide an explanation of this result in terms of the coherence of this classical motion [9], which constitute the natural global extension of the local hyperbolic structure around the PO. Furthermore, our numerical results indicate that the observed oscillatory behavior do not vanish ash → 0.It should be remarked that, contrary to the previously described results on scar theory, our study implies dynamics beyond the Ehrenfest time (∼ | lnh|), when the PO manifolds start to cross and the homoclinic tangle develops, giving rise to subtle quantum interference effects. Other interesting papers, also considering these longer times, are due to Tomsovic and Heller [10], who showed how to construct a valid semiclassical approximation to wavefunctions past this l...

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Quantum chaos in floppy molecular systems: The LiCN molecule

Borondo

Benito

Non-Linear Dynamics and Fundamental Interactions

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Scars in Groups of Eigenstates in a Classically Chaotic System

Cited by 90 publications

References 26 publications

Localization properties of groups of eigenstates in chaotic systems

Localization properties of groups of eigenstates in chaotic systems

Signatures of Homoclinic Motion in Quantum Chaos

Quantum chaos in floppy molecular systems: The LiCN molecule

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