Semiclassical construction of resonances with hyperbolic structure: the scar function

Vergini, Eduardo G.; Carlo, Gabriel G.

doi:10.1088/0305-4470/34/21/308

Cited by 47 publications

(93 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…(11) of Ref. [23]. After one period of time, μ T ≡ μ is given by the winding number of the PO, which equals the number of half-turns made by the manifold directions as they move along the PO.…”

Section: The Tube Functionsmentioning

confidence: 99%

“…(9) can be found in Ref. [23]. Finally, the semiclassically allowed BS quantized energies can be obtained by transforming Eq.…”

Section: The Bohr-sommerfeld Quantization Rulesmentioning

confidence: 99%

“…Polavieja et al averaged groups of eigenstates using the short-time true quantum dynamics of the system [21]. Vergini and coworkers used the short PO theory [22] and obtained scar functions by combination of resonances of POs over which the condition of minimum energy dispersion is imposed, thus including the semiclassical dynamics around the scarring PO up to the Ehrenfest time [23]. Sibert et al [24] and Revuelta et al [25] extended the method to smooth potential systems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Using basis sets of scar functions

et al. 2013

View full text Add to dashboard Cite

We present a method to efficiently compute the eigenfunctions of classically chaotic systems. The key point is the definition of a modified Gram-Schmidt procedure which selects the most suitable elements from a basis set of scar functions localized along the shortest periodic orbits of the system. In this way, one benefits from the semiclassical dynamical properties of such functions. The performance of the method is assessed by presenting an application to a quartic two-dimensional oscillator whose classical dynamics are highly chaotic. We have been able to compute the eigenfunctions of the system using a small basis set. An estimate of the basis size is obtained from the mean participation ratio. A thorough analysis of the results using different indicators, such as eigenstate reconstruction in the local representation, scar intensities, participation ratios, and error bounds, is also presented.

show abstract

Section: The Tube Functionsmentioning

confidence: 99%

“…(9) can be found in Ref. [23]. Finally, the semiclassically allowed BS quantized energies can be obtained by transforming Eq.…”

Section: The Bohr-sommerfeld Quantization Rulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Using basis sets of scar functions

et al. 2013

View full text Add to dashboard Cite

show abstract

“…These have been investigated extensively for the Schroedinger equation and use has been made of high frequency (energy) semiclassical techniques [24], [25], [26], [27], [28]. Recently there has appeared discussions of bounding levels at interior foci [29].…”

Section: Introductionmentioning

confidence: 99%

Two dimensional unstable scar statistics.

Warne¹,

Jorgenson²,

Kotulski³

et al. 2006

View full text Add to dashboard Cite

This report examines the localization of time harmonic high frequency modal fields in two dimensional cavities along periodic paths between opposing sides of the cavity. The cases where these orbits lead to unstable localized modes are known as scars. This paper examines the enhancements for these unstable orbits when the opposing mirrors are both convex and concave. In the latter case the construction includes the treatment of interior foci.3 4

show abstract

“…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.…”

mentioning

confidence: 99%

Signatures of Homoclinic Motion in Quantum Chaos

et al. 2005

View full text Add to dashboard Cite

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...

show abstract

Semiclassical construction of resonances with hyperbolic structure: the scar function

Cited by 47 publications

References 28 publications

Using basis sets of scar functions

Using basis sets of scar functions

Two dimensional unstable scar statistics.

Signatures of Homoclinic Motion in Quantum Chaos

Contact Info

Product

Resources

About