Quantum eigenvalues from classical periodic orbits

Tanner, Gregor; Scherer, Philipp W.; Eb, Bogomolny; Eckhardt, Bruno; Wintgen, Dieter

doi:10.1103/physrevlett.67.2410

Cited by 93 publications

(62 citation statements)

References 19 publications

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“…Furthermore, the measure of the surface of section based on the symbolic dynamics is multifractal with respect to the usual Liouville measure. Since the periodic orbits can be effectively enumerated, the energy levels of the quantum counterpart can be approximately evaluated by means of analytical techniques [53]. A numerical study of the spectral correlations of highly excited states of this system was carried out in [52].…”

Section: The Anisotropic Kepler Problemmentioning

confidence: 99%

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

García-García

Verbaarschot²

2003

Phys. Rev. E

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We investigate the spectral properties of a generalized GOE (Gaussian Orthogonal Ensemble) capable of describing critical statistics. The joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the Calogero-Sutherland Hamiltonian (C-S). Taking advantage of the correspondence between C-S particles and eigenvalues, we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model. Such linear behavior is a signature of critical statistics. This random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self-similar/fractal Poincaré section of its classical counterpart. This is shown in detail for two examples: the anisotropic Kepler problem and a kicked particle in a well potential. In both cases the number variance and the ∆ 3 -statistic is accurately described by our analytical results.

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Section: The Anisotropic Kepler Problemmentioning

confidence: 99%

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

García-García

Verbaarschot²

2003

Phys. Rev. E

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“…As discussed previously in the context of positionspace semiclassical propagation, direct comparison between quantum and semiclassical evolution at long times for a chaotic system, or between quantum and semiclassical stationary properties for such a system, faces the obstacle of the exponential proliferation of classical paths [1]; an analogous problem of exponential growth in the number of periodic orbits exists in the energy domain [14,8,23]. This proliferation seemingly makes longtime semiclassical propagation in a classically chaotic system an exponentially harder problem than the full quantum evolution, puts into question the convergence of long-time semiclassical dynamics to any stationary behavior, and prevents the comparison of semiclassical and quantum stationary properties for smallh.…”

Section: Semiclassical Accuracy a Chaotic Dynamicsmentioning

confidence: 99%

Semiclassical accuracy in phase space for regular and chaotic dynamics

Kaplan

2004

Phys. Rev. E

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A phase-space semiclassical approximation valid to O(h) at short times is used to compare semiclassical accuracy for long-time and stationary observables in chaotic, stable, and mixed systems. Given the same level of semiclassical accuracy for the short time behavior, the squared semiclassical error in the chaotic system grows linearly in time, in contrast with quadratic growth in the classically stable system. In the chaotic system, the relative squared error at the Heisenberg time scales linearly withh eff , allowing for unambiguous semiclassical determination of the eigenvalues and wave functions in the high-energy limit, while in the stable case the eigenvalue error always remains of the order of a mean level spacing. For a mixed classical phase space, eigenvalues associated with the chaotic sea can be semiclassically computed with greater accuracy than the ones associated with stable islands.

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“…The calculation of the matrix elements M(k) lm,l ′ m ′ is rather complicated, as compared to the simple result in Eq. (2). Explicit expressions are given in Ref.…”

Section: The Four-sphere Scattering Systemmentioning

confidence: 99%

“…However, the number of periodic orbits and the numerical effort needed to find them usually increases very rapidly with increasing dimension of the phase space. As a matter of fact, Gutzwiller's periodic orbit theory has been applied predominantly to systems with two degrees of freedom, e.g., the anisotropic Kepler problem [1,2], the hydrogen atom in a magnetic field [3], and two-dimensional billiards [4,5,6]. For these systems direct quantum mechanical computations are usually more powerful and efficient than the semiclassical calculation of spectra by means of periodic orbit theory.…”

Section: Introductionmentioning

confidence: 99%

Superiority of semiclassical over quantum mechanical calculations for a three-dimensional system

et al. 2002

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In systems with few degrees of freedom modern quantum calculations are, in general, numerically more efficient than semiclassical methods. However, this situation can be reversed with increasing dimension of the problem. For a three-dimensional system, viz. the hyperbolic four-sphere scattering system, we demonstrate the superiority of semiclassical versus quantum calculations. Semiclassical resonances can easily be obtained even in energy regions which are unattainable with the currently available quantum techniques.

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Quantum eigenvalues from classical periodic orbits

Cited by 93 publications

References 19 publications

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

Semiclassical accuracy in phase space for regular and chaotic dynamics

Superiority of semiclassical over quantum mechanical calculations for a three-dimensional system

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