Chaos in Classical and Quantum Mechanics

Gutzwiller, Martin C.; José, Jörge V.

doi:10.1007/978-1-4612-0983-6

Cited by 2,756 publications

(913 citation statements)

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“…The horizontal axis shows wavenumber k = E 1/2 . Spectral rigidity ensures that a single missing or extra mode can be detected [28]; this would be visible as a permanent jump of size 1 (the gap between the dotted horizontal lines). No such jump occurs.…”

Section: Classical Autocorrelation Argument (Fp)mentioning

confidence: 99%

“…GivenÃ(ω) we use (28), with the T given above, to estimateC A (ω). We smooth in ω by a Gaussian of width ω sm = 0.03.…”

Section: A Classical Power Spectrummentioning

confidence: 99%

“…The field now called 'quantum chaos' is the study of such quantized systems in the short wavelength (semiclassical, → 0 or high energy) limit, and has become a fruitful area of enquiry for both physicists (for reviews see [28,30]) and mathematicians (see [43,44,63]) in recent decades. In contrast to those of integrable classical systems, eigenfunctions are irregular.…”

Section: Introductionmentioning

confidence: 99%

“…Here we envisage an energy window width L(E) = O(E 1/2 ), that is, a wavenumber window of width O(1): this contains O(E 1/2 ) eigenvalues by Weyl's Law [28,63]. For practical reasons (Section 4.1) we will in fact use other L(E) widths which nevertheless contain many (≈ 10 3 ) eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Barnett

2006

Comm. Pure Appl. Math.

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The Quantum Unique Ergodicity (QUE) conjecture of RudnickSarnak is that every eigenfunction φ n of the Laplacian on a manifold with uniformly-hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue E n → ∞), that is, 'strong scars' are absent. We study numerically the rate of equidistribution for a uniformly-hyperbolic Sinai-type planar Euclidean billiard with Dirichlet boundary condition (the 'drum problem') at unprecedented high E and statistical accuracy, via the matrix elements φ n ,Âφ m of a piecewise-constant test function A. By collecting 30000 diagonal elements (up to level n ≈ 7 × 10 5 ) we find that their variance decays with eigenvalue as a power 0.48±0.01, close to the estimate 1/2 of FeingoldPeres (FP). This contrasts the results of existing studies, which have been limited to E n a factor 10 2 smaller. We find strong evidence for QUE in this system. We also compare off-diagonal variance, as a function of distance from the diagonal, against FP at the highest accuracy (0.7%) thus far in any chaotic system. We outline the efficient scaling method used to calculate eigenfunctions.

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Section: Classical Autocorrelation Argument (Fp)mentioning

confidence: 99%

“…GivenÃ(ω) we use (28), with the T given above, to estimateC A (ω). We smooth in ω by a Gaussian of width ω sm = 0.03.…”

Section: A Classical Power Spectrummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Barnett

2006

Comm. Pure Appl. Math.

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“…Recently, there has been considerable renewed interest in the various aspects of the Semi-Classical Approximation (SCA), a powerful motivation behind that being the problem of the so-called quantum chaos (see for example references [1,2,3,4,5]). An important aspect is represented by the quantum energy levels, and in this connection one recent work [6] shows that the predictions of individual levels by SCA (by this we mean the Bohr-Sommerfeld formula, or one of its generalizations to the non-integrable case, such as EBK; see e.g.…”

mentioning

confidence: 99%

Accuracy of the semi-classical approximation: the Pullen-Edmonds Hamiltonian

1994

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A test on the numerical accuracy of the semiclassical approximation as a function of the principal quantum number has been performed for the Pullen-Edmonds model, a two-dimensional, non-integrable, scaling invariant perturbation of the resonant harmonic oscillator. A perturbative interpretation is obtained of the recently observed phenomenon of the accuracy decrease on the approximation of individual energy levels at the increase of the principal quantum number. Moreover, the accuracy provided by the semiclassical approximation formula is on the average the same as that provided by quantum perturbation theory.

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Comparison of line search minimization algorithms for exploring topography of multidimensional potential energy surfaces: Mg+Arn case

et al. 1997

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The most robust numerical algorithms for unconstrained optimization that involve a line search are tested in the problem of locating stable structures and transition states of atomic microclusters. Specifically, the popular quenching technique is compared with conjugate gradient and variable metric algorithms in the Mg+Arn clusters. It is found that the variable metric method BFGS combined with an approximate line minimization routine is the most efficient, and it shows global convergence properties. This technique is applied to find a few hundred stationary points of Mg+Ar12 and to locate isomerization paths between the two most stable icosahedral structures found for Mg+Ar12. The latter correspond to a solvated and a nonsolvated ion, respectively. © 1997 John Wiley & Sons, Inc. J Comput Chem 18:1011–1022, 1997

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Chaos in Classical and Quantum Mechanics

Cited by 2,756 publications

References 0 publications

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Accuracy of the semi-classical approximation: the Pullen-Edmonds Hamiltonian

Comparison of line search minimization algorithms for exploring topography of multidimensional potential energy surfaces: Mg+Arn case

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