Linear and Nonlinear Theory of Eigenfunction Scars

Kaplan, L.; Heller, Eric J.

doi:10.1006/aphy.1997.5773

Cited by 152 publications

(204 citation statements)

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“…Heller [29,30] put forward a semiclassical explanation based on enhanced short-time return probability for wavepackets launched along the least unstable periodic orbits (UPOs), which has been elaborated [15,13,1,37]. Although the meaning of 'scar' varied historically, it is now taken to mean any deviation from the random wave prediction of eigenfunction intensity near a UPO [37]. Scar 'strength' depends on what test function you use to measure it [38]: in physics this test function is commonly not held fixed as the limit E → ∞ is taken, rather it is chosen to collapse microlocally onto a UPO with a coordinate-space width ∼ E −1/4 .…”

Section: Discussion On Existence Of Scarsmentioning

confidence: 99%

“…Heller's numerical demonstrations of apparently strong scarring were done at mode numbers n ≈ 2 × 10 3 , which, in light of our work, is well below the asymptotic regime. It is now believed by physicists that in 2D Anosov billiards strong scarring does not persist [1,37,33], but there still exist controversies about the width of scars [33], and in related quantum models the mechanism of scarring is an active research area [45]. Mathematically, the issue remains open.…”

Section: Discussion On Existence Of Scarsmentioning

confidence: 99%

“…With the definitions of Lemma B.1, for all j ≥ 1, 1 2E j Γ r n (n · ∇φ j ) 2 ds = 1 (37) This identity is a special case of Lemma 3.1 found by substituting Ω A = Ω, u = v = φ j , and E = E j , and recognising r · ∇φ j = r n n · ∇φ j . The maximum |ω i | in which levels of useful accuracy are found is of order 0.2/R where R is the largest radius of the domain.…”

Section: Lemma B2 (Rellich)mentioning

confidence: 95%

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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: Discussion On Existence Of Scarsmentioning

confidence: 99%

Section: Discussion On Existence Of Scarsmentioning

confidence: 99%

Section: Lemma B2 (Rellich)mentioning

confidence: 95%

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Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Barnett

2006

Comm. Pure Appl. Math.

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“…We recall that the variation of S j (x, E) with respect to the independent variables q and p are given in Eq. (12). Hence, the first member of Eq.…”

Section: Correlations Of Ergodic Wave Functions For Differentmentioning

confidence: 97%

Semiclassical spatial correlations in chaotic wave functions

Toscano

Lewenkopf²

2002

Phys. Rev. E

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We study the spatial autocorrelation of energy eigenfunctions ψ n (q) corresponding to classically chaotic systems in the semiclassical regime. Our analysis is based on the Weyl-Wigner formalism for the spectral average, defined as the average over eigenstates within an energy window ε centered at E. In this framework C ε is the Fourier transform in momentum space of the spectral Wigner function W (x, E; ε). Our study reveals the chord structure that C ε inherits from the spectral Wigner function showing the interplay between the size of the spectral average window, and the spatial separation scale. We discuss under which conditions is it possible to define a local system independent regime for C ε . In doing so, we derive an expression that bridges the existing formulae in the literature and find expressions for C ε (q + , q − , E) valid for any separation size |q + − q − |.

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“…However, a more sensitive test -and still a great challenge -is the semiclassical representation of single eigenfunctions. This includes the study of the scar phenomena [4][5][6][7][8][9][10] and the eventual deviations from uniformity of eigenfunctions in accordance to the Berry-Voros hypothesis [11,12] and Schnirelman's theorem [13].…”

Section: Introductionmentioning

confidence: 99%