Ergodicit� et fonctions propres du laplacien

Verdìère, Yves Colin de

doi:10.1007/bf01209296

Cited by 491 publications

(455 citation statements)

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“…Theorem 1 (Quantum Ergodicity Theorem (QET) [46,55,21,62]) Let Ω ∈ R 2 be a 2D compact domain with piecewise smooth boundary whose classical flow is ergodic. Then for all n except a subsequence of vanishing density, φ n ,Âφ n − A → 0 as n → ∞,…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Barnett

2006

Comm. Pure Appl. Math.

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“…In the case when the geodesic flow on the unit sphere bundle of M is ergodic, Colin de Verdière [2] proved a remarkable theorem. Suppose that A is a zero'th order pseudodifferential operator with symbol a.…”

Section: Introductionmentioning

confidence: 99%

“…The study of the stadium billiard by Heller [3] is closely related to the ideas presented here. Prior to the definitive paper by Colin de Verdière [2], there were important works by Shnirelman [5] and Zelditch [6].…”

Section: Introductionmentioning

confidence: 99%

Quantum unique ergodicity

Donnelly

2002

Proc. Amer. Math. Soc.

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Abstract. Consider a compact Riemannian manifold with ergodic geodesic flow. Quantum ergodicity is generalized from orthonormal bases of eigenfunctions of the Laplacian to packets of eigenfunctions. It is shown that this more general result is sharp. Namely, there may exist exceptional packets of eigenfunctions which concentrate on a submanifold. IntroductionLet M be a compact Riemannian manifold. The Laplacian ∆ of M is a second order self-adjoint differential operator. Moreover, there exist orthonormal bases forIn the case when the geodesic flow on the unit sphere bundle of M is ergodic, Colin de Verdière [2] proved a remarkable theorem. Suppose that A is a zero'th order pseudodifferential operator with symbol a. Then there exists a subsequence φ ki , of density one in φ k , so that, for any A,Here dω is the normalized Liouville measure on the unit sphere bundle S 1 M . An important open problem is whether the equality (0.1) holds for the full orthonormal basis. We contribute to this discussion by broadening the perspective. Definition 1.1 introduces the concept of packets of eigenfunctions, which are certain finite linear combinations of eigenfunctions. Our Theorem 1.4 shows that the result of Colin de Verdière generalizes to such packets of eigenfunctions. The rest of the paper is devoted to establishing the sharpness of this more general result. Namely, there may exist exceptional subsequences, of density zero, where (0.1) fails. Theorem 2.2 constructs manifolds with ergodic geodesic flow which have nonpositive curvature and contain flat cylinders. The proof of Theorem 2.2 is an application of the method of Burns and Gerber [1]. In Theorem 3.5, we produce exceptional sequences of eigenfunction packets supported on the manifolds provided by Theorem 2.2. These packets may be localized to arbitrarily small intervals on the spectrum while maintaining a fixed normalization for the average level spacings.

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“…If, in addition, Γ\H is compact the results of Shnirelman/Colin de Verdiere/Zelditch [Shn74,Col85,Zel87] for every Jordan region F ⊆ Γ\H, with the possible exclusion of a set of λ n of density 0. This result has been extended to non-compact surfaces such as the modular surface P SL(2, Z)\H; cf.…”

Section: Introductionmentioning

confidence: 99%

Computations of Eisenstein series on Fuchsian groups

Avelin

2008

Math. Comp.

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Abstract. We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series E(z; s) on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of E(z; s) as Re s = 1/2, Im s → ∞ and also, on non-arithmetic groups, a complex Gaussian limit distribution for E(z; s) when Re s > 1/2 near 1/2 and Im s → ∞, at least if we allow Re s → 1/2 at some rate. Furthermore, on non-arithmetic groups and for fixed s with Re s ≥ 1/2 near 1/2, our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.

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Ergodicit� et fonctions propres du laplacien

Cited by 491 publications

References 2 publications

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

Quantum unique ergodicity

Computations of Eisenstein series on Fuchsian groups

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