Entropy and the localization of eigenfunctions

Anantharaman, Nalini

doi:10.4007/annals.2008.168.435

Cited by 118 publications

(285 citation statements)

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“…We conclude this section by discussing some very recent results of N. Anantharaman which shed some light on quantum limits in full generality (and not just in the arithmetic context). It can be deduced from her paper [1] that if M is a compact manifold with negative sectional curvature then every quantum limit has positive ergodic theoretic entropy. In the case of surfaces of constant curvature −1 Anantharaman actually proves that for any δ > 0 any quantum limit has a positive measure of ergodic components with ergodic theoretic entropy ≥ (d−1)/2−δ; in this normalization the ergodic theoretic entropy of the uniform measure m SM is d − 1.…”

Section: 3mentioning

confidence: 99%

Diagonalizable flows on locally homogeneous spaces and number theory

Einsiedler¹,

Lindenstrauss²

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Section: 3mentioning

confidence: 99%

Diagonalizable flows on locally homogeneous spaces and number theory

Einsiedler¹,

Lindenstrauss²

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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“…One can for example study semiclassical measures as in Gérard-Leichtnam [9], Zelditch [19], Zelditch-Zworski [20], Anantharaman [1], Anantharaman-Koch-Nonnenmacher [2], Anantharaman-Nonnenmacher [3]. The aim of these studies is generally to prove non-concentration theorems under geometric conditions on the geodesic flow (such as Anosov flow).…”

Section: Introductionmentioning

confidence: 99%

Semiclassical L p Estimates of Quasimodes on Curved Hypersurfaces

Hassell

Tacy

2010

J Geom Anal

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Abstract. Let M be a compact manifold of dimension n, P = P (h) a semiclassical pseudodifferential operator on M , andIn a previous article, the second-named author proved estimates on the L p norms, p ≥ 2, of u restricted to H, under the assumption that the u are semiclassically localised and under some natural structural assumptions about the principal symbol of P . These estimates are of the form Ch −δ(n,k,p) where k = dim H (except for a logarithmic divergence in the case k = n − 2, p = 2). When H is a hypersurface, i.e. k = n − 1, we have δ(n, n − 1, 2) = 1/4, which is sharp when M is the round n-sphere and H is an equator.In this article, we assume that H is a hypersurface, and make the additional geometric assumption that H is curved (in the sense of Definition 2.4 below) with respect to the bicharacteristic flow of P . Under this assumption we improve the estimate from δ = 1/4 to 1/6, generalising work of Burq-Gérard-Tzvetkov and Hu for Laplace eigenfunctions. To do this we apply the MelroseTaylor theorem, as adapted by Pan and Sogge, for Fourier integral operators with folding canonical relations.

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“…A large body of recent work focuses on semiclassical measures (see for example Anatharaman [1], Gérard-Leichtnam [6], Zelditch [12] and Zelditch-Zworski [13]). Sogge's work [11] on spectral clusters give estimates for ||u|| L p (M ) of the form ||u|| L p (M ) λ −δ(n,p) where λ is the eigenvalue of u.…”

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SemiclassicalL^pEstimates of Quasimodes on Submanifolds

Tacy

2010

Communications in Partial Differential Equations

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Let P = P (h) be a semiclassical pseudodifferential operator on a Riemannian manifold M . Suppose that u(h) is a localised, L 2 normalised family of functions such that P (h)u(h) is O(h) in L 2 , as h → 0. Then, for any submanifold Y ⊂ M , we obtain estimates on the L p norm of u(h) restricted to Y , with exponents that are sharp for h → 0. These results generalise those of Burq, Gérard and Tzvetkov [4] on L p norms for restriction of Laplacian eigenfunctions. As part of the technical development we prove some extensions of the abstract Strichartz estimates of Keel and Tao [7].Let P = P (h) be a semiclassical pseudodifferential operator on a Riemannian manifold M . We will assume that P has a real principal symbol, and that its full symbol is smooth in the semiclassical parameter h. Other more technical assumptions on P are given in Definition 1.6. We prove estimates for approximate solutions u = u(h) to the equation P (h)u(h) = 0. As usual in semiclassical analysis we assume that u(h) is defined at least for a sequence h n tending to zero.Our precise definition of approximate solution, or quasimode, is thatThis definition is natural with respect to localisation: if, and χ is a pseudodifferential operator of order zero (with a symbol smooth in h), then P (χu) is also O L 2 (h). We will make the assumption that u(h) can be localised, see Definition 1.3, and therefore will be able to reduce the problem to one of local analysis.Given a submanifold Y of M , we estimate the L p norm of the restriction of u to Y , assuming the normalisation condition u L 2 (M ) = 1. These estimates are of the form u L p (Y ) ≤ Ch −δ where δ depends on the dimension n of M , the dimension k of Y and p (except for one case where there is a logarithmic divergence) -see Theorem 1.7. In every case the exponent δ(n, k, p) given by Theorem 1.7 is optimal. Figure 1 shows the exponent δ for a hypersurface and, for comparison, the L p estimates over the whole manifold (Sogge [11] for spectral clusters and Koch-Tataru-Zworski [9] for semiclassical operators). The potential growth/concentration of the quasimodes of a semiclassical operator is of great interest due to the connection to Quantum Mechanics. It is from Quantum Mechanics that we get the important set of motivating examples,here ∆ g is the (positive) Laplace-Beltrami operator associated with the metric g. We can transition between this picture and the usual eigenfunction picture of Quantum Mechanics by dividing the eigenfunction equationwhere P is as in (1) with a potential term of h 2 V 0 (x) − 1. Therefore the higher eigenvalue asymptotics of eigenfunctions of Quantum Mechanical systems corresponds to the h → 0 limit in semiclassical analysis. When V 0 (x) = 0 this problem reduces to estimating the size of Laplacian eigenfunctions restricted to a submanifold. A complete set of estimates for Laplacian eigenfunctions on compact manifolds is given by Burq, Gérard and Tzvetkov [4].A

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Entropy and the localization of eigenfunctions

Cited by 118 publications

References 34 publications

Diagonalizable flows on locally homogeneous spaces and number theory

Diagonalizable flows on locally homogeneous spaces and number theory

Semiclassical L p Estimates of Quasimodes on Curved Hypersurfaces

SemiclassicalL^pEstimates of Quasimodes on Submanifolds

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Entropy and the localization of eigenfunctions

Cited by 118 publications

References 34 publications

Diagonalizable flows on locally homogeneous spaces and number theory

Diagonalizable flows on locally homogeneous spaces and number theory

Semiclassical L p Estimates of Quasimodes on Curved Hypersurfaces

SemiclassicalLpEstimates of Quasimodes on Submanifolds

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SemiclassicalL^pEstimates of Quasimodes on Submanifolds