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25
Global Strichartz estimates for nontrapping perturbations
 of the Laplacian, Comm. Partial Differential Equations
"... The purpose of this paper is to establish estimates of Strichartz type, globally in space and time, for solutions to certain nontrapping, spatially compact perturbations of the Minkowski wave equation. Precisely, we consider the following wave equation on the exterior domain Ω to a compact obstacle, ..."
Abstract

Cited by 61 (7 self)
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The purpose of this paper is to establish estimates of Strichartz type, globally in space and time, for solutions to certain nontrapping, spatially compact perturbations of the Minkowski wave equation. Precisely, we consider the following wave equation on the exterior domain Ω to a compact obstacle,
From Quasimodes to Resonances
"... this paper we work in the semiclassical setting of "black box scattering" introduced by Sjostrand and the second author in [10] and then extended by Sjostrand in [9]. In the modification of the argument of Stefanov and Vodev we replace the Phragm'enLindelof principle by a local application of the ..."
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Cited by 36 (13 self)
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this paper we work in the semiclassical setting of "black box scattering" introduced by Sjostrand and the second author in [10] and then extended by Sjostrand in [9]. In the modification of the argument of Stefanov and Vodev we replace the Phragm'enLindelof principle by a local application of the maximum principle adapted to the semiclassical setting. The global bound (1) is replaced by a local bound given in Lemma 1 below and coming essentially from the recent work of Sjostrand on the local trace formula for resonances [9]. It is interesting to note that the global trace formula for resonances (established in successive generality by BardosGuillotRalston, Melrose, SjostrandZworski and S'a BarretoZworski  see [9],[15] and references given there) can be proved using the minimum modulus theorem as in the proof of (1)  see [3],[15], while for the more generally valid local formula of [9], the local estimates of the type used here are needed. Finally we remark that the results of Stefanov and Vodev, [11],[12], and the more recent results of Popov and Vodev [7] are also concerned with situations in which one cannot construct quasimodes in the standard sense. By proceeding as in their papers and using the methods of this paper one could generalize their results to even dimensions and to suitable noncompactly supported perturbations
The Semiclassical Trace Formula and Propagation of Wave Packets
 J. Funct. Anal
, 1994
"... We study spectral and propagation properties of operators of the form S ¯ h = P N j=0 ¯h j P j where 8j P j is a differential operator of order j on a manifold M , asymptotically as ¯h ! 0. The estimates are in terms of the flow fOE t g of the classical Hamiltonian H(x; p) = P N j=0 oe P j (x; ..."
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Cited by 21 (4 self)
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We study spectral and propagation properties of operators of the form S ¯ h = P N j=0 ¯h j P j where 8j P j is a differential operator of order j on a manifold M , asymptotically as ¯h ! 0. The estimates are in terms of the flow fOE t g of the classical Hamiltonian H(x; p) = P N j=0 oe P j (x; p) on T M , where oe P j is the principal symbol of P j . We present two sets of results. (I) The "semiclassical trace formula", on the asymptotic behavior of eigenvalues and eigenfunctions of S ¯ h in terms of periodic trajectories of H . (II) Associated to certain isotropic submanifolds ae T M we define families of functions f/ ¯ h g and prove that 8t fexp(\Gammait¯hS h )(/ ¯ h )g is a family of the same kind associated to OE t (). Introduction and description of results. In this paper we present some results concerning spectral and propagation properties of a class of differential operators "with small parameter", ¯h (Planck's constant). We have in mind operators of the form a(x...
QUASIMODES AND RESONANCES: Sharp Lower Bounds
 VOL. 99, NO. 1 DUKE MATHEMATICAL JOURNAL
, 1999
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Geometric Properties Of Eigenfunctions
 Russian Math. Surveys
"... We give an overview of some new and old results on geometric properties of eigenfunctions of Laplacians on Riemannian manifolds. We discuss the properties of nodal sets and critical points, the number of nodal domains, as well as asymptotic properties of eigenfunctions in the high energy limit ( ..."
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Cited by 16 (3 self)
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We give an overview of some new and old results on geometric properties of eigenfunctions of Laplacians on Riemannian manifolds. We discuss the properties of nodal sets and critical points, the number of nodal domains, as well as asymptotic properties of eigenfunctions in the high energy limit (such as weak* limits, the rate of growth of L p norms, and the relationship between positive and negative parts of eigenfunctions). 1. introduction It is wellknown that on a compact Riemannian manifold M one can choose an orthonormal basis of L 2 (M) consisting of eigenfunctions ' j of satisfying ' j + j ' j = 0; (1) where 0 = 0 < 1 2 : : : are the eigenvalues. The purpose of this survey paper is to present some recent (and not so recent) results about the asymptotics of Laplace eigenfunctions on compact manifolds. We focus here mainly on results about the nodal sets, asymptotic L p bounds and the problem of determining weaklimits of expected values (i.e. quantum ...
Équilibre instable en régime semiclassique  II: Conditions de BohrSommerfeld
, 1997
"... Dans ce travail, nous tudions les valeurs propres de l'oprateur de Schrdinger en dimension 1 qui sont proches d'un maximum local du potentiel. Il fait suite [2] o nous tudiions la concentration des fonctions propres associes. Nous montrons en particulier comment s'e#ectue la transition, dans le cas ..."
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Cited by 12 (2 self)
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Dans ce travail, nous tudions les valeurs propres de l'oprateur de Schrdinger en dimension 1 qui sont proches d'un maximum local du potentiel. Il fait suite [2] o nous tudiions la concentration des fonctions propres associes. Nous montrons en particulier comment s'e#ectue la transition, dans le cas du double puits symtrique, entre les doublets de valeurs exponentiellement proches et les valeurs rgulirement espaces lorsque l'nergie augmente. 1
Classical Limits Of Eigenfunctions For Some Completely Integrable Systems
 IMA Vol. Math. Appl
, 1997
"... . We give an overview of some old results on weak* limits of eigenfunctions and prove some new ones. We first show that on M = (S n ; can) every probability measure on S M which is invariant under the geodesic flow and time reversal is a weak* limit of a sequence of Wigner measures correspondi ..."
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Cited by 8 (3 self)
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. We give an overview of some old results on weak* limits of eigenfunctions and prove some new ones. We first show that on M = (S n ; can) every probability measure on S M which is invariant under the geodesic flow and time reversal is a weak* limit of a sequence of Wigner measures corresponding to eigenfunctions of \Delta. We next show that joint eigenfunctions of \Delta and a single Hecke operator on S n cannot scar on a single closed geodesic. We finally use the estimates of [Z3] on the rate of quantum ergodicity to prove that adding a \PsiDO of order \Gamman + 2 doesn't change the level spacings distribution of \Delta (if the former is well defined) on a compact negatively curved manifold of dimension n. In dimension two this shows that the level spacings distributions of quantizations of certain Hamiltonians do not depend on the quantization. 1. Introduction A general theme of semiclassical analysis is to find relations between the asymptotic properties of the eigenfunct...
Note on quantum unique ergodicity
 Proc. Amer. Math. Soc
"... The purpose of this note is to record an observation about quantum unique ergodicity (QUE) which is relevant to the recent construction of H. Donnelly [D] of quasimodes on certain nonpositively curved surfaces, and to similar quasimode constructions known for many years as bouncing ball modes on ..."
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Cited by 7 (0 self)
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The purpose of this note is to record an observation about quantum unique ergodicity (QUE) which is relevant to the recent construction of H. Donnelly [D] of quasimodes on certain nonpositively curved surfaces, and to similar quasimode constructions known for many years as bouncing ball modes on Bunimovich stadia [BSS, H, BZ1, BZ2]. Our new observation (Proposition 0.1) is the asymptotic vanishing of near offdiagonal matrix elements for eigenfunctions of QUE systems. As a corollary, we find that quantum ergodic (QE) systems possessing quasimodes with singular limits and with a limited number of frequencies cannot be QUE. We begin by recalling that QUE (for Laplacians) concerns the matrix elements 〈Aϕi, ϕj〉 of pseudodifferential operators relative to an orthonormal basis {ϕj} of eigenfunctions ∆ϕj = λ 2 j ϕj, 〈ϕj, ϕk 〉 = 0. of the Laplacian ∆ of a compact Riemannian manifold (M, g). We denote the spectrum of ∆ by Sp(∆). By definition, ∆ is QUE if 〈Aϕj, ϕj 〉 → σAdL (1) where dL is the (normalized) Liouville measure on the unit (co)tangent bundle. The term ‘unique ’ indicates that no subsequence of density zero of eigenfunctions need be removed when taking the limit. The main result of this note is that all offdiagonal terms of QUE systems tend to zero if the eigenvalue gaps tend to zero. This strengthens the conclusion of [Z] that almost all offdiagonal terms (with vanishing gaps) tend to zero in general QE situations. As will be seen below, it also provides evidence that Donnelly’s examples are nonQUE and establishes a localization statement of HellerO’Connor [HO]. Proposition 0.1. Suppose that ∆ is QUE. Suppose that {(λir, λjr), ir = jr} is a sequence of pairs of eigenvalues of √ ∆ such that λir − λjr → 0 as r → ∞. Then dΦir,jr → 0. S ∗ M Proof. We define the distributions dΦi,j ∈ D ′ (S ∗ M) by 〈Op(a)ϕi, ϕj 〉 = S ∗ adΦi,u M where a ∈ C ∞ (S∗M). Let {λi, λj} be any sequence of pairs with the gap λi − λj → 0. It is then known that any weak * limit dν of the sequence {dΦi,j} is a measure invariant under the geodesic flow [Z, D]. The weak limit is defined by the property that