Results 1  10
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21
Uniform Estimates of the Resolvent of the Laplace–Beltrami Operator on Infinite Volume Riemannian Manifolds with Cusps.II
, 2001
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Local smoothing for scattering manifolds with hyperbolic trapped sets
 Comm. Math. Phys
, 2009
"... Abstract. We prove a resolvent estimate for the LaplaceBeltrami operator on a scattering manifold with a hyperbolic trapped set, and as a corollary deduce local smoothing. We use a result of NonnenmacherZworski to provide an estimate near the trapped region, a result of Burq and CardosoVodev to p ..."
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Cited by 15 (5 self)
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Abstract. We prove a resolvent estimate for the LaplaceBeltrami operator on a scattering manifold with a hyperbolic trapped set, and as a corollary deduce local smoothing. We use a result of NonnenmacherZworski to provide an estimate near the trapped region, a result of Burq and CardosoVodev to provide an estimate near infinity, and the microlocal calculus on scattering manifolds to combine the two. 1.
GLUING SEMICLASSICAL RESOLVENT ESTIMATES VIA PROPAGATION OF SINGULARITIES
"... Abstract. We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application we prove estimates for the analytic continuation of the resolvent of a Schrödinger operator for certain asymptotically hyperbolic manifolds in the presence ..."
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Cited by 10 (5 self)
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Abstract. We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application we prove estimates for the analytic continuation of the resolvent of a Schrödinger operator for certain asymptotically hyperbolic manifolds in the presence of trapping which is sufficiently mild in one of several senses. As a corollary we obtain local exponential decay for the wave propagator and local smoothing for the Schrödinger propagator. 1.
PROPAGATION THROUGH TRAPPED SETS AND SEMICLASSICAL RESOLVENT ESTIMATES
"... Abstract. Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single iso ..."
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Cited by 9 (3 self)
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Abstract. Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting. 1.
STRICHARTZ ESTIMATES FOR LONG RANGE PERTURBATIONS
, 2006
"... Abstract. We study local in time Strichartz estimates for the Schrödinger equation associated to long range perturbations of the flat Laplacian on the euclidean space. We prove that in such a geometric situation, outside a large ball centered at the origin, the solutions of the Schrödinger equation ..."
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Cited by 8 (4 self)
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Abstract. We study local in time Strichartz estimates for the Schrödinger equation associated to long range perturbations of the flat Laplacian on the euclidean space. We prove that in such a geometric situation, outside a large ball centered at the origin, the solutions of the Schrödinger equation enjoy the same Strichartz estimates as in the non perturbed situation. The proof is based on the IsozakiKitada parametrix construction. If in addition the metric is non trapping, we prove that the Strichartz estimates hold in the whole space.
Spectral Distributions for Long Range Perturbations
, 2002
"... We study distributions which generalize the concept of spectral shift function, for pseudodifferential operators on R^d. These distributions are called spectral distributions. Relations between relative scattering determinants and spectral distributions are established; they lead to the definition o ..."
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Cited by 5 (2 self)
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We study distributions which generalize the concept of spectral shift function, for pseudodifferential operators on R^d. These distributions are called spectral distributions. Relations between relative scattering determinants and spectral distributions are established; they lead to the definition of regularized scattering phases. These relations are analogous to the one valid for the usual spectral shift function. We give several asymptotic properties in the high energy and semiclassical limits where both non trapping and trapping cases are considered. In particular, we prove BreitWigner formulae for the regularized scattering phases, for semiclassical Schrödinger operators with long range potentials.
Sharp Upper Bounds on the Number of Resonances Near the Real Axis for Trapped Systems
, 2001
"... This paper is devoted to a detailed study of the behavior of the resonances and resonant states near the real axis. We work mainly in the semiclassical setting but most results can be easily translated into the classical one. By resonances near the real axis we mean resonances in a "box" ## h) = [a ..."
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Cited by 5 (1 self)
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This paper is devoted to a detailed study of the behavior of the resonances and resonant states near the real axis. We work mainly in the semiclassical setting but most results can be easily translated into the classical one. By resonances near the real axis we mean resonances in a "box" ## h) = [a 0 , b 0 ] + i[S(h), 0], where 0 < S(h) = O(h 1. Such resonances may exist only for trapping geometries. We accept the convention here that resonances lie in the lower halfplane
Exponential lower bounds for quasimodes of semiclassical Schrödinger operators
 In preparation
"... In this paper we establish quantitative unique continuation results for the semiclassical Schrödinger operator on smooth, compact domains. To be specific, we consider a smooth, open, bounded, and connected domain Ω of Rn, and we let G = (gij) ∈ C ∞ (Ω) n2 be a positive definite symmetric matrix wit ..."
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Cited by 3 (2 self)
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In this paper we establish quantitative unique continuation results for the semiclassical Schrödinger operator on smooth, compact domains. To be specific, we consider a smooth, open, bounded, and connected domain Ω of Rn, and we let G = (gij) ∈ C ∞ (Ω) n2 be a positive definite symmetric matrix with real entries.
RESONANCE FREE REGIONS FOR NONTRAPPING MANIFOLDS WITH CUSPS
"... Abstract. We prove resolvent estimates for nontrapping manifolds with cusps which imply the existence of arbitrarily wide resonance free strips, local smoothing for the Schrödinger equation, and resonant wave expansions. We obtain lossless limiting absorption and local smoothing estimates, but the e ..."
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Cited by 1 (1 self)
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Abstract. We prove resolvent estimates for nontrapping manifolds with cusps which imply the existence of arbitrarily wide resonance free strips, local smoothing for the Schrödinger equation, and resonant wave expansions. We obtain lossless limiting absorption and local smoothing estimates, but the estimates on the holomorphically continued resolvent exhibit losses. We prove that these estimates are optimal in certain respects. Resonance free regions near the essential spectrum have been extensively studied since the foundational work of LaxPhillips and Vainberg. Their size is related to the dynamical structure of the set of trapped classical trajectories. More trapping typically results in a smaller region, and the largest resonance free regions exist when there is no trapping. Example. Let H 2 be the hyperbolic upper half plane. Let (X, g) be a nonpositively curved, compactly supported metric perturbation of the quotient space 〈z ↦ → z + 1〉\H 2. As we show in §2.4, such a surface has no trapped geodesics (that is, all geodesics are unbounded). Let (X, g) be as above, or as in §2.1, with dimension n + 1 and Laplacian ∆ ≥ 0. The resolvent ( ∆ − n 2 /4 − σ 2) −1 is holomorphic for Im σ> 0, except at any σ ∈ iR such that σ 2 + n 2 /4 is an eigenvalue, and has essential spectrum {Im σ = 0}: see Figure 1.1.