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40
A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds
 Comm. Partial Differential Equations
"... Abstract. We obtain the Strichartz inequality ..."
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
The Schrödinger propagator for scattering metrics
, 2003
"... Abstract. Let g be a scattering metric on a compact manifold X with boundary, i.e., a smooth metric giving the interior X ◦ the structure of a complete Riemannian manifold with asymptotically conic ends. An example is any compactly supported perturbation of the standard metric on Rn. Consider the op ..."
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Cited by 12 (3 self)
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Abstract. Let g be a scattering metric on a compact manifold X with boundary, i.e., a smooth metric giving the interior X ◦ the structure of a complete Riemannian manifold with asymptotically conic ends. An example is any compactly supported perturbation of the standard metric on Rn. Consider the operator H = 1 ∆+V, where ∆ is the positive 2 Laplacian with respect to g and V is a smooth realvalued function on X vanishing to second order at ∂X. Assuming that g is nontrapping, we construct a global parametrix U(z, w, t) for the kernel of the Schrödinger propagator U(t) = e−itH, where z, w ∈ X ◦ and t ̸ = 0. The parametrix is such that the difference between U and U is smooth and rapidly decreasing both as t → 0 and as z → ∂X, uniformly for w on compact subsets of X ◦. Let r = x−1, where x is a boundary defining function for X, be an asymptotic radial variable, and let W(t) be the kernel e−ir2 /2tU(t). Using the parametrix, we show that W(t) belongs to a class of ‘Legendre distributions ’ on X×X ◦ ×R�0 previously considered by HassellVasy. When the metric is trapping, then the parametrix construction goes through microlocally in the nontrapping part of the phase space. We apply this result to obtain a microlocal characterization of the singularities of U(t)f, for any tempered distribution f and any fixed t ̸ = 0, in terms of the oscillation of f near ∂X. If the metric is nontrapping, then we obtain a complete characterization; more generally we need to assume that f is microsupported in the nontrapping part of the phase space. This generalizes results of CraigKappelerStrauss and Wunsch. 1.
Sharp Strichartz estimates on nontrapping asymptotically conic manifolds
 Amer. J. Math
, 2006
"... Abstract. We obtain the Strichartz inequalities ‖u ‖ q Lt Lr x ([0,1]×M) ≤ C‖u(0)‖L2 (M) for any smooth ndimensional Riemannian manifold M which is asymptotically conic at infinity (with either shortrange or longrange metric perturbation) and nontrapping, where u is a solution to the Schrödinge ..."
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Cited by 12 (2 self)
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Abstract. We obtain the Strichartz inequalities ‖u ‖ q Lt Lr x ([0,1]×M) ≤ C‖u(0)‖L2 (M) for any smooth ndimensional Riemannian manifold M which is asymptotically conic at infinity (with either shortrange or longrange metric perturbation) and nontrapping, where u is a solution to the Schrödinger equation iut + 1 2∆Mu = 0, and 2 < q, r ≤ ∞ are admissible Strichartz exponents ( 2 q + n n =). This corresponds with the estimates available for Euclidean space r 2 (except for the endpoint (q, r) = (2, 2n) when n> 2). These estimates imply n−2 existence theorems for semilinear Schrödinger equations on M, by adapting arguments from Cazenave and Weissler [4] and Kato [14]. This result improves on our previous result in [10], which was an L4 t,x Strichartz estimate in three dimensions. It is closely related to the results in [22], [1], [26], [19], which consider the case of asymptotically flat manifolds. Contents
Longtime decay estimates for Schrodinger equations on manifolds (submitted
"... Abstract. In this paper we develop a quantitative version of Enss ’ method to establish globalintime decay estimates for solutions to Schrödinger equations on manifolds. To simplify the exposition we shall only consider Hamiltonians of the form H: = − 1 2 ∆M, where ∆M is the LaplaceBeltrami oper ..."
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Cited by 9 (1 self)
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Abstract. In this paper we develop a quantitative version of Enss ’ method to establish globalintime decay estimates for solutions to Schrödinger equations on manifolds. To simplify the exposition we shall only consider Hamiltonians of the form H: = − 1 2 ∆M, where ∆M is the LaplaceBeltrami operator on a manifold M which is a smooth compact perturbation of threedimensional Euclidean space R3 which obeys the nontrapping condition. We establish a globalintime local smoothing estimate for the Schrödinger equation ut = −iHu. The main novelty here is the globalintime aspect of the estimates, which forces a more detailed analysis on the low and medium frequencies of the evolution than in the localintime theory. In particular, to handle the medium frequencies we require the RAGE theorem (which reflects the fact that H has no embedded eigenvalues), together with a quantitative version of Enss ’ method decomposing the solution asymptotically into incoming and outgoing components, while to handle the low frequencies we need a Bernstein inequality (which reflects the fact that H has no eigenfunctions or resonances at zero). 1.
Propagation of the homogeneous wave front set for Schrödinger equations
, 2003
"... In this paper we study the propagation of singularity for Schrödingertype equations with variable coefficients. We introduce a new notion of wave propagation set, the homogeneous wave front set, and it propagates along straight lines with finite speed away from x � = 0. Then we show that it is relat ..."
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Cited by 9 (6 self)
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In this paper we study the propagation of singularity for Schrödingertype equations with variable coefficients. We introduce a new notion of wave propagation set, the homogeneous wave front set, and it propagates along straight lines with finite speed away from x � = 0. Then we show that it is related to the wave front set in a natural way. These results may be considered as a refinement of the microlocal smoothing property of CraigKappelerStrauss under more general assumptions. 1
Strichartz estimates and local smoothing estimates for asympototically flat Schrödinger equations
 J. Funct. Anal
"... Abstract. In this article we study globalintime Strichartz estimates for the Schrödinger evolution corresponding to longrange perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [28] of the third author, where it is proved that local smoothing estimates im ..."
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Cited by 9 (7 self)
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Abstract. In this article we study globalintime Strichartz estimates for the Schrödinger evolution corresponding to longrange perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article [28] of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. By [28] the local smoothing estimates are known to hold for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without nontrapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with nontrapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum. 1.
LINEAR VS. NONLINEAR EFFECTS FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL
"... Abstract. We review some recent results on nonlinear Schrödinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite prec ..."
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Cited by 8 (5 self)
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Abstract. We review some recent results on nonlinear Schrödinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite precisely. This includes semiclassical régimes, as well as finite time blowup and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs. 1.
GLOBAL EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH QUADRATIC POTENTIALS
"... Abstract. We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The p ..."
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Cited by 7 (0 self)
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Abstract. We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to use continuity arguments and to control the nonlinear effects. 1.
Fibrations, Compactifications And Algebras Of Pseudodifferential Operators
"... . Some recent, and some new, results on the structure of algebras of pseudodifferential operators on compact manifolds with boundary are discussed. In particular their relationship to compactifications of noncompact spaces is emphasized. It is shown how these relationships allow the methods of pseud ..."
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Cited by 6 (0 self)
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. Some recent, and some new, results on the structure of algebras of pseudodifferential operators on compact manifolds with boundary are discussed. In particular their relationship to compactifications of noncompact spaces is emphasized. It is shown how these relationships allow the methods of pseudodifferential operators ("microlocalization") to be applied to problems in scattering and spectral theory. Euclidean space The first algebra of pseudodifferential operators on a general compact manifold with boundary, that I will discuss here, comes directly from the very familiar case of Euclidean space. Namely consider the standard calculus of pseudodifferential operators on R n : Since there are several variants of this `standard' calculus, let me be precise. An element of the space \Psi m (R n ) is an operator on S(R n ) which can be written in the form (1) Au(x) = (2ß) \Gamman Z R n e ix\Delta¸ a(x; ¸)u(¸); u(¸) = Z R n e \Gammaiy \Delta¸ u(y)dy with the (leftreduc...