2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

Abstract. We obtain the Strichartz inequality

Abstract. We obtain the Strichartz inequalities ‖u ‖ q Lt Lr x ([0,1]×M) ≤ C‖u(0)‖L2 (M) for any smooth n-dimensional Riemannian manifold M which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and non-trapping, 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 semi-linear 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

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 real-valued function on X vanishing to second order at ∂X. Assuming that g is non-trapping, 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 Hassell-Vasy. When the metric is trapping, then the parametrix construction goes through microlocally in the non-trapping 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 non-trapping, 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 Craig-Kappeler-Strauss and Wunsch. 1.

Abstract. In this paper we develop a quantitative version of Enss ’ method to establish global-in-time 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 Laplace-Beltrami operator on a manifold M which is a smooth compact perturbation of three-dimensional Euclidean space R3 which obeys the non-trapping condition. We establish a global-in-time local smoothing estimate for the Schrödinger equation ut = −iHu. The main novelty here is the global-in-time aspect of the estimates, which forces a more detailed analysis on the low and medium frequencies of the evolution than in the local-in-time 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.

. 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 (left-reduc...

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 semi-classical régimes, as well as finite time blow-up and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs. 1.

Abstract. We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n ≥ 6. 1.

Partially supported by NSF and IBERDROLA program of Profesores Visitantes Partially supported by NSF and IBERDROLA program of Profesores Visitantes Partially supported by a MECyT grant 1

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 Craig-Kappeler-Strauss under more general assumptions. 1