Exact propagators for two classes of degenerate hyperbolic equations can be obtained from explicit Green's functions for certain degenerate elliptic equations. The latter, in turn, trace back to analogous results for some subelliptic operators associated to weakly pseudoconvex domains.

The purpose of this talk is to present some recent results obtained in collaboration with P. Gérard and N. Tzvetkov (Université Paris Sud-Orsay). On the well posedness of non-linear Schródinger equations on domains (with Dirichlet boundary conditions):

1)  In the ball, we show that if the non linearity has a gauge invariance (typically F(u) = |u|^p u), the problem is not well posed, even for initial data in some Sobolev spaces above the scaling critical index.

2)  If the domain is the exterior of a bounded obstacle satisfying a non-trapping condition, we show that the problem is locally well posed for any initial data in H¹₀(W) or L²(W) (hence globally well posed in the case of defocusing non-linearities) for a large class of non linearities.

We study surfaces with (hyperbolic) cusp ends. The generator, B, of the Lax-Phillips semigroup has spectrum given in terms of the eigenvalues of the Laplacian and the poles of the scattering matrix. We show that away from the continuous spectrum of the Laplacian, the norm of the resolvent of B+1/2 is comparable (in the non-physical plane) to the norm of the scattering matrix. In particular, for the modular surface this means that the norm of the resolvent of B+1/2 is comparable to |z(2s)|^-1 when 0 < e £ Âs £ 1/2-e. This is joint work with M. Zworski.

This talk will describe a new result establishing global existence and scattering of solutions for the cubic defocusing nonlinear Schrodinger equation in three space dimensions. The main ingredient is a new Morawetz-type inequality which provides a global spacetime L⁴ bound. This talk concerns joint work with Keel, Staffilani, Takaoka and Tao.

For a variety of nonlinearities, the nonlinear Schroedinger equation is known to possess localized quasistationary solutions (``standing waves''). We prove that in the generic situation the standing wave of minimal energy among all other standing waves is unstable. This case was falling out of the scope of the classical paper by Grillakis, Shatah, and Strauss on orbital stability of standing waves. An interesting feature of the problem is the absence of (exponential) instability in the linearized system; in this sense, the resulting instability is ``purely nonlinear''. Essentially, the instability is caused by higher algebraic degeneracy of zero eigenvalue in the spectrum of the linearized system. The result can be generalized to abstract Hamiltonian systems with U(1) symmetry.

Subelliptic PDE's induce the geometric ideas of subRiemannian geometry. Using these subRiemannian geometric invariants I shall construct some examples of fundamental solutions to subelliptic PDE's.

We discuss several results on critical points and L_p norms of eigenfunctions of Laplacians on Riemannian manifolds.

I shall present a simple proof of sharp eigenfunction estimates for manifolds with boundary. Using these and the finite propagation speed for the wave equation one can prove some sharp estimates in harmonic analysis, such as the L¹ mapping properties of Riesz means. I shall also discuss some open problems.

We suppose that it possesses stable solitary wave solutions and we investigate their asymptotic stability, that is the long-time behavior of solutions whose initial conditions are close to a stable solitary wave.

The method, initiated in [1], is based on the spectral decomposition of solutions on the eigenspaces associated to the discrete and continuous spectrum of the linearized operator near the solitary wave. Under some hypothesis on the structure of the spectrum, we prove that, asymptotically in time, the solution decomposes into a solitary wave and a dispersive part described by the free Schrödinger equation. We calculate explicitly the time behavior of the correction.

This is a joint work with V. Buslaev.

References

[1]

V. Buslaev and G. Perleman, Scattering for the nonlinear Schrödinger equation: states close to a soliton. St. Petersburg Math. J. 4(1993), 1111-1142.

I will talk about the use of methods from many-body scattering in the study of the Laplacian on higher rank symmetric spaces. I focus on the relationship of three-body scattering and SL(3,R)/SO(3,R). I will describe the asymptotics of the Green's function at infinity (Taylor series at infinity), extending results of Anker, Guivarch, Ji and Taylor. I also describe the analytic continuation of the resolvent in the spectral parameter through the continuous spectrum. The new results presented are joint work with Rafe Mazzeo.

We discuss the construction of a parametrix for the time-dependent Schrödinger equation in non-trapping regions of a manifold X with asymptotically conic ends. The construction, in the framework of the ``Legendrian distributions'' of Melrose-Zworski (generalized by Hassell and Vasy) involves a phase function parametrizing a certain relation between points in the cosphere bundle of X and the (rescaled) cotangent bundle of the boundary of the compactification. We call this relation the `sojourn time' owing to its similarity to the sojourn time in scattering theory introduced by Guillemin. As a consequence, we are able to prove some new results on propagation of singularities for the Schrödinger operator.

The purpose of my talk is to outline a proof of a new result obtained jointly with Andrew Hassell (ANU) that L²-normalized boundary values (i.e. Cauchy data) u_j^\flat of eigenfunctions of the Laplacian on piecewise smooth convex domains W with corners and with ergodic billiards are quantum ergodic. In other words, that